# questions

title: int-2xleftfprimexfx-log-2right-d-x-is
slug: int-2xleftfprimexfx-log-2right-d-x-is-10072
question: <div class="question">\(\int 2^x\left(f^{\prime}(x)+f(x) \log 2\right) d x\) is</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2^x \mathrm{f}^{\prime}(\mathrm{x})+\mathrm{C}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(2^x f(x)+C\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2^x(\log 2) f(x)+C\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((\log 2) f(x)+C\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2^x f(x)+C\)</span> </div>

answer:
<div class="solution">Hints: \(I=\int 2^x f^{\prime}(x) d x+\int 2^x f(x) \log 2 d x\) \(=2^x f(x)\)</div>


title: int-fraccos-2-xcos-x-d-x
slug: int-fraccos-2-xcos-x-d-x-77145
question: <div class="question">\(\int \frac{\cos 2 x}{\cos x} d x=\)</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 \sin x+\log |\sec x+\tan x|+C\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(2 \sin x-\log |\sec x-\tan x|+c\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \sin x-\log |\sec x+\tan x|+C\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2 \sin x+\log |\sec x-\tan x|+C\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \sin x-\log |\sec x+\tan x|+C\)</span> </div>

answer:
<div class="solution">Hints: \(\int \frac{2 \cos ^2 x-1}{\cos x} d x=2 \sin x-\log |\sec x+\tan |\)</div>


title: int-fracsin-8-xcos-8-x12-sin-2-x-cos-2-x-d-x
slug: int-fracsin-8-xcos-8-x12-sin-2-x-cos-2-x-d-x-89538
question: <div class="question">\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x\)</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(-\frac{1}{2} \sin 2 x+C\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{2} \sin 2 \mathrm{x}+\mathrm{C}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{1}{2} \sin x+C\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-\frac{1}{2} \sin x+C\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-\frac{1}{2} \sin 2 x+C\)</span> </div>

answer:
<div class="solution">Hints: \(\int\left(\sin ^2 x-\cos ^2 x\right) d x=-\int \cos ^{2 x} d x=-\frac{1}{2} \sin 2 x+C\)</div>


title: int-fracx3-d-x1x8
slug: int-fracx3-d-x1x8-95729
question: <div class="question">\(\int \frac{x^3 d x}{1+x^8}=\)</div>
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(4 \tan ^{-1} \mathrm{x}^3+\mathrm{c}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{4} \tan ^{-1} x^4+c\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(x+4 \tan ^{-1} x^4+c\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(x^2+\frac{1}{4} \tan ^{-1} x^4+c\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{4} \tan ^{-1} x^4+c\)</span> </div>

answer:
<div class="solution">Hints: \(\int \frac{x^3 d y}{1+\left(x^4\right)^2}=\frac{1}{4} \tan ^{-1}\left(x^4\right)\)</div>


title: int02-pi-theta-sin-6-theta-cos-theta-d-theta-is-equal-to
slug: int02-pi-theta-sin-6-theta-cos-theta-d-theta-is-equal-to-63404
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">\(\int_0^{2 \pi} \theta \sin ^6 \theta \cos \theta d \theta\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{\pi}{16}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{3 \pi}{16}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{16 \pi}{3}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Hint : Let \(I=\int_0^{2 \pi} \theta \sin ^6 \theta \cos \theta d \theta \ldots \ldots .(I)\)<br/>applying King's Property,<br/>\(\int_0^{2 \pi}(2 \pi-\theta) \sin ^6 \theta \cdot \cos \theta d \theta \ldots \ldots .(II)\)<br/>adding (I) and (II), \(2 \mathrm{I}=2 \pi \int_0^{2 \pi} \sin ^6 \theta \cos \theta d \theta\)<br/>\(\Rightarrow I=\pi \int_0^{2 \pi} \sin ^6 \theta \cos \theta \ldots \ldots .(III)\)<br/>Using Queen's Property, \(\int_0^{2 a} f(x) d x=2 \int_0^a f(x) d x\), when \(f(2 a-x)=f(x)\)<br/>applying Queen's Property on (III), I \(=2 \pi \int_0^\pi \sin ^6 \theta \cos \theta d \theta\)<br/>again, using Queen's Property, \(\int_0^{2 a} f(x) d x=0\), when, \(f(2 a-x)=-f(x)\)<br/>\(I=0\)</div>


title: integrating-factor-if-of-the-defferential-equation-fracd-yd-xfrac3-x2-y1x3fracsin-2x1x-text-is
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
slug:
integrating-factor-if-of-the-defferential-equation-fracd-yd-xfrac3-x2-y1x3fracsin-2x1x-text-is-69780

question:
<div class="question">Integrating Factor (I.F.) of the defferential equation<br/>\(\frac{d y}{d x}-\frac{3 x^2 y}{1+x^3}=\frac{\sin ^2(x)}{1+x} \text { is }\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{e}^{1+\mathrm{x}^3}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\log \left(1+x^3\right)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1+x^3\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{1+x^3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{1+x^3}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: If } e^{\int p d x}=e^{-\int \frac{3 x^2 d x}{1+x^3}}=e^{-\log \left(1+x^3\right)}=e^{\log \left(1+x^3\right)^{-1}} \\<br/>&amp; =\left(1+x^3\right)^{-1}=\frac{1}{1+x^3}<br/>\end{aligned}\)</div>


title: intpi16-pisin-x-d-x
slug: intpi16-pisin-x-d-x-25898
question: <div class="question">\(\int_\pi^{16 \pi}|\sin x| d x=\)</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
answer: <div class="solution">Hints: \(15 \int_0^\pi \sin x d x=15(-\cos x)_0^\pi=30\)</div>
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">32</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">30</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">28</span> </li> </ul>


title: ionic-solids-with-schottky-defect-may-contain-in-their-structure
slug: ionic-solids-with-schottky-defect-may-contain-in-their-structure-37499
question: <div class="question">Ionic solids with Schottky defect may contain in their structure</div>
tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">cation vacancies only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">cation vacancies and interstitial cations</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">equal number of cation and anion vacancies</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">anion vacancies and interstitial anions</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">equal number of cation and anion vacancies</span> </div>

answer:
<div class="solution">In Schotthy defect, equal number of cations and anions are missing creating equal number of cation and anion vacancies.</div>


tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
ionisation-potential-values-of-noble-gases-decrease-down-the-group-with-increase-in-atomic-size-xenon-forms-binary-fluorides-by-the-direct-reaction-of

slug:
ionisation-potential-values-of-noble-gases-decrease-down-the-group-with-increase-in-atomic-size-xenon-forms-binary-fluorides-by-the-direct-reaction-of-88929

question:
<div class="question">Ionisation potential values of noble gases decrease down the group with increase in atomic size. Xenon forms binary fluorides by the direct reaction of elements. Identify the correct statement(s) from below.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Only the heavier noble gases form such compounds</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">It happens because the noble gases have higher ionisation energies</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">It happens because the compounds are formed with electronegative ligands</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Octet of electrons provide the stable arrangements</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">Only the heavier noble gases form such compounds, It happens because the compounds are formed with electronegative ligands</span> </div>

answer:
<div class="solution">Ionisation potential values of noble gases decrease down the group with increase in atomic size. Xenon forms binary fluorides by the direct reaction of elements.
<br/>The reason is that only the heavier noble gases form such compounds. Octet of electrons provide the stable arrangements but this statement is not very much valid in cases of higher element which can be ionised easily due to large distance between nucleus and valence electrons.</div>


title: is-an-imaginary-cube-root-of-unity-and-x-2-1-2-x-1-1-x-2-0-then-one-of-the-values-of-x-is
slug: is-an-imaginary-cube-root-of-unity-and-x-2-1-2-x-1-1-x-2-0-then-one-of-the-values-of-x-is-79762
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">$\omega$ is an imaginary cube root of unity and $\left|\begin{array}{ccc}x+\omega^2 &amp; \omega &amp; 1 \\ \omega &amp; \omega^2 &amp; 1+x \\ 1 &amp; x+\omega &amp; \omega^2\end{array}\right|=0$ then one of the values of $x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\text { Hints: } &amp; \stackrel{C_1^{\prime} \rightarrow C_1+C_2+C_3}{\longrightarrow}\left|\begin{array}{ccc}<br/>x &amp; \omega &amp; 1 \\<br/>x &amp; \omega^2 &amp; 1+x \\<br/>x &amp; x+\omega &amp; \omega^2<br/>\end{array}\right|=x\left|\begin{array}{ccc}<br/>1 &amp; \omega &amp; 1 \\<br/>1 &amp; \omega^2 &amp; 1+x \\<br/>1 &amp; x+\omega &amp; \omega^2<br/>\end{array}\right| \\<br/>\quad &amp; =x\left|\begin{array}{ccc}<br/>1 &amp; \omega &amp; 1 \\<br/>0 &amp; \omega^2-\omega &amp; x \\<br/>0 &amp; x &amp; \omega^2-1<br/>\end{array}\right|=x\left\{\left(\omega^2-\omega\right)\left(\omega^2-1\right)-x^2\right\}=0 \quad \Rightarrow x=0 \quad \text { One value of } x=0<br/>\end{aligned}<br/>$$</div>


title: it-is-difficult-to-cook-rice-in-an-open-vessel-by-boiling-it-at-high-altitudes-because-of
slug: it-is-difficult-to-cook-rice-in-an-open-vessel-by-boiling-it-at-high-altitudes-because-of-59622
tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">At high altitude pressure is low and boiling point also low</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">It is difficult to cook rice in an open vessel by boiling it at high altitudes because of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">low boiling point and high pressure</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">high boiling point and low pressure</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">low boiling point and low pressure</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">high boiling point and high pressure</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">low boiling point and low pressure</span> </div>


tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
k-f-water-186-k-kg-mol-1-the-temperature-at-which-ice-begins-to-separate-from-a-mixture-of-10-mass-ethylene-glycol-is

slug:
k-f-water-186-k-kg-mol-1-the-temperature-at-which-ice-begins-to-separate-from-a-mixture-of-10-mass-ethylene-glycol-is-31938

question:
<div class="question">$\mathrm{K}_{f}$ (water) $=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$. The temperature at which ice begins to separate from a mixture of 10 mass $\%$ ethylene glycol is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-1.86^{\circ} \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-3.72^{\circ} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-3.3^{\circ} \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3^{\circ} \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-3.3^{\circ} \mathrm{C}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\Delta \mathrm{T}_{\mathrm{f}}=\mathrm{K}_{\mathrm{f}} \times \mathrm{m}$
<br/>$=1.86\left(\frac{10}{62} \times \frac{1000}{90}\right)$
<br/>$=3.3^{\circ} \mathrm{C}$
<br/>$\therefore$ Temparature at which ice begins to separate $=3.3^{\circ} \mathrm{C}$</div>


title: ketoenol-taut-omerism-is-not-observed-in
slug: ketoenol-taut-omerism-is-not-observed-in-79567
question: <div class="question">Keto-enol taut omerism is not observed in</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">as contains no $\alpha-\mathrm{H}$</div>
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COC}_6 \mathrm{H}_5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COCH}=\mathrm{CH}_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COCH}_2 \mathrm{COCH}_3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_3 \mathrm{COCH}_2 \mathrm{COCH}_3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COC}_6 \mathrm{H}_5$</span> </div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
ldentify-the-correct-method-for-the-synthesis-of-lhe-compound-shown-below-from-the-following-alternatives

slug:
ldentify-the-correct-method-for-the-synthesis-of-lhe-compound-shown-below-from-the-following-alternatives-38532

question:
<div class="question">ldentify the correct method for the synthesis of lhe compound shown below from the following alternatives.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80_opta.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80_optb.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80_optc.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q80_optb.png"/></span> </div>

answer:
<div class="solution">The sequence of reaction is as follows:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a80.png"/></div>


title: le-the-relation-be-defined-on-r-by-a-b-holds-if-and-only-if-a-b-is-zero-or-irrational-then
slug: le-the-relation-be-defined-on-r-by-a-b-holds-if-and-only-if-a-b-is-zero-or-irrational-then-38235
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Le the relation $\rho$ be defined on $\mathbb{R}$ by a $\rho$ b holds if and only if $a-b$ is zero or irrational, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\rho$ is equivalence relation</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\rho$ is reflexive $\&amp;$ symmetric but is not transitive</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\rho$ is reflexive and transitive but is not symmetric</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\rho$ is reflexive only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho$ is reflexive $\&amp;$ symmetric but is not transitive</span> </div>

answer:
<div class="solution">Hint: 
<br/>If $a-b=0$ then $b-a=0$, if $a-b$ is irrational then $b-a$ is irrational
<br/>$\therefore a \rho b \Rightarrow b \rho a \Rightarrow$ symmetric
<br/>$\forall a \in \mathbb{R}, a-a=0 a \rho a \Rightarrow$ reflexive
<br/>If $a=2, b=\sqrt{2}, c=3$, then
<br/>a $\rho$ b, b $\rho$ c but a $\rho$ c is not true $\Rightarrow$ not transitive</div>


title: leftbeginarraylllab-bc-ca-bc-ca-ab-ca-ab-bcendarrayright
slug: leftbeginarraylllab-bc-ca-bc-ca-ab-ca-ab-bcendarrayright-77571
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">\(\left|\begin{array}{lll}a-b &amp; b-c &amp; c-a \\ b-c &amp; c-a &amp; a-b \\ c-a &amp; a-b &amp; b-c\end{array}\right|=\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Hints : \(\mathrm{c}_1 \rightarrow \mathrm{c}_1+\mathrm{c}_2+\mathrm{c}_3\)</div>


title: let-0-1-then-lim-n-k-1-n-1-k-1-k-a-1-x-d-x-is
slug: let-0-1-then-lim-n-k-1-n-1-k-1-k-a-1-x-d-x-is-71077
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $0    &lt; \alpha &lt; \beta &lt; 1$. Then $\lim {n \rightarrow \infty} \sum_{k=1}^{n} \int_{1/k+\beta}^{1 /(k+a)} \frac{d x}{1+x}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log _{e} \frac{\beta}{\alpha}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\log _{e} \frac{1+\beta}{1+\alpha}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\log _{e} \frac{1+\alpha}{1+\beta}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\infty$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log _{e} \frac{1+\beta}{1+\alpha}$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}[\log |1+x|]_{\frac{1}{k+\beta}} \frac{1}{k+a}$
<br/>$=\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\log \left(1+\frac{1}{k+\alpha}\right)-\log \left(1+\frac{1}{k+\beta}\right)\right)$
<br/>$=\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\log \left(\frac{k+\alpha+1}{k+\alpha}\right)-\log \left(\frac{k+\beta+1}{k+\beta}\right)\right)$
<br/>$=\log \left(\frac{\beta+1}{\alpha+1}\right)$</div>


title: let-1-and-2-be-two-equivalence-relations-defined-on-a-nonvoid-set-s-then
slug: let-1-and-2-be-two-equivalence-relations-defined-on-a-nonvoid-set-s-then-69383
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2020']
answer: <div class="solution">Hint: <br/>Union of two transitive may or may not be transitive</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\rho_{1}$ and $\rho_{2}$ be two equivalence relations defined on a non-void set $S$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">both $\rho_{1} \cap \rho_{2}$ and $\rho_{1} \cup \rho_{2}$ are equivalence relations</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\rho_{1} \cap \rho_{2}$ is equivalence relation but $\rho_{1} \cup \rho_{2}$ is not so.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\rho_{1} \cup \rho_{2}$ is equivalence relation but $\rho_{1} \cap \rho_{2}$ is not so</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">neither $\rho_{1} \cap \rho_{2}$ nor $\rho_{1} \cup \rho_{2}$ is equivalence relation.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho_{1} \cap \rho_{2}$ is equivalence relation but $\rho_{1} \cup \rho_{2}$ is not so.</span> </div>


title: let-1-x-3-x-12-d-x-3-2-g-f-x-c-then
slug: let-1-x-3-x-12-d-x-3-2-g-f-x-c-then-63285
tags: ['Mathematics', 'Indefinite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\int \frac{x^{1 / 2}}{\sqrt{1-x^3}} d x=\frac{2}{3} g(f(x))+c$; then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)=\sqrt{x}, g(x)=x^{3 / 2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)=\sqrt{x}, g(x)=\sin ^{-1} x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad f(x)=\sin ^{-1} x, g(x)=x^{3 / 2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x$</span> </div>

answer:
<div class="solution">$\int \frac{x^{1 / 2} d x}{\sqrt{1-\left(x^{3 / 2}\right)^2}}=2 / 3 \int \frac{d t}{\sqrt{1-t^2}}$<br/>$x^{3 / 2}=t$<br/>$3 / 2 x^{1 / 2} d x=d t$<br/>$=2 / \sin ^{-1}(t)+c$<br/>$=2 / 3 \sin ^{-1}\left(x^{3 / 2}\right)+c$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-1-x-10-r-0-10-c-r-x-r-and-1-x-7-r-0-7-d-r-x-r-if-p-r-0-5-c-2-and-q-r-0-3-d-2-r-1-then-q-p-is-equal-to

slug:
let-1-x-10-r-0-10-c-r-x-r-and-1-x-7-r-0-7-d-r-x-r-if-p-r-0-5-c-2-and-q-r-0-3-d-2-r-1-then-q-p-is-equal-to-21388

question:
<div class="question">Let $(1+x)^{10}=\sum_{r=0}^{10} c_{r} x^{r}$ and $(1+x)^{7}=\sum_{r=0}^{7} d_{r} x^{r}$
<br/>If $P=\sum_{r=0}^{5} c_{2}$ and $Q=\sum_{r=0}^{3} d_{2 r+1},$ then $\frac{P}{Q}$ is equal
<br/>to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">32</span> </li> </ul>

answer:
<div class="solution">$P=\sum_{r=0}^{5} C_{2 r}$
<br/>$={ }^{10} C_{0}+{ }^{10} C_{2}+\quad+{ }^{10} C_{10}=\frac{2^{0}}{2}=2^{9}$
<br/>$Q=\sum_{r=0}^{3} d_{2 r+1}=d_{1}+d_{3}+d_{5}+d_{7}$
<br/>$={ }^{7} C_{1}+{ }^{7} C_{3}+{ }^{7} C_{5}+{ }^{7} C_{7}=\frac{2^{7}}{2}=2^{6}$
<br/>$\therefore \frac{P}{\Omega}=\frac{2^{9}}{3^{6}}=2^{3}=8$</div>


title: let-1-x-x-2-9-a-0-a-1-x-a-2-x-2-a-18-x-18-then
slug: let-1-x-x-2-9-a-0-a-1-x-a-2-x-2-a-18-x-18-then-69912
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\left(1+x+x^{2}\right)^{9}=a_{0}+a_{1} x+a_{2} x^{2}$
<br/>$+\ldots+a_{18} x^{18} \cdot$ Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a_{0}+a_{2}+\ldots+a_{18}=a_{1}+a_{3}+\ldots+a_{17}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a_{0}+a_{2}+\ldots+a_{18}$ is even</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a_{0}+a_{2}+\ldots+a_{18}$ is divisible by 9</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a_{0}+a_{2}+\ldots+a_{18}$ is divisible by 3 but not by 9</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a_{0}+a_{2}+\ldots+a_{18}$ is even</span> </div>

answer:
<div class="solution">$\left(1+x+x^{2}\right)^{9}=a_{0}+a_{1} x+a_{2} x^{2}+ \dots+a_{18}x^{18}$
<br/>$$
<br/>Put $x=-1,$ we get
<br/>$$
<br/>\begin{array}{l}
<br/>(1-1+1)^{9}=a_{0}-a_{1}+a_{2}+\ldots+a_{18} \\
<br/>\Rightarrow 1=a_{0}-a_{1}+a_{2}+\ldots+a_{18}
<br/>\end{array}
<br/>$$
<br/>Put $x=1,$ we get
<br/>$$
<br/>(1+1+1)^{9}=a_{0}+a_{1}+a_{2}+\ldots+a_{14}
<br/>$$
<br/>$\Rightarrow 3^{9}=a_{0}+a_{1}+a_{2}+\ldots+a_{18}$
<br/>On adding Eqs. (i) and (ii), we get
<br/>$$
<br/>\begin{array}{c}
<br/>3^{9}+1=2\left(a_{0}+a_{2}+\ldots+a_{10}\right) \\
<br/>a_{0}+a_{2}+a_{4}+\ldots+a_{18}=\frac{3^{0}+1}{2} \\
<br/>=\frac{19683+1}{2} \\
<br/>=\frac{19684}{2}
<br/>\end{array}
<br/>$$
<br/>$=9842$ which is even number.</div>


title: let-16-x-2-3-y-2-32-x-12-y-44-represents-a-hyperbola-then
slug: let-16-x-2-3-y-2-32-x-12-y-44-represents-a-hyperbola-then-17815
question: <div class="question">Let $16 x^{2}-3 y^{2}-32 x-12 y=44$ represents a hyperbola. Then,</div>
tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">length of the transverse axis is $2 \sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">length of each latusrecturn is $32 / \sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">eccentricity is $\sqrt{19 / 3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">equation of a directrix is $x=\frac{\sqrt{19}}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">length of the transverse axis is $2 \sqrt{3}$, length of each latusrecturn is $32 / \sqrt{3}$, eccentricity is $\sqrt{19 / 3}$</span> </div>

answer:
<div class="solution">Given equation of hyperbola is $16 x^{2}-3 y^{2}-32 x-12 y=44$
<br/>$\Rightarrow 16 x^{2}+16-32 x-3 y^{2}-12-12 y$
<br/>$=44+4$
<br/>$\Rightarrow 16(x-1)^{2}-3(y+2)^{2}=48$
<br/>$\Rightarrow \quad \frac{(x-1)^{2}}{3}-\frac{(y+2)^{2}}{16}=1$
<br/>On comparing the equation with standard equation of hyperbola, we get $a=\sqrt{3}, b=4$
<br/>Now, length of transverse axis $=2 a=2 \sqrt{3}$
<br/>and length of latusrectum $=\frac{2 b^{2}}{a}=\frac{2 \times 16}{\sqrt{3}}=\frac{32}{\sqrt{3}}$
<br/>$\therefore$ Eccentricity $(e)=\sqrt{1+\frac{b^{2}}{a^{2}}}$
<br/>$=\sqrt{1+\frac{16}{3}}=\sqrt{\frac{19}{3}}$
<br/>Equation of directrix is $x=\pm \frac{a}{e}$ $\Rightarrow x-1=\pm \frac{\sqrt{3} \times \sqrt{3}}{\sqrt{19}}$
<br/>$\Rightarrow \quad x=1 \pm \frac{3}{\sqrt{19}}$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
let-a-1-0-0-0-cost-sin-t-0-sint-cos-t-let-1-2-3-be-the-roots-of-det-a-i-3-0-where-i-3-denotes-the-identity-matrix-if-1-2-3-2-1-then-the-set-of-possibl

slug:
let-a-1-0-0-0-cost-sin-t-0-sint-cos-t-let-1-2-3-be-the-roots-of-det-a-i-3-0-where-i-3-denotes-the-identity-matrix-if-1-2-3-2-1-then-the-set-of-possibl-25919

question:
<div class="question">Let $A=\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; \text { cost } &amp; \text { sint } \\ 0 &amp; -\sin t &amp; \cos t\end{array}\right)$
<br/>Let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be the roots of $\operatorname{det}\left(A-\lambda I_{3}\right)=0$, where $I_{3}$ denotes the identity matrix. If $\lambda_{1}+\lambda_{2}+\lambda_{3}=\sqrt{2}+1$, then the set of possible values of $t,-\pi \leq t     &lt; \pi$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a void set</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left\{\frac{\pi}{4}\right\}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left\{-\frac{\pi}{4}, \frac{\pi}{4}\right\}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left\{-\frac{\pi}{3}, \frac{\pi}{3}\right\}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left\{-\frac{\pi}{4}, \frac{\pi}{4}\right\}$</span> </div>

answer:
<div class="solution">$\left|\begin{array}{ccc}1-\lambda &amp; 0 &amp; 0 \\ 0 &amp; \cos t-\lambda &amp; \sin t \\ 0 &amp; -\sin t &amp; \cos t-\lambda\end{array}\right|=0$
<br/>$\Rightarrow(1-\lambda)(\cos t-\lambda)^{2}+\sin ^{2} t=0$
<br/>$\Rightarrow(1-\lambda)\left(\lambda^{2}-2 \lambda \cos t+\cos ^{2} t\right)+\sin ^{2} t=0$
<br/>$\Rightarrow \lambda^{2}-2 x \cos t+\cos ^{2} t-\lambda^{3}+2 \lambda^{2} \cos t+\lambda \cos ^{2} t+\sin ^{2} t=0$
<br/>$\Rightarrow-\lambda^{3}+\lambda^{2}(1+2 \cos t)+\lambda\left(\cos ^{2} t-2 \cos t\right)+1=0$
<br/>$\lambda_{1}+\lambda_{2}+\lambda_{3}=1+2 \operatorname{cost}=1+\sqrt{2}$
<br/>$\therefore$ cost $=\frac{1}{\sqrt{2}} \Rightarrow \mathrm{t}=\frac{\pi}{4},-\frac{\pi}{4}$</div>


title: let-a-1-0-0-1-1-0-1-1-1-then-for-positive-integer-n-a-n-is
slug: let-a-1-0-0-1-1-0-1-1-1-then-for-positive-integer-n-a-n-is-82452
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $A=\left(\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right)$ Then, for positive integer
<br/>$n, A^{n}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\begin{array}{lll}1 &amp; n &amp; n^{2} \\ 0 &amp; n^{2} &amp; n \\ 0 &amp; 0 &amp; n\end{array}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\begin{array}{lll}1 &amp; n &amp; n\left(\frac{n+1}{2}\right) \\ 0 &amp; 1 &amp; n \\ 0 &amp; 0 &amp; 1\end{array}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\begin{array}{lll}1 &amp; n^{2} &amp; n \\ 0 &amp; n &amp; n^{2} \\ 0 &amp; 0 &amp; n^{2}\end{array}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\begin{array}{ccc}1 &amp; n &amp; 2 n-1 \\ 0 &amp; \frac{n+1}{2} &amp; n^{2} \\ 0 &amp; 0 &amp; \frac{n+1}{2}\end{array}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\begin{array}{lll}1 &amp; n &amp; n\left(\frac{n+1}{2}\right) \\ 0 &amp; 1 &amp; n \\ 0 &amp; 0 &amp; 1\end{array}\right)$</span> </div>

answer:
<div class="solution">We have, $A=\left[\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$\therefore \quad A^{2}=A \cdot A$
<br/>$=\left[\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right]\left[\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$=\left[\begin{array}{lll}1+0+0 &amp; 1+1+0 &amp; 1+1+1 \\ 0+0+0 &amp; 0+1+0 &amp; 0+1+1 \\ 0+0+0 &amp; 0+0+0 &amp; 0+0+1\end{array}\right]$
<br/>$=\left[\begin{array}{lll}1 &amp; 2 &amp; 3 \\ 0 &amp; 1 &amp; 2 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\left[\begin{array}{lll}1 &amp; 2 &amp; \frac{2(2+1)}{2} \\ 0 &amp; 1 &amp; 2 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>Again, $A^{3}=A^{2} \cdot A$
<br/>$=\left[\begin{array}{lll}1 &amp; 2 &amp; 3 \\ 0 &amp; 1 &amp; 2 \\ 0 &amp; 0 &amp; 1\end{array}\right]\left[\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right]$
<br/>$\begin{aligned} &amp;=\left[\begin{array}{lll}1+0+0 &amp; 1+2+0 &amp; 1+2+3 \\ 0+0+0 &amp; 0+1+0 &amp; 0+1+2 \\ 0+0+0 &amp; 0+0+0 &amp; 0+0+1\end{array}\right] \\=\left[\begin{array}{lll}1 &amp; 3 &amp; 6 \\ 0 &amp; 1 &amp; 3 \\ 0 &amp; 0 &amp; 1\end{array}\right]=\left[\begin{array}{lll}1 &amp; 3 &amp; \frac{3(3+1)}{2} \\ 0 &amp; 1 &amp; 3 \\ 0 &amp; 0 &amp; 1\end{array}\right] \\\therefore A^{n} =\left[\begin{array}{lll}1 &amp; n &amp; \frac{n(n+1)}{2} \\ 0 &amp; 1 &amp; n \\ 0 &amp; 0 &amp; 1\end{array}\right] \end{aligned}$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-a-1-0-and-b-2-0-be-two-points-a-point-m-moves-in-the-plane-in-such-a-way-that-mb-a-2-m-a-b-then-the-point-m-moves-along

slug:
let-a-1-0-and-b-2-0-be-two-points-a-point-m-moves-in-the-plane-in-such-a-way-that-mb-a-2-m-a-b-then-the-point-m-moves-along-18506

question:
<div class="question">Let $A(-1,0)$ and $B(2,0)$ be two points. $A$ point $M$ moves in the plane in such a way that $\angle M B A=2 \angle M A B .$ Then, the point $M$ moves along</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a straight line</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a parabola</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">an ellipse</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a hyperbola</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a hyperbola</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a55.png"/> <br/> <br/>$$
<br/>\begin{array}{l}
<br/>\text { Let } \angle M A B=\theta \text { , then } \angle M B A=2 \theta \\
<br/>\text { tan } \theta=\frac{k}{1+h} \text { and } \tan 2 \theta=\frac{k}{2-h} \\
<br/>\text { then } \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}=\frac{k}{2-h} \\
<br/>\Rightarrow \quad \frac{2(k / 1+h)}{1-(k / 1+h)^{2}}=\frac{k}{2-h} \\
<br/>\Rightarrow \quad \frac{2}{1+h} \times \frac{(1+h)^{2}}{(1+h)^{2}-k^{2}}=\frac{1}{2-h} \\
<br/>\Rightarrow \quad \frac{2(1+h)}{(1+h)^{2}-k^{2}}=\frac{1}{2-h} \\
<br/>\Rightarrow \quad 1+h^{2}+2 h-k^{2}=2(1+h)(2-h) \\
<br/>\Rightarrow \quad 1+h^{2}+2 h-k^{2}=2\left(2-h+2 h-h^{2}\right) \\
<br/>\Rightarrow \quad 1+h^{2}+2 h-k^{2}=2\left(2+h-h^{2}\right) \\
<br/>\Rightarrow \quad 1+h^{2}+2 h-k^{2}=4+2 h-2 h^{2} \\
<br/>\Rightarrow \quad 1+3 h^{2}-k^{2}=4 \\
<br/>\Rightarrow \quad 3 h^{2}-k^{2}=3
<br/>\end{array}
<br/>$$
<br/>which represents hyperbola (d).</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-a-2-3-and-b-2-1-be-two-angular-points-of-a-bc-if-the-centroid-of-the-triangle-moves-on-the-line-2-x-3-y-1-then-the-locus-of-the-angular-point-c-is

slug:
let-a-2-3-and-b-2-1-be-two-angular-points-of-a-bc-if-the-centroid-of-the-triangle-moves-on-the-line-2-x-3-y-1-then-the-locus-of-the-angular-point-c-is-66061

question:
<div class="question">Let $A(2,-3)$ and $B(-2,1)$ be two angular points of $\Delta A B C$. If the centroid of the triangle moves on the line $2 x+3 y=1$, then the locus of the angular point $C$ is given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 x+3 y=9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x-3 y=9$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 x+2 y=5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x-2 y=3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 x+3 y=9$</span> </div>

answer:
<div class="solution">Let the coordinates of $C$ be $(\alpha, \beta)$.
<br/>$\therefore$ Coordinates of centroid
<br/>$$
<br/>\begin{array}{l}
<br/>=\left(\frac{2-2+\alpha}{3}, \frac{-3+1+\beta}{3}\right) \\
<br/>=\left(\frac{\alpha}{3}, \frac{\beta-2}{3}\right)
<br/>\end{array}
<br/>$$
<br/>Since, centroid lie on $2 x+3 y=1$ 
<br/>$\therefore \quad \frac{2 \alpha}{3}+3\left(\frac{\beta-2}{3}\right)=1$
<br/>$\Rightarrow \quad \frac{2 \alpha}{3}+\frac{3 \beta-6}{3}=1$
<br/>$\Rightarrow \quad 2 \alpha+3 \beta-6=3 \Rightarrow 2 \alpha+3 \beta=9$
<br/>$\therefore$ Locus of point $C$ will be $2 x+3 y=9$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
let-a-3-0-3-0-3-0-3-0-3-then-the-roots-of-the-equation-det-a-i-3-0-where-i-3-is-the-identity-matrix-of-order-3-are

slug:
let-a-3-0-3-0-3-0-3-0-3-then-the-roots-of-the-equation-det-a-i-3-0-where-i-3-is-the-identity-matrix-of-order-3-are-62205

question:
<div class="question">Let $A=\left(\begin{array}{lll}3 &amp; 0 &amp; 3 \\ 0 &amp; 3 &amp; 0 \\ 3 &amp; 0 &amp; 3\end{array}\right) .$ Then, the roots of the equation $\operatorname{det}\left(A-\lambda I_{3}\right)=0$ (where $I_{3}$ is the identity matrix of order 3 ) are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3,0,3</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0,3,6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1,0,-6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3,3,6</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">0,3,6</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>A=\left[\begin{array}{lll}
<br/>3 &amp; 0 &amp; 3 \\
<br/>0 &amp; 3 &amp; 0 \\
<br/>3 &amp; 0 &amp; 3
<br/>\end{array}\right]
<br/>$$
<br/>Again, we have
<br/>$$
<br/>\left|\begin{array}{ccc}
<br/>3-\lambda &amp; 0 &amp; 3 \\
<br/>0 &amp; 3-\lambda &amp; 0 \\
<br/>3 &amp; 0 &amp; 3-\lambda
<br/>\end{array}\right|=0
<br/>$$
<br/>$\Rightarrow(3-\lambda)\left[(3-\lambda)^{2}-0\right]-0+3[0-3(3-\lambda)]=0$
<br/>$\Rightarrow (3-\lambda)^{3}-9(3-\lambda)=0$
<br/>$\Rightarrow (3-\lambda)\left[(3-\lambda)^{2}-9\right]=0$
<br/>$\Rightarrow (3-\lambda)\left(9+\lambda^{2}-6 \lambda-9\right)=0$
<br/>$\Rightarrow (3-\lambda)\left(\lambda^{2}-6 \lambda\right)=0$
<br/>$\Rightarrow \quad(3-\lambda) \lambda(\lambda-6)=0$
<br/>${\lambda}=3,0,6$</div>


title: let-a-3-t-1-0-1-3-t-1-0-1-0-and-det-a-5-then
slug: let-a-3-t-1-0-1-3-t-1-0-1-0-and-det-a-5-then-58227
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\left(\begin{array}{ccc}3-\mathrm{t} &amp; 1 &amp; 0 \\ -1 &amp; 3-\mathrm{t} &amp; 1 \\ 0 &amp; -1 &amp; 0\end{array}\right)$ and $\operatorname{det} \mathrm{A}=5$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{t}=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{t}=2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$t=-1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{t}=-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{t}=-2$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$|\mathrm{A}|=\left|\begin{array}{ccc}3-\mathrm{t} &amp; 1 &amp; 0 \\ -1 &amp; 3-\mathrm{t} &amp; 1 \\ 0 &amp; -1 &amp; 0\end{array}\right|=3-\mathrm{t}=5, \mathrm{t}=-2$</div>


title: let-a-12-x-1-24-6-2-5-2-3-the-value-of-x-for-which-the-matrix-a-is-not-invertible-is
slug: let-a-12-x-1-24-6-2-5-2-3-the-value-of-x-for-which-the-matrix-a-is-not-invertible-is-39737
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\left(\begin{array}{ccc}12 &amp; 24 &amp; 5 \\ \mathrm{x} &amp; 6 &amp; 2 \\ -1 &amp; -2 &amp; 3\end{array}\right) .$ The value of $\mathrm{x}$ for which the matrix $\mathrm{A}$ is not invertible is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>If matrix is not invertible $\Rightarrow|\mathrm{A}|=0$
<br/>$\therefore|\mathrm{A}|=\left|\begin{array}{ccc}12 &amp; 24 &amp; 5 \\ \mathrm{x} &amp; 6 &amp; 2 \\ -1 &amp; -2 &amp; 3\end{array}\right|=0 \Rightarrow \mathrm{x}=3$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
let-a-a-c-b-d-be-a-2-2-real-matrix-with-det-a-1-if-the-equation-det-a-i-2-0-has-imaginary-roots-i-2-be-the-identity-matrix-of-order-2-then

slug:
let-a-a-c-b-d-be-a-2-2-real-matrix-with-det-a-1-if-the-equation-det-a-i-2-0-has-imaginary-roots-i-2-be-the-identity-matrix-of-order-2-then-77364

question:
<div class="question">Let $\mathrm{A}=\left(\begin{array}{ll}\mathrm{a} &amp; \mathrm{b} \\ c &amp; \mathrm{~d}\end{array}\right)$ be a $2 \times 2$ real matrix with $\operatorname{det} \mathrm{A}=1$. If the equation $\operatorname{det}\left(\mathrm{A}-\lambda \mathrm{I}{2}\right)=0$ has imaginary roots $\left(\mathrm{I}{2}\right.$ be the
<br/>Identity matrix of order 2), then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(a+d)^{2}    &lt; 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(a+d)^{2}=4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(a+d)^{2}&gt;4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(a+d)^{2}=16$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(a+d)^{2}    &lt; 4$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$|\mathrm{A}|=0 \therefore \mathrm{ad}-\mathrm{bc}=1$
<br/>$\left|\mathrm{~A}-\lambda \mathrm{I}_{2}\right|=0$
<br/>$\left|\begin{array}{cc}\mathrm{a}-\lambda &amp; \mathrm{b} \\ \mathrm{c} &amp; \mathrm{d}-\lambda\end{array}\right|=0$
<br/>$\therefore \mathrm{ad}-(\mathrm{a}+\mathrm{d}) \lambda+\lambda^{2}-\mathrm{bc}=0$
<br/>$\lambda^{2}-(\mathrm{a}+\mathrm{d}) \lambda+1=0$
<br/>$\therefore(\mathrm{a}+\mathrm{d})^{2}    &lt; 4$</div>


title: let-a-and-b-are-orthogonal-matrices-and-operatornamedet-aoperatornamedet-b0-then
slug: let-a-and-b-are-orthogonal-matrices-and-operatornamedet-aoperatornamedet-b0-then-18452
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let \(A\) and \(B\) are orthogonal matrices and \(\operatorname{det} A+\operatorname{det} B=0\). Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(A+B\) is singular</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(A+B\) is non-singular</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(A+B\) is orthogonal</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(A+B\) is skew symmetric</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(A+B\) is singular</span> </div>

answer:
<div class="solution">Hint: \(A A^{\top}=I\) and \(B B^{\top}=I\)<br/>\(\Rightarrow \operatorname{det}\left(A A^{\top}\right)=1\) and \(\operatorname{det}\left(B B^{\top}\right)=1\)<br/>\(\operatorname{det}(A)=-\operatorname{det}(B)\) (Given) \( \Rightarrow(\operatorname{det}(A))^2=1 \Rightarrow(\operatorname{det}(B))^2=1\)<br/>\(\therefore \operatorname{det}(A+B)=\operatorname{det}\left(A\left(B^{\top}+A^{\top}\right) B\right)\)<br/>\(=-\operatorname{det}\left(B^{\top}+A^{\top}\right) \quad[\because(\operatorname{det}(A))(\operatorname{det}(B))=-1]\) as \(\operatorname{det}(A)=-\operatorname{det}(B)\)<br/>\(=-\operatorname{det}(B+A)^{\top}\)<br/>\(=-\operatorname{det}(A+B)^{\top}\)<br/>\(=-\operatorname{det}(A+B)\)<br/>\(\Rightarrow 2 \operatorname{det}(A+B)=0\)<br/>\(\Rightarrow \operatorname{det}(A+B)=0\)<br/>\(\therefore(A+B)\) is singular</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-a-and-b-are-two-independent-events-the-probability-that-both-a-and-b-happen-is-frac112-and-probability-that-neither-a-nor-b-happen-is-frac12-then

slug:
let-a-and-b-are-two-independent-events-the-probability-that-both-a-and-b-happen-is-frac112-and-probability-that-neither-a-nor-b-happen-is-frac12-then-86843

question:
<div class="question">Let \(A\) and \(B\) are two independent events. The probability that both \(A\) and \(B\) happen is \(\frac{1}{12}\) and probability that neither A nor \(B\) happen is \(\frac{1}{2}\). Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(P(A)=\frac{1}{3}, P(B)=\frac{1}{4}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(P(A)=\frac{1}{2}, P(B)=\frac{1}{6}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(P(A)=\frac{1}{6}, P(B)=\frac{1}{2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(P(A)=\frac{2}{3}, P(B)=\frac{1}{8}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(P(A)=\frac{1}{3}, P(B)=\frac{1}{4}\)</span> </div>

answer:
<div class="solution">\(\begin{array}{ll}<br/>\text {Hint : } P(A \cap B)=\frac{1}{12} &amp; \Rightarrow P(A) P(B)=\frac{1}{12} \cdots (1) \\<br/>P\left(A^{\prime} \cap B^{\prime}\right)=\frac{1}{2} &amp; \Rightarrow P(A \cup B)=1-\frac{1}{2}=\frac{1}{2}<br/>\end{array}\)<br/>\(\begin{aligned} &amp; \Rightarrow P(A)+P(B)-P(A) P(B)=\frac{1}{2} \\ &amp; \Rightarrow P(A)+P(B)=\frac{1}{2}+\frac{1}{12}=\frac{7}{12}\end{aligned}\)<br/>\(\begin{aligned} &amp; \text { From (1), } \quad P(A)\left(\frac{7}{12}-P(A)\right)=\frac{1}{12} \\ &amp; \Rightarrow P(A)^2-\frac{7}{12} P(A)+\frac{1}{12}=0 \\ &amp; \Rightarrow 12 P(A)^2-7 P(A)+1=0 \\ &amp; \Rightarrow 12 P(A)^2-4 P(A)-3 P(A)+1=0 \\ &amp; \Rightarrow(3 P(A)-1)-(4 P(A)-1)=0\end{aligned}\)<br/>\(\begin{aligned} &amp; \therefore P(A)=\frac{1}{3} \text { or } \quad P(A)=\frac{1}{4} \\ &amp; \therefore \quad P(B)=\frac{1}{12} \times 3 \Rightarrow  P(B)=\frac{1}{12} \times 4 \\ &amp; \qquad\qquad=\frac{1}{4} \qquad \qquad \qquad=\frac{1}{3} \\ &amp; \therefore P(A)=\frac{1}{3}, P(B)=\frac{1}{4}\end{aligned}\)</div>


title: let-a-and-b-be-two-events-such-that-p-a-b-6-1-p-a-b-45-31-and-p-b-10-7-then
slug: let-a-and-b-be-two-events-such-that-p-a-b-6-1-p-a-b-45-31-and-p-b-10-7-then-87603
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ be two events such that $P(A \cap B)=\frac{1}{6}, P(A \cup B)=\frac{31}{45}$ and $P(\bar{B})=\frac{7}{10}$ then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$A$ and $B$ are independent</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A and $B$ are mutually exclusive</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$P\left(\frac{A}{B}\right)    &lt; \frac{1}{6}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$P\left(\frac{B}{A}\right)    &lt; \frac{1}{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A$ and $B$ are independent</span> </div>

answer:
<div class="solution">Given, $P(A \cap B)=\frac{1}{6}$
<br/>$P(A \cup B)=\frac{31}{45}$
<br/>and $\quad P(\bar{B})=\frac{7}{10}$
<br/>$\therefore \quad \quad P(B)=1-\frac{7}{10}=\frac{3}{10}$
<br/>Now, $\quad P(A \cup B)=P(A)+P(B)-P(A \cap B)$
<br/>$\Rightarrow \quad \frac{31}{45}=P(A)+\frac{3}{10}-\frac{1}{6}$
<br/>$\therefore \quad P(A)=\frac{31}{45}+\frac{1}{6}-\frac{3}{10}$ $=\frac{5}{9}$
<br/>Then. $\quad P\left(\frac{B}{A}\right)=\frac{P(A \cap B)}{P(A)}=\frac{1}{5}=\frac{3}{10}&gt;\frac{1}{6}$
<br/>and $\quad P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{6}}{\frac{3}{3}}=\frac{5}{9}&gt;\frac{1}{6}$
<br/>and, $P(A) \times P(B)=\frac{5}{9} \times \frac{3}{10}=\frac{1}{6}=P(A \cap B)$
<br/>Hence, $A$ and $B$ are independent.</div>


title: let-a-and-b-be-two-events-with-p-a-c-03-p-b-04-and-p-a-b-c-05-then-p-b-a-b-c-is-equal-to
slug: let-a-and-b-be-two-events-with-p-a-c-03-p-b-04-and-p-a-b-c-05-then-p-b-a-b-c-is-equal-to-63213
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $A$ and $B$ be two events with $P\left(A^{C}\right)=0.3$ $P(B)=0.4 \quad$ and $\quad P\left(A \cap B^{C}\right)=0.5 . \quad$ Then
<br/>$P\left(B \mid A \cup B^{c}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{4}$</span> </div>

answer:
<div class="solution">$P\left(\frac{B}{A \cup B^{c}}\right)=\frac{P\left(B \cap\left(A \cup B^{e}\right)\right)}{P\left(A \cup B^{c}\right)}$
<br/>$=\frac{P(A \cap B)}{P(A)+P\left(B^{c}\right)-P\left(A \cap B^{c}\right)}$
<br/>$=\frac{P(A)-P\left(A \cap B^{c}\right)}{P(A)+P\left(B^{c}\right)-P\left(A \cap B^{c}\right)}$
<br/>$=\frac{0.7-0.5}{0.7+0.6-0.5}=\frac{0.2}{0.8}=\frac{1}{4}$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
let-a-and-b-be-two-nonsingular-skew-symmetric-matrices-such-that-a-b-b-a-then-a-2-b-2-a-b-1-a-b-1-is-equal-to

slug:
let-a-and-b-be-two-nonsingular-skew-symmetric-matrices-such-that-a-b-b-a-then-a-2-b-2-a-b-1-a-b-1-is-equal-to-36754

question:
<div class="question">Let $A$ and $B$ be two non-singular skew symmetric matrices such that $A B=B A$, then $A^{2} B^{2}\left(A^{\top} B\right)^{-1}\left(A B^{-1}\right)^{\top}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\mathrm{B}^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\mathrm{A}^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{AB}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\mathrm{A}^{2}$</span> </div>

answer:
<div class="solution">$\mathrm{AB}=\mathrm{BA}, \mathrm{A}^{\top}=-\mathrm{A}, \mathrm{B}^{\top}=-\mathrm{B}$ 
<br/>$\because$ matrices are non singular 
<br/>$\therefore$ order is even 
<br/>$\mathrm{A}^{2} \mathrm{~B}^{2}\left(\mathrm{~A}^{\top} \mathrm{B}\right)^{-1}\left(\mathrm{AB}^{-1}\right)^{\top}$
<br/>$\therefore$ order is even
<br/>$A^{2} B^{2}\left(A^{\top} B\right)^{-1}\left(A B^{-1}\right)^{\top}$
<br/>$=A^{2} B^{2}(-A B)^{-1}\left(A B^{-1}\right)^{\top}$
<br/>$=-A^{2} B^{2} B^{-1} A^{-1}\left(B^{\top}\right)^{-1} A^{\top}$
<br/>$=-A^{2} B A^{-1} B^{-1} A$
<br/>$=-A^{2} B(B A)^{-1} A$
<br/>$=-A^{2} B(A B)^{-1} A=-A^{2} B B^{-1} A^{-1} A=-A^{2}$</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
let-a-and-b-be-two-square-matrices-of-order-3-and-a-b-o-3-where-o-3-denotes-the-null-matrix-of-order-3-then

slug:
let-a-and-b-be-two-square-matrices-of-order-3-and-a-b-o-3-where-o-3-denotes-the-null-matrix-of-order-3-then-21459

question:
<div class="question">Let $A$ and $B$ be two square matrices of order 3 and $A B=O_{3}$, where $O_{3}$ denotes the null matrix of order 3. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">must be $A=O_{3}, B=O_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">if $A \neq O_{3}$, must be $B \neq O$,</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">if $A=O_{3}$. must be $B \neq O_{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">may be $A \neq O_{3} . B \neq O_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">may be $A \neq O_{3} . B \neq O_{3}$</span> </div>

answer:
<div class="solution">Since, product of two non-null matrix can be a null matrix.
<br/>Therefore, may be
<br/>$$
<br/>A \neq O_{3}, B \neq O_{3}
<br/>$$</div>


title: let-a-b-0-and-f-n-a-1-n-b-1-n-j-n-a-b-1-n-for-all-n-2-then
slug: let-a-b-0-and-f-n-a-1-n-b-1-n-j-n-a-b-1-n-for-all-n-2-then-13766
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $a&gt;b&gt;0$ and $f(n)=a^{1 / n}-b^{1 / n}$,$J(n)=(a-b)^{1 / n}$ for all $n \geq 2,$ Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(n)    &lt; J(n)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(n)&gt;J(n)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(n)=J(n)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(n)+J(n)=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(n)    &lt; J(n)$</span> </div>

answer:
<div class="solution">Let $a=4, b=1$ and $n=2$
<br/>$\begin{array}{l}
<br/>\therefore \quad I(n)=a^{1 / n}-b^{1 / n} \\
<br/>=4^{1 / 2}-1^{1 / 2}=2-1=1 \\
<br/>\text { and } J(n)=(a-b)^{1 / n} \\
<br/>=(4-1)^{1 / 2}=\sqrt{3} \\
<br/>\therefore \quad J(n)&gt;I(n)
<br/>\end{array}$
<br/>Similarly, we can prove for other values of $a, b, n$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
let-a-b-be-two-distinct-points-on-the-parabola-y-2-4-x-if-the-axis-of-the-parabola-touches-a-circle-of-radius-r-having-a-b-as-diameter-the-slope-of-th

slug:
let-a-b-be-two-distinct-points-on-the-parabola-y-2-4-x-if-the-axis-of-the-parabola-touches-a-circle-of-radius-r-having-a-b-as-diameter-the-slope-of-th-29205

question:
<div class="question">Let $A, B$ be two distinct points on the parabola $y^{2}=4 x$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as diameter, the slope of the line $A B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{r}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{r}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2}{r}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{r}$</span> </div>

answer:
<div class="solution">Centre of circle $=\left(\frac{t_{1}^{2}+t_{2}^{2}}{2},\left(t_{1}+t_{2}\right)\right)$
<br/>since, circle touch the $x$ -axis, so equation of tangent is $y=0$
<br/>$\because$ Radius = Perpendicular distance from centre to the tangent
<br/>$\Rightarrow$ Radius $=\left|t_{1}+t_{2}\right|=r$
<br/>Slope of $A B=\frac{2}{t_{1}+t_{2}}=\frac{2}{\pm r}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a46.png"/></div>


title: let-a-b-c-and-d-be-any-four-real-numbers-then-a-n-b-n-c-n-d-n-holds-for-any-natural-number-n-if
tags: ['Mathematics', 'Mathematical Induction', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
slug:
let-a-b-c-and-d-be-any-four-real-numbers-then-a-n-b-n-c-n-d-n-holds-for-any-natural-number-n-if-99220

question:
<div class="question">Let $a, b, c$ and $d$ be any four real numbers. Then. $a^{n}+b^{n}=c^{n}+d^{n}$ holds for any natural number $n$. if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a+b=c+d$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a-b=c-d$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a+b=c+d \cdot d^{2}+b^{2}=c^{2}+d^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$a-b=c-d \cdot d^{2}-b^{2}=c^{2}-d^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a-b=c-d \cdot d^{2}-b^{2}=c^{2}-d^{2}$</span> </div>

answer:
<div class="solution">From option (d), we have <br/>$\begin{aligned}<br/>a-b &amp;=c-d \\<br/>a^{2}-b^{2} &amp;=c^{2}-d^{2}<br/>\end{aligned}$<br/>Consider, $a^{2}-b^{2}=c^{2}-d^{2}$<br/>$\Rightarrow(a+b)(a-b)=(c-d)(c+d)$<br/>$\Rightarrow \quad a+b=c+d$<br/>On adding Eqs we get<br/>$2 a=2 c \Rightarrow a=c$<br/>$\Rightarrow \quad b=d$<br/>Thus, $a^{n}+b^{n}=c^{n}+d^{n}$ for all $n \in N$</div>


title: let-a-b-c-are-subsets-of-set-x-then-consider-the-validity-of-the-following-set-theoretic-statement
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
slug:
let-a-b-c-are-subsets-of-set-x-then-consider-the-validity-of-the-following-set-theoretic-statement-70205

question:
<div class="question">Let \(A, B, C\) are subsets of set \(X\). Then consider the validity of the following set theoretic statement :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(A \cup(B \backslash C)=(A \cup B) \backslash(A \cup C)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\((A \backslash B) \backslash C=A \backslash(B \cup C)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((A \cup B) \backslash A=A \backslash B\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(A \backslash C=B \backslash C\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((A \backslash B) \backslash C=A \backslash(B \cup C)\)</span> </div>

answer:
<div class="solution">Hint:<br/>\(\begin{aligned}<br/>:(A \backslash B) \backslash C &amp; =\left(A \cap B^{\prime}\right) \cap C^{\prime} \\<br/>&amp; =A \cap\left(B^{\prime} \cap C^{\prime}\right) \\<br/>&amp; =A \cap(B \cup C)^{\prime} \\<br/>&amp; =A \backslash(B \cup C)<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">27</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-a-b-c-be-real-numbers-each-greater-than-1-such-that-3-2-lo-g-b-a-5-3-lo-g-c-b-2-5-lo-g-a-c-3-if-the-value-of-b-is-9-then-the-value-of-a-must-be

slug:
let-a-b-c-be-real-numbers-each-greater-than-1-such-that-3-2-lo-g-b-a-5-3-lo-g-c-b-2-5-lo-g-a-c-3-if-the-value-of-b-is-9-then-the-value-of-a-must-be-78985

question:
<div class="question">Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be real numbers, each greater than 1 , such that $\frac{2}{3} \log _{\mathrm{b}} \mathrm{a}+\frac{3}{5} \log _{\mathrm{c}} \mathrm{b}+\frac{5}{2} \log _{\mathrm{a}} \mathrm{c}=3$. If the value of $b$ is 9 , then the value of ' $a$ ' must be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt[3]{81}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{27}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">18</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">27</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\frac{2 \ln a}{3 \ln b}+\frac{3 \ln b}{5 \ln c}+\frac{5 \ln c}{2 \ln a}=3$
<br/>By A.M $=$ G.M
<br/>$\frac{2 \ln a}{3 \ln b}=1$
<br/>$\Rightarrow a^{2}=b^{3} \Rightarrow a=\left(3^{6}\right)^{1 / 2}=3^{3} \Rightarrow a=27$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
let-a-b-c-be-real-numbers-such-that-a-b-c-0-and-the-quadratic-equation-a-x-2-b-x-c-0-has-imaginary-roots-then

slug:
let-a-b-c-be-real-numbers-such-that-a-b-c-0-and-the-quadratic-equation-a-x-2-b-x-c-0-has-imaginary-roots-then-50543

question:
<div class="question">Let $a, b, c$ be real numbers such that $a+b+c    &lt; 0$ and the quadratic equation $a x^{2}+b x+c=0$ has imaginary roots. Then.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a $&gt;0, c&gt;0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a&gt;0 . c    &lt; 0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a    &lt; 0, c&gt;0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$a    &lt; 0, c &lt; 0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a    &lt; 0, c &lt; 0$</span> </div>

answer:
<div class="solution">Let $f(x)=a x^{2}+b x+c$
<br/>$\Rightarrow \quad f(1)=a+b+c    &lt; 0$
<br/>Again, $f(x)$ has imaginary zeros. So, $a &lt; 0$. Also, $f(0)=c .$ since $f(x)$ is downward parabola. So, $c &lt; 0$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a17.png"/></div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-a-b-c-be-such-that-b-a-c-0-if-a-b-c-a-1-b-1-c-1-a-1-b-1-c-1-a-1-a-1-1-n-2-a-b-1-b-1-1-n-1-b-c-1-c-1-1-n-c-0-then-the-value-of-n-is-1

slug:
let-a-b-c-be-such-that-b-a-c-0-if-a-b-c-a-1-b-1-c-1-a-1-b-1-c-1-a-1-a-1-1-n-2-a-b-1-b-1-1-n-1-b-c-1-c-1-1-n-c-0-then-the-value-of-n-is-1-91437

question:
<div class="question">Let $a, b, c$ be such that $b(a+c) \times 0$
<br/>If $\left|\begin{array}{ccc}a &amp; a+1 &amp; a-1 \\ -b &amp; b+1 &amp; b-1 \\ c &amp; c-1 &amp; c+1\end{array}\right|$
<br/>$+\left|\begin{array}{ccc}a+1 &amp; b+1 &amp; c-1 \\ a-1 &amp; b-1 &amp; c+1 \\ (-1)^{n+2} a &amp; (-1)^{n+1} b &amp; (-1)^{n} c\end{array}\right|=0$
<br/>then the value of $n$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">any integer</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">any even integer</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">any odd integer</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">any odd integer</span> </div>

answer:
<div class="solution">We have,
<br/>$\left|\begin{array}{ccc}a &amp; a+1 &amp; a-1 \\ -b &amp; b+1 &amp; b-1 \\ c &amp; c-1 &amp; c+1\end{array}\right|$ + $\left|\begin{array}{ccc}a+1 &amp; b+1 &amp; c-1 \\ a-1 &amp; b-1 &amp; c+1 \\ (-1)^{n+2} a &amp; (-1)^{n+1} b &amp; (-1)^{n} c\end{array}\right|=0$
<br/>$\Rightarrow\left|\begin{array}{ccc}a &amp; a+1 &amp; a-1 \\ -b &amp; b+1 &amp; b-1 \\ c &amp; c-1 &amp; c+1\end{array}\right|$ $+\left|\begin{array}{ccc}a+1 &amp; a-1 &amp; (-1)^{n+2} a \\ b+1 &amp; b-1 &amp; (-1)^{n+1} b \\ c-1 &amp; c+1 &amp; (-1)^{n} c\end{array}\right|=0$
<br/>$\Rightarrow\left|\begin{array}{ccc}a &amp; a+1 &amp; a-1 \\ -b &amp; b+1 &amp; b-1 \\ c &amp; c-1 &amp; c+1\end{array}\right|$$+\left|\begin{array}{ccc}(-1)^{n+2} a &amp; a+1 &amp; a-1 \\ (-1)^{n+1} b &amp; b+1 &amp; b-1 \\ (-1)^{n} c &amp; c-1 &amp; c+1\end{array}\right|=0$
<br/>$\Rightarrow\left|\begin{array}{ccc}a\left(1+(-1)^{n+2}\right) &amp; a+1 &amp; a-1 \\ b\left(-1+(-1)^{n+1}\right) &amp; b+1 &amp; b-1 \\ c\left(1+(-1)^{n}\right) &amp; c-1 &amp; c+1\end{array}\right|=0$
<br/>$\therefore n$ is any odd integer.</div>


tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
let-a-b-c-be-three-nonvoid-subsets-of-set-s-let-a-c-b-c-where-c-denote-the-complement-of-set-c-in-s-then

slug:
let-a-b-c-be-three-nonvoid-subsets-of-set-s-let-a-c-b-c-where-c-denote-the-complement-of-set-c-in-s-then-24356

question:
<div class="question">Let $A, B, C$ be three non-void subsets of set $S$. Let $(A \cap C) \cup\left(B \cap C^{\prime}\right)=\phi$ where $C^{\prime}$ denote the complement of set $C$ in S. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$A \cap B=\phi$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A \cap B \neq \phi$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{A} \cap \mathrm{C}=\mathrm{A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{A} \cup \mathrm{C}=\mathrm{A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A \cap B=\phi$</span> </div>

answer:
<div class="solution">$(A \cap C) \cup\left(B \cap C^{\prime}\right)=\phi$
<br/>$\Rightarrow A \cap C=\phi$ and $B \cap C^{\prime}=\phi$ ...(i)
<br/>$\Rightarrow \mathrm{B} \subseteq \mathrm{C}$ ...(ii)
<br/>from (i) and (ii)
<br/>$A \cap B=\phi$</div>


title: let-a-b-c-p-q-and-r-be-positive-real-numbers-such-that-a-b-and-c-are-in-gp-and-apbqcr-then
slug: let-a-b-c-p-q-and-r-be-positive-real-numbers-such-that-a-b-and-c-are-in-gp-and-apbqcr-then-91340
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let \(a, b, c, p, q\) and \(r\) be positive real numbers such that \(a, b\) and \(c\) are in GP and \(a^{p}=b^{q}=c^{r}\). Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(p, q, r\) are in G.P.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(p, q, r\) are in A.P.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(p, q, r\) are in H.P.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(p^{2}, q^{2}, r^{2}\) are in A.P.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(p, q, r\) are in H.P.</span> </div>

answer:
<div class="solution">Let $a^{p}=b^{4}=c^{r}=k$ $\therefore \quad a=k^{1 / p}, b=k^{1 / q}, c=k^{1 / r}$
<br/>since, $a, b, c$ are in $\mathrm{GP} .$
<br/>$\therefore$
<br/>$\frac{b}{a}=\frac{c}{b}$
<br/>$\frac{k^{1 / q}}{k^{1 / P}}=\frac{k^{1 / r}}{k^{1 / q}}$
<br/>$k^{1 / q-1 / p}=k^{1 / r-1 / 4}$
<br/>$\frac{1}{q}-\frac{1}{p}=\frac{1}{r}-\frac{1}{q}$
<br/>$\therefore \quad \frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ are in GP.
<br/>$\Rightarrow \quad p, q, r$ in $\mathrm{HP}$</div>


title: let-a-be-a-3-3-matrix-and-b-be-its-adjoint-matrix-if-b-64-then-a-is-equal-to
slug: let-a-be-a-3-3-matrix-and-b-be-its-adjoint-matrix-if-b-64-then-a-is-equal-to-36102
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $A$ be a $3 \times 3$ matrix and $B$ be its adjoint matrix. If $|B|=64,$ then $|A|$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Â±2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pm 4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pm 8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Â±12</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pm 8$</span> </div>

answer:
<div class="solution">We have,
<br/>$|B|=64$
<br/>$\Rightarrow |\operatorname{adj} A \mid=64$
<br/>$\Rightarrow|A|^{2}=64$
<br/>$|A|=\pm 8$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-a-be-a-set-containing-n-elements-a-subset-p-of-a-is-chosen-and-the-set-a-is-reconstructed-by-replacing-the-elements-of-p-a-subset-q-of-a-is-chosen

slug:
let-a-be-a-set-containing-n-elements-a-subset-p-of-a-is-chosen-and-the-set-a-is-reconstructed-by-replacing-the-elements-of-p-a-subset-q-of-a-is-chosen-41719

question:
<div class="question">Let \(A\) be a set containing \(n\) elements. \(A\) subset \(P\) of \(A\) is chosen, and the set \(A\) is reconstructed by replacing the elements of \(P\). A subset \(Q\) of \(A\) is chosen again. The number of ways of choosing \(P\) and \(Q\) such the \(Q\) contains just one element more than \(P\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\({ }^{2 n} C_{n-1}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\({ }^{2 n} C_n\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\({ }^{2 n} C_{n+2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2^{2 n+1}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\({ }^{2 n} C_{n-1}\)</span> </div>

answer:
<div class="solution">Hint : Required number of ways<br/>\(\begin{aligned}<br/>&amp; ={ }^{\mathrm{n}} \mathrm{C}_0 \cdot{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_1 \cdot{ }^{\mathrm{n}} \mathrm{C}_2+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-1} \cdot{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \\<br/>&amp; ={ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}-1}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
let-a-be-a-square-matrix-of-order-3-whose-all-entries-are-1-and-let-i-3-be-the-identity-matrix-of-order-3-then-the-matrix-a-3-i-3-is

slug:
let-a-be-a-square-matrix-of-order-3-whose-all-entries-are-1-and-let-i-3-be-the-identity-matrix-of-order-3-then-the-matrix-a-3-i-3-is-60930

question:
<div class="question">Let $A$ be a square matrix of order 3 whose all entries are 1 and let $I_{3}$ be the identity matrix of order $3 .$ Then, the matrix $A-3 I_{3}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">invertible</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">orthogonal</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">non-invertible</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">real Skew Symmetric matrix</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">non-invertible</span> </div>

answer:
<div class="solution">Given, that $A-3 J_{3}$
<br/>$$
<br/>=\left[\begin{array}{lll}
<br/>1 &amp; 1 &amp; 1 \\
<br/>1 &amp; 1 &amp; 1 \\
<br/>1 &amp; 1 &amp; 1
<br/>\end{array}\right]-3\left[\begin{array}{lll}
<br/>1 &amp; 0 &amp; 0 \\
<br/>0 &amp; 1 &amp; 0 \\
<br/>0 &amp; 0 &amp; 1
<br/>\end{array}\right]
<br/>$$
<br/>$=\left[\begin{array}{lll}1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 1 \\ 1 &amp; 1 &amp; 1\end{array}\right]-\left[\begin{array}{ccc}3 &amp; 0 &amp; 0 \\ 0 &amp; 3 &amp; 0 \\ 0 &amp; 0 &amp; 3\end{array}\right]$
<br/>$=\left[\begin{array}{ccc}-2 &amp; 1 &amp; 1 \\ 1 &amp; -2 &amp; 1 \\ 1 &amp; 1 &amp; -2\end{array}\right]$
<br/>$=[-2(4-1)-1(-2-1)+1(1+2)]$
<br/>$=[-2(3)-1(-3)+1(3)]$
<br/>$=[-6+3+3]=0$
<br/>$\mathrm{Now}, \mathrm{det}=0$
<br/>So, matrix $A-3 I_{3}$ is non-invertible matrix.</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
let-a-be-the-centre-of-the-circle-x-2-y-2-2-x-4-y-20-0-let-b-1-7-and-d-4-2-be-two-points-on-the-circle-such-that-tangents-at-b-and-d-meet-at-c-the-are

slug:
let-a-be-the-centre-of-the-circle-x-2-y-2-2-x-4-y-20-0-let-b-1-7-and-d-4-2-be-two-points-on-the-circle-such-that-tangents-at-b-and-d-meet-at-c-the-are-45868

question:
<div class="question">Let $A$ be the centre of the circle $x^{2}+y^{2}-2 x-4 y-20=0 .$ Let $B(1,7)$ and
<br/>$D(4,-2)$ be two points on the circle such that tangents at $B$ and $D$ meet at $C$. The area of the quadrilateral $A B C D$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$150 \mathrm{sq}$ units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">50 sq units.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$75 \mathrm{sq}$ units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$70 \mathrm{sq}$ units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$75 \mathrm{sq}$ units</span> </div>

answer:
<div class="solution">Given, equation of circle is
<br/>$$
<br/>x^{2}+y^{2}-2 x-4 y-20=0
<br/>$$
<br/>Center $\left(1,2\right).$ and radius $=\sqrt{(1)^{2}+\left(2)^{2}+20\right.}=5$
<br/>Coordinate of intersecting point of tangents at $B$ and $D$ is $C(06,7)$ 
<br/>Area of quadrilateral $A B C D$
<br/>$$
<br/>=2 \times \operatorname{ar}(\Delta A B C)
<br/>$$
<br/>$=2 \times \frac{1}{2} \times 15 \times 5=75 \mathrm{sq}$ units
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a61.png"/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
let-a-be-the-point-0-4-and-b-be-a-moving-point-on-x-axis-let-m-be-the-midpoint-of-a-b-and-let-the-perpendicular-bisector-of-a-b-meets-the-y-axis-at-r-

slug:
let-a-be-the-point-0-4-and-b-be-a-moving-point-on-x-axis-let-m-be-the-midpoint-of-a-b-and-let-the-perpendicular-bisector-of-a-b-meets-the-y-axis-at-r-53449

question:
<div class="question">Let $A$ be the point $(0,4)$ and $B$ be a moving point on $x$-axis. Let $M$ be the midpoint of $A B$ and let the perpendicular bisector of $A B$ meets the $y$-axis at $R$. The locus of the midpoint $P$ of $M R$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y+x^{2}=2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+(y-2)^{2}=\frac{1}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(y-2)^{2}-x^{2}=\frac{1}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{2}+y^{2}=16$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y+x^{2}=2$</span> </div>

answer:
<div class="solution">Let B $=(2 \alpha, 0)$
<br/>$\therefore \mathrm{M}=(\alpha, 2)$
<br/>$\mathrm{m}_{\mathrm{AB}}=\frac{4}{-2 \alpha}=-\frac{2}{\alpha}$
<br/>$\therefore \mathrm{m}_{\mathrm{MR}}=\frac{\alpha}{2}, \therefore \mathrm{MR} \quad \mathrm{y}-2=\frac{\alpha}{2}(\mathrm{x}-\alpha)$
<br/>Put, $\mathrm{x}=0 \Rightarrow \mathrm{y}=2-\frac{\alpha^{2}}{2}$
<br/>$\therefore \mathrm{R} \equiv\left(0,2-\frac{\alpha^{2}}{2}\right)$
<br/>$\therefore \mathrm{P} \equiv\left(\frac{\alpha}{2}, \frac{2-\alpha^{2} / 2+2}{2}\right) \equiv \left(\frac{\alpha}{2}, 2-\frac{\alpha^{2}}{4}\right)$
<br/>$\therefore \mathrm{y}=2-\mathrm{x}^{2} \Rightarrow \mathrm{y}+\mathrm{x}^{2}=2$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/1sDn1CBgMLUHOE7bKfO4A1d3UjqdvrpBATuFIPfFzww.original.fullsize.png"/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-a-be-the-point-04-in-the-xyplane-and-let-b-be-the-point-2-t-0-let-l-be-the-midpoint-of-a-b-and-let-the-perpendicular-bisector-of-a-b-meet-the-yaxi

slug:
let-a-be-the-point-04-in-the-xyplane-and-let-b-be-the-point-2-t-0-let-l-be-the-midpoint-of-a-b-and-let-the-perpendicular-bisector-of-a-b-meet-the-yaxi-80911

question:
<div class="question">Let \(A\) be the point \((0,4)\) in the xy-plane and let \(B\) be the point \((2 t, 0)\). Let \(L\) be the midpoint of \(A B\) and let the perpendicular bisector of \(A B\) meet the \(y\)-axis \(M\). Let \(N\) be the midpoint of LM. Then locus of \(N\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a circle</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">a parabola</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a straight line</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a hyperbola</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a parabola</span> </div>

answer:
<div class="solution">Hint :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/qNKw4XizkTf8Pj6wEryy1Wf5HqfbGZ7gvfJg5emvORg.original.fullsize.png"/><br/><br/>Equation of LM<br/>\(\begin{aligned}<br/>&amp; y-2=\frac{t}{2}(x-t) \\<br/>&amp; \Rightarrow M\left(0, \frac{4-t^2}{2}\right)<br/>\end{aligned}\)<br/>Midpoint of LM : N(h, k)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow \quad 2 h=t, 2 k=4-\frac{t^2}{2} \\<br/>&amp; \Rightarrow \quad x^2=2-y<br/>\end{aligned}\)</div>


title: let-a-min-x-2-2-x-3-x-r-and-b-lim-0-2-1-c-o-s-then-r-0-n-a-r-b-n-r-is
slug: let-a-min-x-2-2-x-3-x-r-and-b-lim-0-2-1-c-o-s-then-r-0-n-a-r-b-n-r-is-90484
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $a=\min \left\{x^{2}+2 x+3: x \in R\right\}$ and $b=\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta^{2}} \cdot$ Then $\sum_{r=0}^{n} a^{r} b^{n-r}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2^{n+1}-1}{3 \cdot 2^{n}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2^{n+1}+1}{3 \cdot 2^{n}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{4^{n+1}-1}{3 \cdot 2^{n}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}\left(2^{n}-1\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4^{n+1}-1}{3 \cdot 2^{n}}$</span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { (c) Let } f(x)=x^{2}+2 x+3\\
<br/>&amp;\begin{aligned}
<br/>a &amp;=f(x)_{\min }=\frac{-D}{4 a}=\frac{-(4-12}{4}=\frac{8}{4}=2 \\
<br/>\text { and } b &amp;=\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta^{2}}=\lim _{\theta \rightarrow 0} \frac{1-1+2 \sin ^{2} \theta / 2}{\theta^{2}} \\
<br/>&amp;=\lim _{\theta \rightarrow 0} \frac{2 \sin ^{2} \theta / 2}{(\theta / 2)^{2} \cdot 4}=\lim _{\theta \rightarrow 0} \frac{1}{2} \cdot\left[\frac{\sin ^{2} \theta / 2}{(\theta / 2)^{2}}\right] \\
<br/>&amp;=\frac{1}{2} \cdot \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta / 2}{(\theta / 2)^{2}} \\
<br/>&amp;=\frac{1}{2} \cdot 1=\frac{1}{2} \\
<br/>b &amp;=\frac{1}{2} \\
<br/>\text { Now, } &amp; \sum_{r=0}^{n} a^{r} \cdot b^{n-r}
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$
<br/>$=\sum_{r=0}^{n}\left(2^{r}\left(\frac{1}{2}\right)^{n-r}\right)$
<br/>$=\sum_{r=0}^{n} 2^{\prime} \cdot 2^{(r-n)}$
<br/>$=\sum_{r=0}^{n} 2^{2 r-n}$
<br/>$=2^{-n} \sum_{r=0}^{n} 2^{2 r}$
<br/>$=2^{-n}\left[1+2^{2}+2^{4}+2^{6}+\ldots+2^{2 n}\right]$
<br/>$=2^{-n}\left[\frac{1 \cdot\left(2^{2}\right)^{n+1}-1}{2^{2}-1}\right]\left\{\because S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}\right\}$
<br/>$=2^{-n}\left[\frac{4^{n+1}-1}{3}\right]$
<br/>$=\frac{4^{n+1}-1}{3 \cdot 2^{n}}$</div>


title: let-a-n-1-2-2-2-n-2-n-and-b-n-n-n-n-then
slug: let-a-n-1-2-2-2-n-2-n-and-b-n-n-n-n-then-54393
question: <div class="question">Let $a_n=\left(1^2+2^2+\ldots n^2\right)^n$ and $b_n=n^n(n !)$. Then</div>
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a_n &lt; b_n \forall n$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a_n&gt;b_n \forall n$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a_n=b_n$ for infinitely many $n$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a_n &lt; b_n$ if $n$ be even and $a_n&gt;b_n$ if $n$ be odd</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a_n&gt;b_n \forall n$</span> </div>

answer:
<div class="solution">Using PMI $\quad a_n&gt;b_n \quad\left[{ }^* a_n&gt;b_n \forall n \geq 2\right.$ as $\left.a_1=b_1\right]$</div>


title: let-a-x-2-3-3-x-x-2-b-x-5-0-x-2-then-all-solutions-of-the-equation-det-a-b-0-is
slug: let-a-x-2-3-3-x-x-2-b-x-5-0-x-2-then-all-solutions-of-the-equation-det-a-b-0-is-41007
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $A=\left(\begin{array}{cc}x+2 &amp; 3 x \\ 3 &amp; x+2\end{array}\right), B=\left(\begin{array}{cc}x &amp; 0 \\ 5 &amp; x+2\end{array}\right)$. Then
<br/>all solutions of the equation det $(A B)=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,-1,0,2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1,4,0,-2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1,-1,4,3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1,4,0,3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1,4,0,-2</span> </div>

answer:
<div class="solution">We know that, $\operatorname{det}(A B)=$ det $(A)$ det $(B)$
<br/>$\therefore$ $\operatorname{det}(A B)=0$
<br/>$\Rightarrow$ det$(A) \cdot$ det $(B)=0$ $\Rightarrow\left|\begin{array}{cc}x+2 &amp; 3 x \\ 3 &amp; x+2\end{array} \| \begin{array}{cc}x &amp; 0 \\ 5 &amp; x+2\end{array}\right|=0$
<br/>$\Rightarrow\left\{(x+2)^{2}-9 x\right\}\{x(x+2)-0\}=0$
<br/>$\Rightarrow \quad\left(x^{2}+4 x+4-9 x\right) \times(x+2)=0$
<br/>$\Rightarrow \quad x(x+2)\left(x^{2}-5 x+4\right)=0$
<br/>$\Rightarrow \quad x(x+2)(x-1)(x-4)=0$
<br/>$\Rightarrow \quad x=0,-2,1,4$</div>


title: let-a-x-r-1-x-1-f-a-a-be-a-mapping-defined-by-f-x-x-x-then-f-is
slug: let-a-x-r-1-x-1-f-a-a-be-a-mapping-defined-by-f-x-x-x-then-f-is-82814
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{A}=\{\mathrm{x} \in \mathbb{R}:-1 \leq \mathrm{x} \leq 1\} \&amp; \mathrm{f}: \mathrm{A} \rightarrow \mathrm{A}$ be a mapping defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}| .$ Then $\mathrm{f}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">injective but not surjective</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">surjective but not injective</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">neither injective nor surjective</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">bijective</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">bijective</span> </div>

answer:
<div class="solution">Hint: 
<br/>$f(x)=x|x|=\left\{\begin{array}{c}-x^{2} x \in[-1,0) \\ x^{2} x \in[0,1]\end{array}\right.$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/R8PXzPFUOpjJX1ZwFSXLcq4vlSBQKHzOV4-O6TFJUXE.original.fullsize.png"/></div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-a1-a2-a3-ldots-an-be-positive-real-numbers-then-the-minimum-value-of-fraca1a2fraca2a3ldots-fracana1-is

slug:
let-a1-a2-a3-ldots-an-be-positive-real-numbers-then-the-minimum-value-of-fraca1a2fraca2a3ldots-fracana1-is-89474

question:
<div class="question">Let \(a_1, a_2, a_3, \ldots, a_n\) be positive real numbers. Then the minimum value of \(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\ldots .+\frac{a_n}{a_1}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{n}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\({ }^n \mathrm{C}_2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{n}\)</span> </div>

answer:
<div class="solution">Hint: \(\frac{\frac{a_1}{a_2}+\frac{a_2}{a_3}+\ldots . .+\frac{a_n}{a_1}}{n} \geq \sqrt[n]{\frac{a_1}{a_2} \times \frac{a_2}{a_3} \times \ldots . \frac{a_n}{a_1}}\)<br/>\(A M \geq G M\) (property)<br/>\(\therefore\) Minimum value \(=\mathrm{n}\)<br/>It is possible when \(a_1=a_2=a_3=\ldots .=a_n\)</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-a2-sec-theta-3-tan-theta-and-b2-sec-phi-3-tan-phi-where-thetaphifracpi2-be-two-points-on-the-hyperbola-fracx24fracy291-if-alpha-beta-is-the-point-

slug:
let-a2-sec-theta-3-tan-theta-and-b2-sec-phi-3-tan-phi-where-thetaphifracpi2-be-two-points-on-the-hyperbola-fracx24fracy291-if-alpha-beta-is-the-point-50246

question:
<div class="question">Let \(A(2 \sec \theta, 3 \tan \theta)\) and \(B(2 \sec \phi, 3 \tan \phi)\) where \(\theta+\phi=\frac{\pi}{2}\) be two points on the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\). If \((\alpha, \beta)\) is the point of intersection of normals to the hyperbola at \(A\) and \(B\), then \(\beta\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{12}{3}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{13}{3}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(-\frac{12}{3}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(-\frac{13}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-\frac{13}{3}\)</span> </div>

answer:
<div class="solution">Hint : Equation of Normal at \(A\)<br/>\(\frac{4 x}{2 \sec \theta}+\frac{9 y}{3 \tan \theta}=13\)<br/>at \(B\)<br/>\(\begin{aligned}<br/>&amp; \frac{4 x}{2 \operatorname{cosec} \theta}+\frac{9 y}{3 \cot \theta}=13 \\<br/>&amp; \Rightarrow(2 \cos \theta) x+(3 \cot \theta) y=13   \quad ----(1)\\<br/>&amp; \&amp;~(2 \sin \theta) x+(3 \tan \theta) y=13 \quad ---(2)\\<br/>&amp; \Rightarrow y=-\frac{13}{3}<br/>\end{aligned}\)</div>


title: let-aleftbeginarrayccc2-0-3-4-7-11-5-4-8endarrayright-then
slug: let-aleftbeginarrayccc2-0-3-4-7-11-5-4-8endarrayright-then-88281
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let \(A=\left(\begin{array}{ccc}2 &amp; 0 &amp; 3 \\ 4 &amp; 7 &amp; 11 \\ 5 &amp; 4 &amp; 8\end{array}\right)\). Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\operatorname{det} A\) is divisible by 11</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\operatorname{det} A\) is not divisible by 11</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\operatorname{det} A=0\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A is orthogonal matrix</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\operatorname{det} A\) is divisible by 11</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint: } \operatorname{det}(A)=2(56-44)+0+3(16-35) \\<br/>&amp; =24-57 \\<br/>&amp; =-33 \text { which is divisible by } 11<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-aleftbeginarraylll0-0-1-1-0-0-0-0-0endarrayright-bleftbeginarraylll0-1-0-0-0-1-0-0-0endarrayright-and-pleftbeginarraylll0-1-0-x-0-0-0-0-yendarrayr

slug:
let-aleftbeginarraylll0-0-1-1-0-0-0-0-0endarrayright-bleftbeginarraylll0-1-0-0-0-1-0-0-0endarrayright-and-pleftbeginarraylll0-1-0-x-0-0-0-0-yendarrayr-53048

question:
<div class="question">Let \(A=\left(\begin{array}{lll}0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0\end{array}\right), B=\left(\begin{array}{lll}0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0\end{array}\right)\) and \(P=\left(\begin{array}{lll}0 &amp; 1 &amp; 0 \\ x &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; y\end{array}\right)\) be an orthogonal matrix such that \(B=PAP^{-1}\) holds. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(x=1=y\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(x=1, y=0\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(x=0, y=1\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(x=-1, y=0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x=1=y\)</span> </div>

answer:
<div class="solution">Hint : \(B=P A P^{-1}\)<br/>\(P^{-1} B P=P^{-1} P A P^{-1} P\)<br/>\(\begin{aligned}<br/>&amp; {\left[\begin{array}{ccc}<br/>0 &amp; 0 &amp; x y \\<br/>x &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right]=A=\left[\begin{array}{lll}<br/>0 &amp; 0 &amp; 1 \\<br/>1 &amp; 0 &amp; 0 \\<br/>0 &amp; 0 &amp; 0<br/>\end{array}\right]} \\<br/>&amp; \therefore x=1=y<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-alpha-beta-be-the-roots-of-the-equation-a-x2b-xc0-a-b-c-real-and-snalphanbetan-and-leftbeginarrayccc3-1s1-1s2-1s1-1s2-1s3-1s2-1s3-1s4endarrayright

slug:
let-alpha-beta-be-the-roots-of-the-equation-a-x2b-xc0-a-b-c-real-and-snalphanbetan-and-leftbeginarrayccc3-1s1-1s2-1s1-1s2-1s3-1s2-1s3-1s4endarrayright-53034

question:
<div class="question">Let \(\alpha, \beta\) be the roots of the equation \(a x^2+b x+c=0, a, b, c\) real and \(s_n=\alpha^n+\beta^n\) and \(\left|\begin{array}{ccc}3 &amp; 1+s_1 &amp; 1+s_2 \\ 1+s_1 &amp; 1+s_2 &amp; 1+s_3 \\ 1+s_2 &amp; 1+s_3 &amp; 1+s_4\end{array}\right|=k \frac{(a+b+c)^2}{a^4}\) then \(k=\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(b^2-4 a c\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(b^2+4 a c\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(b^2+2 a c\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(4 a c-b^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(b^2-4 a c\)</span> </div>

answer:
<div class="solution">Hint: \(\left|\begin{array}{ccc}3 &amp; 1+s_1 &amp; 1+s_2 \\ 1+s_1 &amp; 1+s_2 &amp; 1+s_3 \\ 1+s_2 &amp; 1+s_3 &amp; 1+s_4\end{array}\right|\)<br/>\(=\left|\begin{array}{ccc}<br/>1+1+1 &amp; 1+\alpha+\beta &amp; 1+\alpha^2+\beta^2 \\<br/>1+\alpha+\beta &amp; 1+\alpha^2+\beta^2 &amp; 1+\alpha^3+\beta^3 \\<br/>1+\alpha^2+\beta^2 &amp; 1+\alpha^3+\beta^3 &amp; 1+\alpha^4+\beta^4<br/>\end{array}\right|\)<br/>\(\begin{aligned} &amp; =\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; \alpha &amp; \beta \\ 1 &amp; \alpha^2 &amp; \beta^2\end{array}\right| \times\left|\begin{array}{ccc}1 &amp; 1 &amp; 1 \\ 1 &amp; \alpha &amp; \alpha^2 \\ 1 &amp; \beta &amp; \beta^2\end{array}\right| \\ &amp; =\{(1-\alpha)(\alpha-\beta)(1-\beta)\}^2 \\ &amp; =(1-(\alpha+\beta)+\alpha \beta)^2(\alpha-\beta)^2 \\ &amp; =\left(1+\frac{b}{a}+\frac{c}{a}\right)^2\left(\frac{b^2}{a^2}-\frac{4 c}{a}\right) \\ &amp; =\frac{(a+b+c)^2}{a^2} \times \frac{b^2-4 a c}{a^2}=\left(b^2-4 a c\right) \frac{(a+b+c)^2}{a^4} \\ &amp; \therefore k=b^2-4 a c\end{aligned}\)</div>


tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$c=a$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-and-be-iwo-distinct-roots-of-a-cos-b-sin-c-where-a-b-c-are-three-real-constants-and-0-0-2-then-is-also-a-root-of-the-same-equation-if

slug:
let-and-be-iwo-distinct-roots-of-a-cos-b-sin-c-where-a-b-c-are-three-real-constants-and-0-0-2-then-is-also-a-root-of-the-same-equation-if-65538

question:
<div class="question">Let $\alpha$ and $\beta$ be iwo distinct roots of $a \cos \theta+b \sin \theta=c,$ where $a, b, c$ are three
<br/>real constants and $0 \in|0,2 \pi|$. Then, $\alpha+\beta$ is also a root of the same equation, if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a+b=c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$b+c=a$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$c+a=b$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$c=a$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given equation is $a \cos \theta+b \sin \theta=c$
<br/>$\Rightarrow a\left(\frac{1-\tan ^{2} \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right)+\frac{2 b \tan \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}=c$
<br/>$\left[\because \cos \theta=\frac{1-\tan ^{2} \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right.$ and $\left.\sin \theta=\frac{2 \tan \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right]$
<br/>$\Rightarrow a\left(1-\tan ^{2} \frac{\theta}{2}\right)+2 b \tan \frac{\theta}{2}=c\left(1+\tan ^{2} \frac{\theta}{2}\right)$
<br/>$\Rightarrow a-\operatorname{atan}^{2} \frac{\theta}{2}+2 b \tan \frac{\theta}{2}-c-c \tan ^{2} \frac{\theta}{2}=0$
<br/>$\Rightarrow(c+a) \tan ^{2} \frac{\theta}{2}-2 b \tan \frac{\theta}{2}+(c-a)=0$
<br/>Let $\alpha$ and $\beta$ be the roots of the equation $\therefore \alpha+\beta=\frac{2 b}{c+a}$ and $\alpha \beta=\frac{c-a}{c+a}$
<br/>Now, $\tan \frac{\alpha+\beta}{2}=\frac{\frac{2 b}{c+a}}{1-\frac{c-a}{c+a}}$
<br/>$=\frac{\frac{c+a}{c+a-c+a}}{c+a}=\frac{b}{a}$
<br/>since, $\frac{b}{c}$ is a root of the equation. $\therefore \quad(c+a) \frac{b^{2}}{a^{2}}-2 b\left(\frac{b}{a}\right)+c-a=0$
<br/>$\Rightarrow b^{2} c+b^{2} a-2 b^{2} a+c a^{2}-a^{3}=0$
<br/>$\Rightarrow-b^{2} a+b^{2} c+c a^{2}-a^{3}=0$
<br/>$\Rightarrow \quad b^{2} c-b^{2} a+c a^{2}-a^{3}=0$
<br/>$\Rightarrow \quad b^{2}(c-a)+a^{2}(c-a)=0$
<br/>$\Rightarrow \quad(c-a)\left(b^{2}+a^{2}\right)=0$
<br/>$\Rightarrow c-a=0 \quad$ or $\quad b^{2}+a^{2}=0$
<br/>$\Rightarrow \quad c=a$ or $b^{2}+a^{2}=0$</div>


title: let-and-be-the-roots-of-x-2-x-1-0-if-n-be-a-positive-integer-then-n-n-is
slug: let-and-be-the-roots-of-x-2-x-1-0-if-n-be-a-positive-integer-then-n-n-is-23448
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\alpha$ and $\beta$ be the roots of $x^{2}+x+1=0 .$ If $n$ be a positive integer, then $\alpha^{n}+\beta^{n}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \cos \frac{2 n \pi}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sin \frac{2 n \pi}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \cos \frac{n \pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sin \frac{n \pi}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \cos \frac{2 n \pi}{3}$</span> </div>

answer:
<div class="solution">We have, $x^{2}+x+1=0$
<br/>$\Rightarrow \quad x=\frac{-1 \pm \sqrt{3} i}{2}$
<br/>$\Rightarrow \quad \alpha=\frac{-1+\sqrt{3} i}{2}$ and $\beta=\frac{-1-\sqrt{3} i}{2}$
<br/>or $\quad \alpha=e^{\frac{i2\pi}{3}}$ and $\beta=e^{\frac{-2\pi i}{3}}$
<br/>$\therefore$
<br/>$\alpha^{n}+\beta^{n}=e^{\frac{2 n i t}{3}}+e^{\frac{-2\pi i}{3}}$
<br/>$=2\left(\frac{e^{\frac{2n\pi i}{3}}+e^{\frac{-2n\pi i}{3}}}{2}\right)$
<br/>$=2 \cos \left(\frac{2 n \pi}{3}\right)$</div>


title: let-be-a-relation-defined-on-n-the-set-of-natural-numbers-as-x-y-n-n-2-x-y-41-then
slug: let-be-a-relation-defined-on-n-the-set-of-natural-numbers-as-x-y-n-n-2-x-y-41-then-81780
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\rho$ be a relation defined on $N$, the set of natural numbers, as $\rho=\{(x, y) \in N \times N: 2 x+y=41\} .$ Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\rho$ is an equivalence relation</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\rho$ is only reflexive relation</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\rho$ is only symmetric relation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\rho$ is not transitive</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\rho$ is not transitive</span> </div>

answer:
<div class="solution">We have, $\rho=\{(x, y) \in N \times N: 2 x+y=41\}$<br/>For reflexive,<br/>$\begin{array}{l}<br/>x p x \Rightarrow 2 x+x=41 \\<br/>\Rightarrow 3 x=41<br/>\end{array}$<br/>$\Rightarrow x=\frac{41}{3} \notin N$<br/>So, $\rho$ is not reflexive. <br/>For symmetric,<br/>$x \rho y \Rightarrow 2 x+y=41$<br/>and $y \rho x \Rightarrow 2 y+x=41$<br/>$\Rightarrow x p y \neq y \rho x$<br/>So, $\rho$ is not symmetric.<br/>For transitive,<br/>$x p y \Rightarrow 2 x+y=41$<br/>and $y \rho z \Rightarrow 2 y+z=41$<br/>$\Rightarrow x p z$<br/>$\Rightarrow \rho$ is not transitive.</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-be-the-roots-of-the-equation-x-2-6-x-2-0-with-if-a-n-n-n-for-n-1-then-the-value-of-2-a-9-a-10-2-a-8-is

slug:
let-be-the-roots-of-the-equation-x-2-6-x-2-0-with-if-a-n-n-n-for-n-1-then-the-value-of-2-a-9-a-10-2-a-8-is-18217

question:
<div class="question">Let $\alpha, \beta$ be the roots of the equation $x^{2}-6 x-2=0$ with $\alpha&gt;\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$, then the value of $\frac{a_{10}-2 a_{8}}{2 a_{9}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">$x^{2}-6 x-2=0$
<br/>$x^{n}-6 x^{n-1}-2 x^{n-2}=0$
<br/>$\Rightarrow x^{n}-2 x^{n-2}=6 x^{n-1}$
<br/>for $n=10$
<br/>$x^{10}-2 x^{8}=6 x^{9}$
<br/>$\alpha^{10}-2 \alpha^{8}=6 \alpha^{9}$
<br/>$\beta^{10}-2 \beta^{8}=6 \beta^{9}$
<br/>$\left(\alpha^{10}-\beta^{10}\right)-2\left(\alpha^{8}-\beta^{8}\right)=6\left(\alpha^{9}-\beta^{9}\right)$
<br/>$\Rightarrow a_{10}-2 a_{8}=6 a_{9}$
<br/>$\Rightarrow \frac{a_{10}-2 a_{8}}{2 a_{9}}=3$</div>


title: let-be-the-roots-of-x-2-x-1-0-and-s-n-n-n-for-all-integers-n-1-then-for-every-integer-n-2
slug: let-be-the-roots-of-x-2-x-1-0-and-s-n-n-n-for-all-integers-n-1-then-for-every-integer-n-2-68539
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\alpha, \beta$ be the roots of $x^{2}-x-1=0$ and $S_{n}=\alpha^{n}+\beta^{n},$ for all integers $n \geq 1 .$ Then, for every integer $n \geq 2$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$S_{n}+S_{n-1}=S_{n+1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$S_{n}-S_{n-1}=S_{n+1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$S_{n-1}=S_{n+1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$S_{n}+S_{n-1}=2 S_{n+1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$S_{n}+S_{n-1}=S_{n+1}$</span> </div>

answer:
<div class="solution">Given, $\alpha$ and $\beta$ are the roots of $x^{2}-x-1=0$
<br/>$\therefore \quad \alpha+\beta=1$
<br/>and $\alpha \beta=-1$
<br/>Now, consider option (a). $S_{n}+S_{n-1}=S_{n+1}$
<br/>Put $n=2$, we get, $\quad S_{2}+S_{1}=S_{3}$ $\therefore \quad\left(\alpha^{2}+\beta^{2}\right)+(\alpha+\beta)=\alpha^{3}+\beta^{3}$
<br/>$\left[\because \cdot S_{n}=\alpha^{n}+\beta^{n},\right.$ given]
<br/>$\Rightarrow(a+\beta)^{2}-2 a \beta+(\alpha+\beta)$
<br/>$=(\alpha+\beta)^{3}-3 \alpha \beta(\alpha+\beta)$
<br/>$\Rightarrow 1^{2}-2(-1)+1=1^{3}-3(-1)(1)$
<br/>$\Rightarrow 2+1=1+3$
<br/>$\Rightarrow \quad 1+2+1=1+3$
<br/>$\Rightarrow$
<br/>$4=4,$ which is true.
<br/>Hence, option (a) is correct.</div>


title: let-be-the-three-unit-vectors-such-that-0-and-the-angle-between-and-is-3-0-then-is
slug: let-be-the-three-unit-vectors-such-that-0-and-the-angle-between-and-is-3-0-then-is-54877
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be the three unit vectors such that $\vec{\alpha} \cdot \vec{\beta}=\vec{\alpha} \cdot \vec{\gamma}=0$ and the angle between $\vec{\beta}$ and $\vec{\gamma}$
<br/>is $30^{\circ} .$ Then $\vec{\alpha}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2(\vec{\beta} \times \vec{\gamma})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2(\vec{\beta} \times \vec{y})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pm 2(\vec{\beta} \times \vec{\gamma})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\vec{\beta} \times \vec{y})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pm 2(\vec{\beta} \times \vec{\gamma})$</span> </div>

answer:
<div class="solution">$\vec{\alpha}=\lambda(\vec{\beta} \times \vec{\gamma})=\lambda\left(|\vec{\beta}||\vec{\gamma}|\left|\sin 30^{\circ}\right|\right.$
<br/>$\Rightarrow \quad|\vec{\alpha}|=|\lambda|\left(|\beta||\vec{\gamma}| \cdot \frac{1}{2}\right)$
<br/>$\begin{array}{lc}\Rightarrow &amp; 1=|\lambda| \cdot 1 \cdot 1 \cdot \frac{1}{2} \\ \Rightarrow &amp; |\lambda|=2 \\ \rightarrow &amp; \lambda-\pm 2\end{array}$
<br/>$\therefore \quad \vec{\alpha}=\pm 2(\vec{\beta} \times \vec{\gamma})$</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
let-be-three-nonzero-vectors-which-are-pairwise-noncollinear-if-3-is-collinear-with-and-2-is-collinear-with-then-3-6-is

slug:
let-be-three-nonzero-vectors-which-are-pairwise-noncollinear-if-3-is-collinear-with-and-2-is-collinear-with-then-3-6-is-84585

question:
<div class="question">Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{\alpha}+3 \vec{\beta}$ is collinear with $\vec{\gamma}$ and $\vec{\beta}+2 \vec{\gamma}$ is collinear with $\vec{\alpha}$, then $\vec{\alpha}+3 \vec{\beta}+6 \vec{\gamma}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\vec{\gamma}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\overrightarrow{0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\vec{\alpha}+\vec{\gamma}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\vec{\alpha}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\overrightarrow{0}$</span> </div>

answer:
<div class="solution">$\vec{\alpha}+3 \vec{\beta}=\mathrm{k}_{1} \vec{\gamma} \Rightarrow \vec{\beta}=\frac{\mathrm{k}_{1}}{3} \vec{\gamma}-\frac{\vec{\alpha}}{3}$
<br/>$\vec{\beta}+2 \vec{\gamma}=\mathrm{k}_{2} \vec{\alpha} \Rightarrow \vec{\beta}=\mathrm{k}_{2} \vec{\alpha}-2 \vec{\gamma}$
<br/>$\Rightarrow \frac{\mathrm{k}_{1} \vec{\gamma}}{3}-\frac{\vec{\alpha}}{3}=\mathrm{k}_{2} \vec{\alpha}-2 \vec{\gamma} \Rightarrow \vec{\alpha}\left(\mathrm{k}_{2}+\frac{1}{3}\right)=\vec{\gamma}\left(\frac{\mathrm{k}_{1}}{3}+2\right) \Rightarrow \mathrm{k}_{2}=-\frac{1}{3}$ and $\frac{\mathrm{k}_{1}}{3}=-2 \Rightarrow \mathrm{k}_{1}=-6$
<br/>$\Rightarrow \vec{\alpha}+3 \vec{\beta}+6 \vec{\gamma}=\overrightarrow{0}$</div>


title: let-be-three-unit-vectors-such-that-2-1-where-if-is-not-parallel-to-then-the-angle-between-and-is
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
slug:
let-be-three-unit-vectors-such-that-2-1-where-if-is-not-parallel-to-then-the-angle-between-and-is-14653

question:
<div class="question">Let $\hat{\alpha}, \hat{\beta}, \hat{\gamma}$ be three unit vectors such that $\hat{\alpha} \times (\hat{\beta} \times \hat{\gamma})=\frac{1}{2}(\hat{\beta}+\hat{\gamma})$ where $\hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=(\hat{\alpha} \cdot \hat{\gamma}) \hat{\beta}-(\hat{\alpha} \cdot \hat{\beta}) \gamma \cdot$ If $\hat{\beta}$ is not parallel to $\hat{\gamma},$ then the angle between $\hat{\alpha}$ and $\hat{\beta}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{5 \pi}{6}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{2 \pi}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \pi}{3}$</span> </div>

answer:
<div class="solution">Given,
<br/>$|\hat{\alpha}|=|\hat{\beta}|=|\hat{\gamma}|=1$
<br/>and $\quad \hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=\frac{1}{2}(\hat{\beta}+\hat{\gamma})$
<br/>$\left.\operatorname{Now}, \hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=\frac{1}{2} \hat{(\hat{\beta}}+\hat{\gamma}\right)$
<br/>$\Rightarrow \quad(\hat{\alpha} \cdot \hat{\gamma}) \hat{\beta}-(\hat{\alpha} \cdot \hat{\beta}) \hat{\gamma}=\frac{1}{2} \hat{\beta}+\frac{1}{2} \hat{\gamma}$
<br/>On comparing we get, $-(\hat{\alpha} \cdot \hat{\beta})=\frac{1}{2}$
<br/>$\Rightarrow|\hat{\alpha}||\hat{\beta}| \cos \theta=-\frac{1}{2}$
<br/>$\Rightarrow \quad \cos \theta=-\frac{1}{2} \quad[\because|\hat{\alpha}|=|\hat{\beta}|=1]$
<br/>$\Rightarrow \quad \theta=120^{\circ}$ or $\frac{2 \pi}{3}$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7: 1</span> </div>
source_db: db\MarksBatch2_P1.db
title:
let-c-1-and-c-2-denote-the-centres-of-the-circles-x-2-y-2-4-and-x-2-2-y-2-1-respectively-and-let-p-and-q-be-their-points-of-intersectionthen-the-areas

slug:
let-c-1-and-c-2-denote-the-centres-of-the-circles-x-2-y-2-4-and-x-2-2-y-2-1-respectively-and-let-p-and-q-be-their-points-of-intersectionthen-the-areas-36254

question:
<div class="question">Let $C_{1}$ and $C_{2}$ denote the centres of the circles $x^{2}+y^{2}=4$ and $(x-2)^{2}+y^{2}=1$ respectively and let $P$ and $Q$ be their points of intersection.Then, the areas of $\Delta C_{1} P Q$ and $C_{2} P Q$ are in the ratio</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3: 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5: 1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7: 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9: 1</span> </li> </ul>

answer:
<div class="solution">$x^{2}+y^{2}=4$
<br/>$$
<br/>\begin{array}{l}
<br/>(x-2)^{2}+y^{2}=1 \\
<br/>C_{1}=(0,0), C_{2}=(2,0)
<br/>\end{array}
<br/>$$
<br/>On solving Eqs. (i) and (ii), we get 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_mat_a65.png"/> <br/> <br/>$$
<br/>\begin{aligned}
<br/>x=\frac{7}{4}, y=\pm \frac{\sqrt{15}}{4} \\
<br/>C_{1} N=\frac{7}{4}, P Q=\frac{2 \sqrt{15}}{4} &amp;=\frac{\sqrt{15}}{2} \\
<br/>\frac{\text { Area }\left(\Delta C_{1} P Q\right)}{\text { Area }\left(\Delta C_{2} P Q\right)}=&amp; \frac{\frac{1}{2} \cdot P Q \cdot C_{1} N}{\frac{1}{2} \cdot P Q \cdot C_{2} N} \\
<br/>&amp;=\frac{C_{1} N}{C_{2} N}=\frac{\frac{7}{4}}{2-\frac{7}{4}}=\frac{\frac{7}{4}}{\frac{1}{4}}=7: 1
<br/>\end{aligned}
<br/>$$
<br/>Hence, the required option is (c).</div>


title: let-c-denote-the-set-of-all-complex-numbers-define-a-z-w-z-w-c-and-z-w-b-z-w-z-w-c-and-z-2-w-2-then
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2021']
answer: <div class="solution">$z^{2}=w^{2}$ <br/>$(z-w)(z+w)$ <br/>$z=w,-w$ <br/>$|z|=|w|$</div>
source_db: db\MarksBatch2_P1.db
slug:
let-c-denote-the-set-of-all-complex-numbers-define-a-z-w-z-w-c-and-z-w-b-z-w-z-w-c-and-z-2-w-2-then-64961

question:
<div class="question">Let $\mathrm{C}$ denote the set of all complex numbers.
<br/>Define $A=\{(z, w) \mid z, w \in C$ and $|z|=|w|\}, B=\left\{(z, w) \mid z, w \in C\right.$ and $\left.z^{2}=w^{2}\right\}$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A=B$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A} \subset \mathrm{B}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{B} \subset \mathrm{A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A \cap B=\varphi$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{B} \subset \mathrm{A}$</span> </div>


tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
let-c-rms-h-2-is-the-rms-speed-of-h-2-at-150-k-at-what-temperature-the-most-probable-speed-of-helium-c-mp-he-will-be-half-of-c-rms-h-2

slug:
let-c-rms-h-2-is-the-rms-speed-of-h-2-at-150-k-at-what-temperature-the-most-probable-speed-of-helium-c-mp-he-will-be-half-of-c-rms-h-2-14730

question:
<div class="question">Let $\left(\mathrm{C}_{\mathrm{rms}}\right)_{\mathrm{H}_2}$ is the r.m.s speed of $\mathrm{H}_2$ at $150 \mathrm{~K}$. At what temperature, the most probable speed of helium $\left[\left(\mathrm{C}_{\mathrm{mp}}\right)_{\mathrm{He}}\right]$ will be half of $\left(\mathrm{C}_{\mathrm{rms}}\right)_{\mathrm{H}_2}$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$75 \mathrm{~K}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$112.5 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$225 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$900 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$112.5 \mathrm{~K}$</span> </div>

answer:
<div class="solution">$C_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}} \quad C_{m p}=\sqrt{\frac{2 R T}{M}}$<br/>As per question $\frac{1}{2} \sqrt{\frac{3 \times R \times 150}{2}}=\sqrt{\frac{2 R T}{4}}$ $\mathrm{T}=112.5 \mathrm{~K}$</div>


title: let-cos-1-b-y-lo-g-n-x-n-then
slug: let-cos-1-b-y-lo-g-n-x-n-then-54971
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}$. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x^{2} y_{2}+x y_{1}+n^{2} y=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x y_{2}-x y_{1}+2 n^{2} y=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2} y_{2}+3 x y_{1}-n^{2} y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x y_{2}+5 x y_{1}-3 y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2} y_{2}+x y_{1}+n^{2} y=0$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\cos ^{-1}\left(\frac{y}{b}\right)=\log \left(\frac{x}{n}\right)^{n}=n \times \log \left(\frac{x}{n}\right)$
<br/>$\Rightarrow-\frac{1}{\sqrt{b^{2}-y^{2}}} \cdot y_{1}=\frac{n}{x}$
<br/>$\Rightarrow x^{2} y_{1}^{2}=n^{2}\left(b^{2}-y^{2}\right) \Rightarrow x^{2} y_{2}+x y_{1}+n^{2} y=0$</div>


title: let-cos-1leftfracybrightlog-eleftfracxnrightn-then-a-y2b-y1c-y0-is-possible-for
slug: let-cos-1leftfracybrightlog-eleftfracxnrightn-then-a-y2b-y1c-y0-is-possible-for-74650
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let \(\cos ^{-1}\left(\frac{y}{b}\right)=\log _e\left(\frac{x}{n}\right)^n\), then \(A y_2+B y_1+C y=0\) is possible for</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{A}=2, \mathrm{~B}=\mathrm{x}^2, \mathrm{C}=\mathrm{n}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(A=x^2, B=x, C=n^2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(A=x, B=2 x, C=3 n+1\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(A=x^2, B=3 x, C=2 n\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(A=x^2, B=x, C=n^2\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hint: } \frac{-1}{\sqrt{1-\frac{y^2}{b^2}}} \times \frac{1}{b} y_1=\left(\frac{n}{n \times \frac{x}{n}}\right) \\<br/>&amp; \Rightarrow \frac{-1 y_1}{\sqrt{b^2-y^2}}=\frac{n}{x} \\<br/>&amp; \Rightarrow y_1 x+n \sqrt{b^2-y^2}=0 \\<br/>&amp; \Rightarrow y_1+x y_2+\frac{n}{2 \sqrt{b^2-y^2}} \cdot(-2 y) y_1=0<br/>\end{aligned}\)<br/>\(\begin{aligned} &amp; \Rightarrow y_1+x y_2+\frac{n^2}{x} y=0 \\ &amp; \Rightarrow y_1 x+x^2 y_2+n^2 y=0 \\ &amp; \Rightarrow x^2 y_2+x y_1+n^2 y=0 \\ &amp; A=x^2, B=x, C=n^2\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-d-1-and-d-2-be-the-lengths-of-the-perpendiculars-drawn-from-any-point-of-the-line-7-x-9-y-10-0-upon-the-lines-3-x-4-y-5-and-12-x-5-y-7-respectivel

slug:
let-d-1-and-d-2-be-the-lengths-of-the-perpendiculars-drawn-from-any-point-of-the-line-7-x-9-y-10-0-upon-the-lines-3-x-4-y-5-and-12-x-5-y-7-respectivel-27256

question:
<div class="question">Let $d_{1}$ and $d_{2}$ be the lengths of the perpendiculars drawn from any point of the line $7 x-9 y+10=0$ upon the lines $3 x+4 y=5$ and $12 x+5 y=7,$ respectively. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$d_{1}&gt;d_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$d_{1}=d_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$d_{1}    &lt; d_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$d_{1}=2 d_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$d_{1}=d_{2}$</span> </div>

answer:
<div class="solution">Let $(h, k)$ be any point on the line $7 x-9 y+10=0,$ then $7 h-9 k+10=0$
<br/>$\Rightarrow \quad 7 h=9 k-10$
<br/>$\Rightarrow \quad h=\frac{9 k-10}{7}$
<br/>Now, perpendicular distance from point
<br/>$(h, k)$ to the line $3 x+4 y=5$ is $d_{1}$ 
<br/>$d_{1}=\frac{3 h+4 k-5}{\sqrt{3^{2}+4^{2}}}$
<br/>$\Rightarrow \quad d_{1}=\frac{3 h+4 k-5}{5}$
<br/>and perpendicular distance from $(h, k)$ to the line $12 x+5 y=7$ is $\bar{d}_{2}$
<br/>$\therefore \quad d_{2}=\frac{12 h+5 k-7}{\sqrt{12^{2}+5^{2}}}$
<br/>$\Rightarrow \quad d_{2}=\frac{12 h+5 k-7}{13}$
<br/>Now, $d_{1}-d_{2}=\frac{3 h+4 k-5}{5}-\frac{12 h+5 k-7}{13}$
<br/>$\Rightarrow d_{1}-d_{2}$
<br/>$=\frac{13(3 h+4 k-5)-5(12 h+5 k-7)}{65}$
<br/>$=\frac{39 h+52 k-65-60 h-25 k+35}{65}$
<br/>$=\frac{-21 h+27 k-30}{65}$
<br/>$=\frac{-21\left(\frac{9 k-10}{7}\right)+27 k-30}{65} \quad[$ from Eq.(i) $]$
<br/>$=\frac{-27 k+30+27 k-30}{65}=0$
<br/>$\Rightarrow d_{1}-d_{2}=0 \Rightarrow d_{1}=d_{2}$</div>


title: let-d-n-denotes-the-number-of-divisors-of-n-including-1-and-itself-then-d-225-d-1125-and-d-640-are
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">in GP</span> </div>
source_db: db\MarksBatch2_P1.db
slug:
let-d-n-denotes-the-number-of-divisors-of-n-including-1-and-itself-then-d-225-d-1125-and-d-640-are-12322

question:
<div class="question">Let $d(n)$ denotes the number of divisors of $n$ including 1 and itself. Then, $d(225), d(1125)$ and $d(640)$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">in AP</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">in HP</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">in GP</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">consecutive integers</span> </li> </ul>

answer:
<div class="solution">Here, $225=3^{2} \times 5^{2}$
<br/>$\Rightarrow d(225)=3 \times 3=9$
<br/>$1126=3^{2} \times 5^{3}$
<br/>$\Rightarrow \quad d(1125)=3 \times 4=12$
<br/>and $640=2^{7} \times 5$
<br/>$\Rightarrow \quad d(640)=8 \times 2=16$
<br/>Hence, 9,12 and 16 are in GP.</div>


title: let-denote-the-cube-roots-of-unity-other-than-1-and-let-s-n-0-1-n-n-then-the-value-of-s-is
slug: let-denote-the-cube-roots-of-unity-other-than-1-and-let-s-n-0-1-n-n-then-the-value-of-s-is-68907
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $\alpha, \beta$ denote the cube roots of unity other than 1 and $\alpha \neq \beta .$ Let $S=\sum_{n=0}^{\infty}(-1)^{n}\left(\frac{\alpha}{\beta}\right)^{n} .$ Then the value of $S$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">either $-2 \omega$ or $-2 \omega^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">either $-2 \omega$ or $2 \omega^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">either $2 \omega$ or $-2 \omega^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">either $2 \omega$ or $2 \omega^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">either $-2 \omega$ or $-2 \omega^{2}$</span> </div>

answer:
<div class="solution">Case I Let $\alpha=\omega$ and $\beta=\omega^{2}$
<br/>$S=\sum_{n=0}^{302}(-1)^{n}\left(\frac{\omega}{\omega^{2}}\right)^{n}$
<br/>$=\sum_{n=0}^{302}(-1)^{n}\left(\omega^{2}\right)^{n}$
<br/>$=1-\omega^{2}+\omega^{4}-\omega^{6}+\omega^{8}-\omega^{10}+\omega^{12}$
<br/>$\quad+\ldots+\omega^{600}-\omega^{602}+\omega^{604}$
<br/>$=1-\omega^{2}+\omega-1+\omega^{2}-\omega+1+\ldots$
<br/>$=0+\ldots+1-\omega^{2}+\omega$
<br/>$=-\omega^{2}-\omega^{2}=-2 \omega^{2} \quad\left[\because 1+\omega+\omega^{2}=0\right]$
<br/>Case II Let $\alpha=\omega^{2}$ and $\beta=\omega$
<br/>$\therefore S=\sum_{n=0}^{\infty_{2}}(-1)^{n}\left(\frac{\omega^{2}}{\omega}\right)^{n}$
<br/>$=\sum_{n=0}^{302}(-1)^{n}\left(\frac{\omega^{4}}{\omega^{3}}\right)^{n}$
<br/>$=\sum_{n=0}^{302}(-1)^{n}(\omega)$
<br/>$=1-\omega+\omega^{2}-\omega^{3}+\omega^{4}-\omega^{5}+\omega^{6}-\ldots$
<br/>$+\omega^{300}-\omega^{301}+\omega^{302}$
<br/>$=1-\omega+\omega^{2}-1+\omega-\omega^{2}+1$
<br/>$=0+\ldots+1+\omega^{2}-\omega$
<br/>$=-\omega-\omega=-2 \omega$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
let-each-of-the-equations-x-2-2-x-y-a-y-2-0-a-x-2-2-x-y-y-2-0-represent-two-straight-lines-passing-through-the-origin-if-they-have-a-common-line-then-

slug:
let-each-of-the-equations-x-2-2-x-y-a-y-2-0-a-x-2-2-x-y-y-2-0-represent-two-straight-lines-passing-through-the-origin-if-they-have-a-common-line-then-37446

question:
<div class="question">Let each of the equations $x^{2}+2 x y+a y^{2}=0 \&amp; a x^{2}+2 x y+y^{2}=0$ represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x-y=0, x-3 y=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x+3 y=0,3 x+y=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 x+y=0,3 x-y=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(3 x-2 y)=0, x+y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+3 y=0,3 x+y=0$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{2}+2\left(\frac{\mathrm{x}}{\mathrm{y}}\right)+\mathrm{a}=0 \&amp; \mathrm{a}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{2}+2\left(\frac{\mathrm{x}}{\mathrm{y}}\right)+1=0$ have exactly one root in common (taking $\frac{\mathrm{x}}{\mathrm{y}}$ as a single
<br/>variable).
<br/>By, $\left(a_{1} b_{2}-a_{2} b_{1}\right)\left(b_{1} c_{2}-b_{2} c_{1}\right)=\left(a_{1} c_{2}-a_{2} c_{1}\right)^{2}$
<br/>We get $: \Rightarrow \mathrm{a}=1$ or $-3$
<br/>a cannot be 1
<br/>Taking $\mathrm{a}=-3$, roots of 1 st equation $: 1,-3$ and 2 nd equation : $1,-\frac{1}{3}$
<br/>So other lines: $\frac{x}{y}=-3$ and $\frac{x}{y}=-\frac{1}{3}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
let-exp-x-denote-exponential-function-ex-if-fxexp-leftxfrac1xright-x0-then-the-minimum-value-of-f-in-the-interval-25-is

slug:
let-exp-x-denote-exponential-function-ex-if-fxexp-leftxfrac1xright-x0-then-the-minimum-value-of-f-in-the-interval-25-is-68777

question:
<div class="question">Let \(\exp (x)\) denote exponential function ex. If \(f(x)=\exp \left(x^{\frac{1}{x}}\right), x&gt;0\) then the minimum value of \(f\) in the interval \([2,5]\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\exp \left(e^{\frac{1}{c}}\right)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\exp \left(2^{\frac{1}{2}}\right)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\exp \left(5^{\frac{1}{5}}\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\exp \left(3^{\frac{1}{3}}\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\exp \left(5^{\frac{1}{5}}\right)\)</span> </div>

answer:
<div class="solution">Given that,
<br/>$$
<br/>f(x)=e^{(x)^{\frac{1}{x}}, x&gt;0}
<br/>$$
<br/>Taking log on both sides, we get
<br/>$$
<br/>\log +(x)=(x)^{\frac{1}{x}}=g(x)
<br/>$$
<br/>Here, $g(x)=x^{\frac{1}{x}}$
<br/>$\Rightarrow \quad \log g(x)=\frac{1}{x} \log x$
<br/>On differentiating w.r.t. $x$, we get
<br/>$$
<br/>\begin{aligned}
<br/>\frac{1}{g(x)} \cdot g^{\prime}(x) &amp;=\frac{x \cdot \frac{1}{x}-\log x}{x^{2}} \\
<br/>&amp;=\left(\frac{1-\log x}{2}\right)
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad g^{\prime}(x)=x^{\left(\frac{1}{x}-2\right)}(1-\log x)$
<br/>For maximum or minimum of $g(x)$ put
<br/>$$
<br/>g^{\prime}(x)=0
<br/>$$
<br/>$\Rightarrow \quad x^{\left(\frac{1}{x}-2\right)}(1-\log x)=0$
<br/>$$
<br/>\begin{array}{lr}
<br/>\Rightarrow &amp; \log x=1=\log e \\
<br/>\Rightarrow &amp; x=e
<br/>\end{array}
<br/>$$
<br/>and $\left.\quad g^{\prime \prime}(x)\right|_{x=e}&gt;0$
<br/>So, $g(x)$ is minimum at $x=e$ $\therefore g(x)$ increases in $(0, e)$ and decreases in $(\theta, \infty)$ it will be minimum at either 2 or 5 $\therefore 2^{\frac{1}{2}}&gt;5^{\frac{1}{5}} \Rightarrow$ Minimum value of $f(x)=e^{(5)^{\frac{1}{5}}}$</div>


title: let-f-1-3-r-be-a-continuous-function-that-is-differentiable-in-13-an-f-x-f-x-2-4-for-all-x-1-3-then
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
slug:
let-f-1-3-r-be-a-continuous-function-that-is-differentiable-in-13-an-f-x-f-x-2-4-for-all-x-1-3-then-83751

question:
<div class="question">Let $f:[1,3] \rightarrow R$ be a continuous function that is differentiable in (1,3) an $f^{\prime}(x)=|f(x)|^{2}+4$ for all $x \in(1,3).$ Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(3)-f(1)=5$ is true</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(3)-f(1)=5$ is false</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(3)-f(1)=7$ is true</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(3)-f(1)    &lt; 0$ only at one point of (1,3)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(3)-f(1)=5$ is false</span> </div>

answer:
<div class="solution">Given that $f:[1,3] \rightarrow R$ be a continuous and differentiable in (1,3) 
<br/>and $f^{\prime}(x)=|f(x)|^{2}+4$
<br/>By applying $L M V T,$ there exist at least one point $c \in(1,3)$ such that
<br/>$$
<br/>\begin{array}{c}
<br/>\frac{f(3)-f(1)}{3-1}=f^{\prime}(c)\left[\because f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\right] \\
<br/>\Rightarrow \quad \frac{f(3)-f(1)}{2}=f^{\prime}(c) \\
<br/>\Rightarrow \quad \frac{f(3)-f(1)}{2}=\left[f|c|^{2}+4\right] \\
<br/>\Rightarrow f^{\prime}(c)=|f(c)|^{2}+4 \\
<br/>\Rightarrow \quad f(3)-f(1)=2 \cdot|f(c)|^{2}+8
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad f(3)-f(1) \geq 8$</div>


title: let-f-1-sin-2-2-sin-2-then-for-all-values-of
slug: let-f-1-sin-2-2-sin-2-then-for-all-values-of-73526
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(\theta)=\left(1+\sin ^{2} \theta\right)\left(2-\sin ^{2} \theta\right) .$ Then, for all
<br/>values of $\theta$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(\theta)&gt;\frac{9}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(\theta)     &lt; 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(\theta)&gt;\frac{11}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 \leq f(\theta) \leq \frac{9}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \leq f(\theta) \leq \frac{9}{4}$</span> </div>

answer:
<div class="solution">Given
<br/>$$
<br/>\begin{aligned}
<br/>f(\theta)=\left(1+\sin ^{2} \theta\right)\left(2-\sin ^{2} \theta\right) \\
<br/>&amp;=2+2 \sin ^{2} \theta-\sin ^{2} \theta-\sin ^{4} \theta \\
<br/>&amp;=-\sin ^{4} \theta+\sin ^{2} \theta+2 \\
<br/>&amp;=-\left(\sin ^{4} \theta-\sin ^{2} \theta-2\right) \\
<br/>&amp;=-\left\{\sin ^{4} \theta-\sin ^{2} \theta+\frac{1}{4}-\frac{9}{4}\right\} \\
<br/>&amp;=+\frac{9}{4}-\left(\sin ^{2} \theta-\frac{1}{2}\right)^{2} \\
<br/>\because \quad-1 \leq \sin \theta \leq 1 \Rightarrow 0 \leq \sin ^{2} \theta \leq 1\\
<br/>\Rightarrow &amp; 0 \geq-\left(\sin ^{2} \theta-\frac{1}{2}\right)^{2} \geq-\frac{1}{4} \\
<br/>\Rightarrow \quad \quad-\frac{1}{2} \leq \sin ^{2} \theta-\frac{1}{2} \leq \frac{1}{2}\\
<br/>\Rightarrow \quad 0 \leq\left(\sin ^{2} \theta-\frac{1}{2}\right)^{2} \leq \frac{1}{4}
<br/>\Rightarrow &amp; \frac{9}{4} \geq \frac{9}{4}-\left(\sin ^{2} \theta-\frac{1}{2}\right)^{2} \geq \frac{9}{4}-\frac{1}{4} \\
<br/>\Rightarrow \quad 2 \leq f(\theta) \leq \frac{9}{4} &amp; \quad[\text { from Eq. }(0)]
<br/>\end{aligned}
<br/>$$</div>


title: let-f-1-x-e-x-f-2-x-e-f-1-x-f-n-1-x-c-f-n-x-for-all-n-1-then-for-any-fixed-n-d-x-d-f-n-x-is
slug: let-f-1-x-e-x-f-2-x-e-f-1-x-f-n-1-x-c-f-n-x-for-all-n-1-then-for-any-fixed-n-d-x-d-f-n-x-is-15731
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f_{1}(x)=e^{x}, f_{2}(x)=e^{f_{1}(x)} \ldots \ldots$
<br/>$f_{n+1}(x)=c^{f n(x)}$ for all $n \geq 1 .$ Then for any fixed
<br/>$n \cdot \frac{d}{d x} f_{n}(x)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f_{n}(x)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f_{n}(x)f_{n-1}(x)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f_{n}(x)f_{n-1}(x)...f_{1}(x)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f_{n}(x)...f_{1}(x)e^{x}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f_{n}(x)f_{n-1}(x)...f_{1}(x)$</span> </div>

answer:
<div class="solution">We have, $f_{1}(x)=e^{x}$
<br/>$$
<br/>\begin{array}{r}
<br/>f_{2}(x)=e^{f_{1}(x)} \\
<br/>\cdots \cdot \cdots \cdots \\
<br/>\cdots f_{n+1}(x)=e^{f_{n}(x)}
<br/>\end{array}
<br/>$$
<br/>$f_{n}(x)=e^{f_{n-1}(x)}$
<br/>On taking log both sides, we get
<br/>$\quad \log \left\{f_{n}(x)\right\}=f_{n-1}(x) \log e$
<br/>$\Rightarrow \quad \frac{d}{d x} \log \left(f_{n}(x)\right)=\frac{d}{d x} f_{n-1}(x) \quad(\because \log e=1)$
<br/>$\Rightarrow \quad \frac{1}{f_{n}(x)} \frac{d}{d x} f_{n}(x)=f_{n-1}^{\prime}(x)$
<br/>$\Rightarrow \quad \frac{d}{d x} f_{n}(x)=f_{n}(x) f_{n-1}^{\prime}(x) \quad \ldots$ (i)
<br/>Now, $\quad f_{1}^{\prime}(x)=e^{x}$
<br/>$\begin{array}{l}\text { and } \quad f_{2}(x)=e^{f_{1}(x)} \\ \Rightarrow \log f_{2}(x)=f_{1}(x) \log e=f_{1}(x) \\ \Rightarrow \quad \frac{1}{f_{2}(x)} \cdot f_{2}^{\prime}(x)=f_{1}^{\prime}(x) \\ \Rightarrow \quad \begin{aligned} f_{2}^{\prime}(x) &amp;=f_{2}(x) \cdot f_{1}^{\prime}(x) \\=f_{2}(x) \cdot e^{x} &amp;\left(\because f_{1}^{\prime}(x)=e^{x}\right) \end{aligned} \\ =f_{2}(x) \cdot f_{1}(x) &amp; {\left[\because e^{x}=f_{1}(x)\right] \ldots(\mathrm{ii})}\end{array}$
<br/>From Eq. (i). $\frac{d}{d x} f_{n}(x)=f_{n}(x) \cdot f_{n-1}(x) \ldots f_{1}(x)[\mathrm{using} \mathrm{Eq}$</div>


title: let-f-2-2-r-be-a-continuous-function-such-that-f-x-assumes-only-irrational-values-if-f-2-2-then
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
slug:
let-f-2-2-r-be-a-continuous-function-such-that-f-x-assumes-only-irrational-values-if-f-2-2-then-60534

question:
<div class="question">Let $f:[-2,2] \rightarrow R$ be a continuous function such that $f(x)$ assumes only irrational values. If $f(\sqrt{2})=\sqrt{2},$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(0)=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(\sqrt{2}-1)=\sqrt{2}-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(\sqrt{2}-1)=\sqrt{2}+1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$f(\sqrt{2}-1)=\sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(\sqrt{2}-1)=\sqrt{2}$</span> </div>

answer:
<div class="solution">If a function $f(x)$ assumes only irrational values which is also continuous, then $f(x)$ must be constant function.
<br/>$\Rightarrow \quad f(x)=\sqrt{2}$
<br/>$$
<br/>[\because f(\sqrt{2})=\sqrt{2} \mid
<br/>$$
<br/>$\therefore \quad f(\sqrt{2}-1)=\sqrt{2}$</div>


title: let-f-a-b-r-be-continuous-in-a-b-differentiable-in-a-b-and-f-a-0-f-b-then
slug: let-f-a-b-r-be-continuous-in-a-b-differentiable-in-a-b-and-f-a-0-f-b-then-41234
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous in $[a, b]$, differentiable in $(a, b)$ and $f(a)=0=f(b)$. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">there exists at least one point $c \in(a, b)$ for which $f^{\prime}(c)=f(c)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f^{\prime}(x)=f(x)$ does not hold at any point of $(a, b)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">at every point of $(a, b), f^{\prime}(x)&gt;f(x)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">at every point of $(a, b), f^{\prime}(x) &lt; f(x)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there exists at least one point $c \in(a, b)$ for which $f^{\prime}(c)=f(c)$</span> </div>

answer:
<div class="solution">$:$ Let $g(x)=e^{-x} f(x)$<br/>$$<br/>\mathrm{g}(\mathrm{a})=\mathrm{g}(\mathrm{b})=0<br/>$$<br/>By Rolle's theorem, for atleast one $c \in(a, b)$ where $\mathrm{g}^{\prime}(\mathrm{c})=0$<br/>or $e^{-c} f^{\prime}(c)-f(c) e^{-c}=0$<br/>or $e^{-c}\left(f^{\prime}(c)-f(c)\right)=0$<br/>or $f^{\prime}(c)=f(c) \quad$ for atleast one $c \in(a, b)$</div>


title: let-f-a-b-r-be-differentiable-on-a-b-and-k-r-let-f-a-0-f-b-also-let-j-x-f-x-k-f-x-then
slug: let-f-a-b-r-be-differentiable-on-a-b-and-k-r-let-f-a-0-f-b-also-let-j-x-f-x-k-f-x-then-31976
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f:[a, b] \rightarrow R$ be differentiable on $[a, b]$ and $k \in R$. Let $f(a)=0=f(b)$
<br/>Also let $J(x)=f'(x)+k f(x) .$ Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$J(x)&gt;0$ for all $x \in[a, b]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$J(x)    &lt; 0$ for all $x \in[a, b]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$J(x)=0$ has at least one root in $(a, b)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$J(x)=0$ through $(a, b)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$J(x)=0$ has at least one root in $(a, b)$</span> </div>

answer:
<div class="solution">We have, $f:[a, b] \longrightarrow R$ be differentiable
<br/>on $[a, b]$ and $k \in R,$ also $f(a)=0=f(b)$ and $\quad J(x)=f(x)+k f(x)$
<br/>Let $g(x)=k x f(x)$ which is continuous in $[a, b]$ and differentiable in $(a, b)$ such that
<br/>$$
<br/>g(a)=0=g(b)
<br/>$$
<br/>Then, for every $c \in(a, b), g^{\prime}(c)=0$
<br/>(by Rolle's theorem) $\begin{array}{lr}\text { Now, } &amp; g^{\prime}(x)=k f(x)+k x f^{\prime}(x) \\ \Rightarrow \quad &amp; g(c)=k f(c)+k c f^{\prime}(c)\end{array}$
<br/>$\Rightarrow \quad k f(c)+k c f(c)=0$
<br/>$\Rightarrow \quad f(x)=0,$ for every $x=c \in(a, b)$
<br/>$\therefore J(x)=0$ has atleast one root in $(a, b)$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
let-f-a-b-r-be-such-that-f-is-differentiable-in-a-b-is-continuous-at-x-a-and-x-b-and-moreover-f-a-0-f-b-then

slug:
let-f-a-b-r-be-such-that-f-is-differentiable-in-a-b-is-continuous-at-x-a-and-x-b-and-moreover-f-a-0-f-b-then-91945

question:
<div class="question">Let $f:[a, b] \rightarrow R$ be such that $f$ is differentiable in $(a, b)$ is continuous at $x=a$ and $x=b$ and moreover $f(a)=0=f(b)$. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">there exists at least one point $c$ in $(a,b)$ such that $f^{\prime}=f$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f^{\prime}(x)=f(x)$ does not hold at any point in $(a, b)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">at every point of $(a, b) f^{\prime}(x)&gt;f(x)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">at every point of $(a, b), f^{\prime}(x)    &lt; f(x)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there exists at least one point $c$ in $(a,b)$ such that $f^{\prime}=f$</span> </div>

answer:
<div class="solution">Let $g(x)=e^{-x} f(x)$
<br/>such that $g(a)=0, g(b)=0$
<br/>and $g(x)$ is continuous and differentiable.
<br/>Then, for atleast one value of $c \in(a, b)$ such that $g(c)=0$
<br/>Now, $g(x)=e^{-x} f^{\prime}(x)+\left(-e^{-x}\right) f(x)$
<br/>$\Rightarrow \quad g(c)=e^{-c} f^{\prime}(c)+\left(-e^{-\ell}\right) f(c)=0$
<br/>$\Rightarrow e^{-c} f^{\prime}(c)=e^{-c} f(c) \Rightarrow f^{\prime}(c)=f(c)$</div>


title: let-f-and-g-be-differentiable-on-the-interval-i-and-let-a-b-i-a-b-then
slug: let-f-and-g-be-differentiable-on-the-interval-i-and-let-a-b-i-a-b-then-62244
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f$ and $g$ be differentiable on the interval $I$ and let $a, b \in I, a    &lt; b$. Then,</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">If $f(a)=0=f(b)$, the equation $f^{\prime}(x)+f(x) g^{\prime}(x)=0$ is solvable in $(a, b)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">If $f(a)=0=f(b)$, the equation $f^{\prime}(x)+f(x) g^{\prime}(x)=0$ may not be solvable in $(a, b)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">If $g(a)=0=g(b)$, the equation $g^{\prime}(x)+k g(x)=0$ is solvable in $(a, b), k \in \mathbb{R}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">If $g(a)=0=g(b)$, the equation $g^{\prime}(x)+k g(x)=0$ may not be solvable in $(a, b), k \in \mathbb{R}$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">If $f(a)=0=f(b)$, the equation $f^{\prime}(x)+f(x) g^{\prime}(x)=0$ is solvable in $(a, b)$, If $g(a)=0=g(b)$, the equation $g^{\prime}(x)+k g(x)=0$ is solvable in $(a, b), k \in \mathbb{R}$</span> </div>

answer:
<div class="solution">From option (a)
<br/>We have,
<br/>$f(a)=f(b)=0$
<br/>$\Rightarrow f^{\prime}(a) \cdot f^{\prime}(b)    &lt; 0$
<br/>Again, let $h(x)=f^{\prime}(x)+f(x) g^{\prime}(x)$
<br/>$\Rightarrow h(a)=f^{\prime}(a)+fg^{\prime}(a)=f^{\prime}(a)$
<br/>and $h(b)=f^{\prime}(b)+f(b) g^{\prime}(b)=f^{\prime}(b)$
<br/>$\therefore \quad h(a) \cdot h(b)=f^{\prime}(a) \cdot f^{\prime}(b) &lt; 0$
<br/>$\therefore h(x)=0$ has root between $(a, b)$
<br/>From option (c) 
<br/>We have, $g(a)=g(b)=0$
<br/>$\Rightarrow \quad g^{\prime}(a)\cdot g^{\prime}(b) &lt; 0$
<br/>Again, let $m(x)=g^{\prime}(x)+k g(x)$
<br/>$\Rightarrow m(a)=g^{\prime}(a)+k g(a)=g^{\prime}(a)$
<br/>$\Rightarrow \quad m(b)=g^{\prime}(b)+k g(b)=g^{\prime}(b)$
<br/>$\therefore \quad m(a) \cdot m(b)=g^{\prime}\cdot g^{\prime}(b) &lt; 0$
<br/>$\therefore \quad m(x)=0$ has root between $(a, b)$ So, option (a) and (c) are correct.</div>


title: let-f-and-g-be-periodic-functions-with-the-periods-t-1-and-t-2-respectively-then-f-g-is
slug: let-f-and-g-be-periodic-functions-with-the-periods-t-1-and-t-2-respectively-then-f-g-is-26167
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2021']
answer: <div class="solution">$\mathrm{f}+\mathrm{g}$ is a periodic function.</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f$ and $g$ be periodic functions with the periods $T_{1}$ and $T_{2}$ respectively. Then $f+g$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">periodic with period $\mathrm{T}_{1}+\mathrm{T}_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">non-periodic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">periodic with the period $\mathrm{T}_{1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">periodic when $T_{1}=T_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">periodic when $T_{1}=T_{2}$</span> </div>


title: let-f-be-a-continuous-function-in-0-1-then-lim-n-j-0-n-n-1-f-n-j-is-1
slug: let-f-be-a-continuous-function-in-0-1-then-lim-n-j-0-n-n-1-f-n-j-is-1-29954
tags: ['Mathematics', 'Definite Integration', 'JEE Main']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f$, be a continuous function in $[0,1]$, then $\lim_{n \rightarrow \infty} \sum_{j=0}^n \frac{1}{n} f\left(\frac{j}{n}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\int_{\frac{1}{2}}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\int_{0}^{1} f(x) d x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\int_{0}^{1} f(x) d x$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/dvs365ftregyuge64ety.png"/> <br/>\(\lim _{n \rightarrow \infty} \sum_{i=0}^{n} \frac{1}{n} f\left(\frac{i}{n}\right)\)
<br/>Let \(1 / n \rightarrow d x\)</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-a-differentiable-function-with-lim-x-f-x-0-if-y-y-f-x-f-x-f-x-0-lim-x-y-x-0-then-where-y-d-x-d-y

slug:
let-f-be-a-differentiable-function-with-lim-x-f-x-0-if-y-y-f-x-f-x-f-x-0-lim-x-y-x-0-then-where-y-d-x-d-y-62085

question:
<div class="question">Let $f$ be a differentiable function with $\lim _{x \rightarrow \infty} f(x)=0 .$ If $y^{\prime}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0, \lim _{x \rightarrow \infty} y(x)=0$, then $\left(\right.$ where $\left.y^{\prime}=\frac{d y}{d x}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y+1=e^{f(x)}+f(x)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y-1=e^{f(x)}+f(x)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y+1=e^{-f(x)}+f(x)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y-1=e^{-f f] x}+f(x)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y+1=e^{-f(x)}+f(x)$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\frac{d y}{d x}+f^{\prime}(x) y=f^{\prime}(x) f(x)$
<br/>$\Rightarrow y \times e^{f(x)}=\int f^{\prime}(x) f(x) e^{f(x)} d x$
<br/>$\Rightarrow y \times e^{f(x)}=e^{f(x)}(f(x)-1)+c \quad[$ Putting $f(x)=0 ; y=0, c=1]$
<br/>$\Rightarrow y \times e^{f(x)}=e^{f(x)}(f(x)-1)+1$
<br/>$\Rightarrow y=f(x)-1+e^{-f(x)}$
<br/>$\Rightarrow y+1=e^{-f(x)}+f(x)$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-a-nonconstant-continuous-function-for-all-x-0-let-f-satisfy-the-relation-f-x-f-a-x-1-for-some-a-r-then-i-0-a-1-f-x-d-x-is-equal-to

slug:
let-f-be-a-nonconstant-continuous-function-for-all-x-0-let-f-satisfy-the-relation-f-x-f-a-x-1-for-some-a-r-then-i-0-a-1-f-x-d-x-is-equal-to-62925

question:
<div class="question">Let $f$ be a non-constant continuous function for all $x \geq 0$. Let $f$ satisfy the relation $f(x) f(a-x)=1$ for some $a \in R^{+}$. Then, $I=\int_{0}^{a} \frac{d x}{1+f(x)}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{a}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{a}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(a)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\frac{a}{2}$</span> </div>

answer:
<div class="solution">$I=\int_{0}^{a} \frac{d x}{1+f(x)}$
<br/>$=\int_{0}^{a} \frac{1}{1+f(a-x)} d x$
<br/>$\quad\left[\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]$
<br/>$=\int_{0}^{a} \frac{d x}{1+\frac{1}{f(x)}} \quad[\because f(x) f(a-x)=1]$
<br/>$I=\int_{0}^{a} \frac{f(x)}{f(x)+1} d x$
<br/>On adding Eqs. (i) and (ii), we get $2 I=\int_{0}^{a} \frac{f(x)+1}{f(x)+1} d x$
<br/>$\Rightarrow \quad 2 I=\int_{0}^{a} 1 d x$
<br/>$\begin{array}{ll}\Rightarrow &amp; 2 t=[x]_{0}^{a} \\ \Rightarrow &amp; 2 I=a \\ \Rightarrow &amp; I=\frac{a}{2}\end{array}$
<br/>72. $(b,$ d) We have, $x=1-2 x$
<br/>$\Rightarrow y=\frac{1-2 x}{x}$
<br/>$\Rightarrow \frac{d y}{d x}=\frac{-2 x-(1-2 x) \cdot 1}{x^{2}}$
<br/>$\quad=\frac{-2 x-1+2 x}{x^{2}}=\frac{-1}{x^{2}}    &lt; 0$
<br/>since, $a x+b y+c=0$ is tangent to the curve $x=1-2 x$
<br/>$\therefore \quad \frac{-a}{b} &lt; 0 \Rightarrow \frac{a}{b}&gt;0$
<br/>$\Rightarrow$ Either $a&gt;a b&gt;0$ or $a &lt; 0, b &lt; 0$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-a-nonnegative-function-defined-in-0-2-f-exists-and-be-continuous-for-all-x-and-0-x-1-f-t-2-d-t-0-x-f-t-d-t-and-f-0-0-then

slug:
let-f-be-a-nonnegative-function-defined-in-0-2-f-exists-and-be-continuous-for-all-x-and-0-x-1-f-t-2-d-t-0-x-f-t-d-t-and-f-0-0-then-73122

question:
<div class="question">Let $f$ be a non-negative function defined in $[0, \pi / 2], f^{\prime}$ exists and be continuous for all $x$ and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} d t=\int_0^x f(t) d t$ and $f(0)=0$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f\left(\frac{1}{2}\right)  &lt; \frac{1}{2}$ and $f\left(\frac{1}{3}\right) &gt; \frac{1}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f\left(\frac{1}{2}\right) &gt; \frac{1}{2}$ and $f\left(\frac{1}{3}\right)  &lt; \frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f\left(\frac{4}{3}\right)  &lt; \frac{4}{3}$ and $f\left(\frac{2}{3}\right) &lt; \frac{2}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}\left(\frac{4}{3}\right) &gt; \frac{4}{3}$ and $\mathrm{f}\left(\frac{2}{3}\right) &gt; \frac{2}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f\left(\frac{4}{3}\right)  &lt; \frac{4}{3}$ and $f\left(\frac{2}{3}\right) &lt; \frac{2}{3}$</span> </div>

answer:
<div class="solution">$\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} d t=\int_0^x f(t) d t$<br/>Using Leibnitz Rule,<br/>$\begin{aligned}<br/>&amp;\sqrt{1-\left(f^{\prime}(x)\right)^2}=f(x) \\<br/>&amp;\therefore f^{\prime}(x)=\pm \sqrt{1-f^2(x)} \Rightarrow \int \frac{d f(x)}{\sqrt{1-f^2(x)}}=\pm \int d x \\<br/>&amp;\sin ^{-1}(f(x))=\pm x+c \\<br/>&amp;\therefore f(x)=\sin (\pm x+C) \\<br/>&amp;\because f(0)=0 \Rightarrow C=0 \\<br/>&amp;\therefore f(x)=\sin x \text { or }-\sin x<br/>\end{aligned}$<br/>But $f$ is non-negative on $[0, \pi / 2]$<br/>$\begin{aligned}<br/>&amp;\therefore \mathrm{f}(\mathrm{x})=\operatorname{Sin} \mathrm{x} \\<br/>&amp;\because \mathrm{f}(\mathrm{x})  &lt; \mathrm{x} \text { for all } \mathrm{x} &gt; 0 \\<br/>&amp;\therefore \mathrm{f}\left(\frac{4}{3}\right) &lt; \frac{4}{3} \text { and } \mathrm{f}\left(\frac{2}{3}\right) &lt; \frac{2}{3}<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-a-nonnegative-function-defined-on-left0-fracpi2right-if-int0xleftfprimetsin-2-tright-d-tintx0-ft-tan-t-d-t-f01-then-int0fracpi2-fx-d-x-is

slug:
let-f-be-a-nonnegative-function-defined-on-left0-fracpi2right-if-int0xleftfprimetsin-2-tright-d-tintx0-ft-tan-t-d-t-f01-then-int0fracpi2-fx-d-x-is-79167

question:
<div class="question">Let \(f\) be a non-negative function defined on \(\left[0, \frac{\pi}{2}\right]\). If \(\int_0^x\left(f^{\prime}(t)-\sin 2 t\right) d t=\int_x^0 f(t) \tan t d t, f(0)=1\), then \(\int_0^{\frac{\pi}{2}} f(x) d x\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(3-\frac{\pi}{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(3+\frac{\pi}{2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{\pi}{2}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\(3-\frac{\pi}{2}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>\text {Hint } &amp; f^{\prime}(x)-\sin 2 x=-f(x) \tan x \\<br/>&amp; \Rightarrow f^{\prime}(x)+\tan x f(x)=\sin 2 x \\<br/>&amp; \Rightarrow \frac{d f(x)}{d x}+\tan x . f(x)=\sin 2 x<br/>\end{aligned}\)<br/>\(\text { If }=e^{f \tan x d x}=e^{\ln |\sec x|}=|\sec x|=\sec x \quad \text { As, } x \in\left[0, \frac{\pi}{2}\right]\)<br/>Solution is,<br/>\(\begin{aligned}<br/>&amp; f(x) \times \sec x=\int \sin 2 x \sec x d x=\int 2 \sin x \cos x \sec x d x=2 \int \sin x d x \\<br/>&amp; f(x) \sec x=-2 \cos x+C \\<br/>&amp; \text { Put } x=0 ; f(0)=1 \\<br/>&amp; f(0) \sec 0=-2 \cos 0+c \\<br/>&amp; \Rightarrow 1=-2+C \quad \therefore C=3<br/>\end{aligned}\)<br/>\(\therefore\) Required solution is;<br/>\(\begin{aligned}<br/>&amp; f(x) \sec x=-2 \cos x+3 \\<br/>&amp; f(x)=-2 \cos ^2 x+3 \cos x<br/>\end{aligned}\)<br/>\(\begin{aligned}<br/>&amp; =-2\left(\frac{1+\cos 2 x}{2}\right)+3 \cos x=-1-\cos 2 x+3 \cos x \\<br/>&amp; f(x)=-1-\cos 2 x+3 \cos x<br/>\end{aligned}\)<br/>Now \(\int_0^{\frac{\pi}{2}} f(x) d x\)<br/>\(\int_0^{\frac{\pi}{2}}(-1-\cos 2 x+3 \cos x) d x=\left[-x-\frac{\sin 2 x}{2}+3 \sin x\right]_0 ^{-\frac{\pi}{2}}=\left(-\frac{\pi}{2}+3\right)-0=3-\frac{\pi}{2}\)<br/>\(\left[-x-\frac{\sin 2 x}{2}+\left.3 \sin x\right|_0 ^{\frac{\pi}{2}}\right.\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-a-strictly-decreasing-function-defined-on-mathbbr-such-that-fx-0-forall-x-in-mathbbr-let-fracx2flefta25-a3rightfracy2fa151-be-an-ellipse-with

slug:
let-f-be-a-strictly-decreasing-function-defined-on-mathbbr-such-that-fx-0-forall-x-in-mathbbr-let-fracx2flefta25-a3rightfracy2fa151-be-an-ellipse-with-42678

question:
<div class="question">Let \(f\) be a strictly decreasing function defined on \(\mathbb{R}\) such that \(f(x) &gt; 0, \forall x \in \mathbb{R}\). Let \(\frac{x^2}{f\left(a^2+5 a+3\right)}+\frac{y^2}{f(a+15)}=1\) be an ellipse with major axis along the \(y\)-axis. The value of ' \(a\) ' can lie in the interval (s)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\((-\infty,-6)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((-6,2)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((2, \infty)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((-\infty, \infty)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\((-\infty,-6)\), \((2, \infty)\)</span> </div>

answer:
<div class="solution">Hint : \(f \Rightarrow\) strictly decreasing function \(\forall x \in R\)<br/>\(\begin{aligned}<br/>&amp; f(x) &gt; 0 \Rightarrow f^{\prime}(x)  &lt; 0 \\<br/>&amp; \frac{x^2}{f\left(a^2+5 a+3\right)}+\frac{y^2}{f(a+15)}=1<br/>\end{aligned}\)<br/>As major axis is \(y\)-axis<br/>\(\begin{aligned}<br/>&amp; f(a+15) &gt; f\left(a^2+5 a+3\right) \\<br/>&amp; \Rightarrow a+15 &lt; a^2+5 a+3 \quad (\because f \text{ is decreasing) }\\<br/>&amp; \Rightarrow a^2+4 a-12 &gt; 0 \\<br/>&amp; \Rightarrow a &lt; -6 \text { or } a &gt; 2<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
let-f-be-any-continuously-differentiable-function-on-a-b-and-twice-differentiable-on-a-b-such-that-f-a-f-a-0-and-f-b-0-then

slug:
let-f-be-any-continuously-differentiable-function-on-a-b-and-twice-differentiable-on-a-b-such-that-f-a-f-a-0-and-f-b-0-then-38227

question:
<div class="question">Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$ Then.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f^{\prime \prime}(a)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f^{\prime}(x)=0$ for some $x \in(a, b)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f^{\prime \prime}(x) \neq 0$ for some $x \in(a, b)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f^{\prime \prime \prime}(x)=0$ for some $x \in(a, b)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f^{\prime}(x)=0$ for some $x \in(a, b)$, $f^{\prime \prime}(x) \neq 0$ for some $x \in(a, b)$</span> </div>

answer:
<div class="solution">We have, $f$ is continuous and differentiably function on $[a, b]$ Also, $f(a)=f(b)=0$
<br/>By Rolle's theorem, there exists $c \in(a, b)$ such the $f^{\prime}(c)=0$
<br/>Thus, there exists $x \in(a, b)$ such that $f^{\prime}(x)=0$ Let at $x=c \in(a, b), f^{\prime}(c)=0$
<br/>Now, $f$ is continuously differentiable on $|a, b|$ $\Rightarrow \quad f^{\prime}$ is continuous on $\left.\mid a, b\right]$ Also, $f$ is twice differentiable on $(a, b)$ $\therefore f^{\prime}$ is differentiable on $(a, b)$
<br/>and $f^{\prime}(a)=0=f^{\prime}(c)$
<br/>By Rolle's theorem, there exists $k \in(a, c)$ such that. $f^{\prime \prime}(k)=0$
<br/>Thus, there exists $x \in(a, c)$ such that $f^{\prime \prime}(x)=0$ So, there exists $x \in(a, b)$ such that $f^{\prime \prime}(x)=0$ Let us consider, $f(x)=(x-a)^{2}(x-b)$
<br/>where $f(a)=f(b)=f^{\prime}(a)=0$ but
<br/>$f^{\prime \prime}(a) \neq 0$ and $f^{m}(x) \neq 0$ for any $x \in(a, b)$</div>


title: let-f-be-derivable-in-0-1-then
slug: let-f-be-derivable-in-0-1-then-78803
question: <div class="question">Let $\mathrm{f}$ be derivable in $[0,1]$, then</div>
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">there exists $c \in(0,1)$ such that $\int_0^c f(x) d x=(1-c) f(c)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">there does not exist any point $d \in(0,1)$ for which $\int_0^d f(x) d x=(1-d) f(d)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\int_0^c f(x) d x$ does not exist, for any $c \in(0,1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\int_0^c f(x) d x$ is independent of $c, c \in(0,1)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">there exists $c \in(0,1)$ such that $\int_0^c f(x) d x=(1-c) f(c)$</span> </div>

answer:
<div class="solution">Let $g(x)=x \int_0^x f(t) d t-\int_0^x f(t) d t$<br/>Now, $g(0)=0$<br/>$g(1)=0$<br/>By Rolle's Theorem<br/>$g^{\prime}(x)=0$<br/>for some $x \in(0,1)$<br/>$g^{\prime}(x)=x f(x)+\int_0^x f(t) d t-f(x)=0$</div>


title: let-f-d-r-where-d-0-1-2-4-be-defined-by-f-x-x-4-x-if-x-0-1-if-x-2-4-then
slug: let-f-d-r-where-d-0-1-2-4-be-defined-by-f-x-x-4-x-if-x-0-1-if-x-2-4-then-90480
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: D \rightarrow R$ where $D=[0,1] \cup[2,4]$ be defined by $f(x)=\left\{\begin{array}{ll}x, &amp; \text { if } x \in[0,1] \\ 4-x, &amp; \text { if } x \in[2,4]\end{array}\right.$. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Rolle's theorem is applicable to $\mathrm{f}$ in $\mathrm{D}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Rolle's theorem is not applicable to $\mathrm{f}$ in $\mathrm{D}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">there exists $\xi \in \mathrm{D}$ for which $\mathrm{f}^{\prime}(\xi)=0$ but Rolle's theorem is not applicable</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f$ is not continuous in $D$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Rolle's theorem is not applicable to $\mathrm{f}$ in $\mathrm{D}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Bg6N08rzYnMkSwBm8bI7khHQzk84i-9DhMEfBxDTSks.original.fullsize.png"/><br/> <br/>$f(x)=\left\{\begin{array}{ll}x, &amp; x \in[0,1] \\ 4-x, &amp; x \in[2,4]\end{array}\right.$
<br/>$f(x)$ is increasing in $[0,1]$
<br/>and $f(x)$ is decreasing in $[2,4]$
<br/>$\therefore$ Rolle's theorem is not applicable to $\mathrm{f}$ in $\mathrm{D}$.</div>


title: let-f-n-2-n-1-g-n-1-n-1-2-n-for-all-n-n-then
slug: let-f-n-2-n-1-g-n-1-n-1-2-n-for-all-n-n-then-78693
question: <div class="question">Let $f(n)=2^{n+1}, g(n)=1+(n+1)^{2 n}$ for all $n \in \mathbb{N}$. Then</div>
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(n) &gt; g(n)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f(n)  &lt; g(n)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(n)$ and $g(n)$ are not comparable</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(n) &gt; g(n)$ if $n$ be even and $f(n)  &lt; g(n)$ if $n$ be odd</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(n)  &lt; g(n)$</span> </div>

answer:
<div class="solution">$g(n)-f(n)=1+(n+1) 2^n-2 \cdot 2^n=1+n \cdot 2^n-2^n=1+(n-1) 2^n &gt; 0$<br/>$\because \mathrm{n} \geq 1 \Rightarrow(\mathrm{n}-1) \geq 0$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
let-f-n-r-be-such-that-f-1-1-and-f-1-2-f-2-3-f-3-n-f-n-n-n-1-f-n-for-all-n-n-n-2-where-n-is-the-set-of-natural-numbers-and-r-is-the-set-of-real-number

slug:
let-f-n-r-be-such-that-f-1-1-and-f-1-2-f-2-3-f-3-n-f-n-n-n-1-f-n-for-all-n-n-n-2-where-n-is-the-set-of-natural-numbers-and-r-is-the-set-of-real-number-68407

question:
<div class="question">Let $f: N \rightarrow R$ be such that $f(1)=1$ and $f(1)+2 f(2)+3 f(3)+\ldots+n f(n)=n(n+1) f(n),$ for all $n \in N, n \geq 2,$ where $N$ is the set of natural numbers and $R$ is the set of real numbers. Then, the value of $f(500)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1000</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">500</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 500$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 1000$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 1000$</span> </div>

answer:
<div class="solution">We have, $f: N \rightarrow R$ such that $f(1)=1$
<br/>$$
<br/>\begin{array}{l}
<br/>\text { and } f(1)+2 f(2)+3 f(3)+\ldots+n f(n) \\
<br/>\quad=n(n+1) f(n), \forall n \geq 2 \\
<br/>\text { Clearly, } f(1)+2 f(2)=2(2+1) f(2) \\
<br/>\Rightarrow \quad f(1)=6 f(2)-2 f(2) \\
<br/>\Rightarrow \quad f(1)=4 f(2) \\
<br/>\Rightarrow \quad f(2)=\frac{f(1)}{4}=\frac{1}{4}
<br/>\end{array}
<br/>$$
<br/>Similarly.
<br/>$$
<br/>f(1)+2 f(2)+3 f(3)=3(3+1) f(3)
<br/>$$
<br/>$\Rightarrow \quad 1+\frac{1}{2}+3 f(3)=12 f(3)$
<br/>$\Rightarrow \quad 9 f(3)=\frac{3}{2} \Rightarrow f(3)=\frac{1}{6}$
<br/>and $f(1)+2 f(2)+3 f(3)+4 f(4)=4(5) f(4)$
<br/>$\Rightarrow \quad 1+\frac{1}{2}+\frac{1}{2}=16 f(4)$
<br/>$\Rightarrow \quad f(4)=\frac{2}{16}=\frac{1}{8}$
<br/>In general, $f(n)=\frac{1}{2 n}$
<br/>$\therefore \quad f(500)=\frac{1}{1000}$</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
let-f-r-3-r-1-be-defined-by-f-x-x-2-x-3-let-g-r-r-be-given-as-g-x-2-x-3-then-the-sum-of-all-the-values-of-x-for-which-f-1-x-g-1-x-13-2-is-equal-to-1

slug:
let-f-r-3-r-1-be-defined-by-f-x-x-2-x-3-let-g-r-r-be-given-as-g-x-2-x-3-then-the-sum-of-all-the-values-of-x-for-which-f-1-x-g-1-x-13-2-is-equal-to-1-83196

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>:</mo><mi>R</mi><mo>-</mo><mo>{</mo><mn>3</mn><mo>}</mo><mo>→</mo><mi>R</mi><mo>-</mo><mo>{</mo><mn>1</mn><mo>}</mo></math> be defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo> </mo><mo>:</mo><mo> </mo><mi>R</mi><mo>→</mo><mi>R</mi></math> be given as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></math>. Then, the sum of all the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>+</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>13</mn><mn>2</mn></mfrac></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&amp;</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>y</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>x</mi><mo>-</mo><mn>3</mn></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>+</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>13</mn><mn>2</mn></mfrac></math></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo>=</mo></math><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/23862ff9-6288-43cd-bf1f-ac996f32453e-image.png" style="width: 58px; height: 89px;"/></p> <br/> <br/><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> sum of roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>5</mn></math></p></div>


title: let-f-r-r-be-a-continuous-function-which-satisfies-f-x-0-x-f-t-d-t-then-the-value-of-f-lo-g-e-5-is
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
slug:
let-f-r-r-be-a-continuous-function-which-satisfies-f-x-0-x-f-t-d-t-then-the-value-of-f-lo-g-e-5-is-22922

question:
<div class="question">Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x)=\int_{0}^{x} f(t) d t .$ Then. the value of $f\left(\log _{e} 5\right)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Given, $\quad f(x)=\int_{0}^{x} f(t) d t$
<br/>Using Leibnitz theorem, we get
<br/>$$
<br/>f^{\prime}(x)=f(x) \Rightarrow f(x)=k e^{x}
<br/>$$
<br/>On putting $x=0$, we get $f(0)=\int_{0}^{0} f(t) d t$
<br/>$\Rightarrow \quad k e^{0}=0$
<br/>$\left.\because \int_{a}^{a} f(x) d x=0\right]$
<br/>$\left.\Rightarrow \quad k=0 \quad \mid \because e^{0}=1\right]$
<br/>$\therefore \quad f(x)=0 \Rightarrow f\left(\log _{e} 5\right)=0$</div>


title: let-f-r-r-be-a-twice-continuously-differentiable-function-such-that-f-0-f-1-f-0-0-then
slug: let-f-r-r-be-a-twice-continuously-differentiable-function-such-that-f-0-f-1-f-0-0-then-28071
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be a twice continuously differentiable function such that $f(0)=f(1)=f^{\prime}(0)=0 .$ Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f^{\prime \prime}(0)=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f^{\prime \prime}=0$ for some $C \in A$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">if c $\neq 0,$ then $f^{\prime \prime}\neq 0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f^{\prime}(x)&gt;0$ for all $x \neq 0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f^{\prime \prime}=0$ for some $C \in A$</span> </div>

answer:
<div class="solution">We have. 
<br/>$$f: \mathbb{R} \rightarrow \mathbb{R} $$ be a twice continuousty differentiable function such that $f(0)=f(1)=f'(0)=0$ 
<br/>Now, for atleast one value of $\epsilon_{1} \in(0,1)$ $f'\left(c_{1}\right)=0 \quad$ (by Rolle's theorem) 
<br/>Again. $\quad f'(0)=0=f(c_{1})$
<br/>$f'(c)=$ ofor some $c \in\left(0, c_{1}\right)$ (by Rolle's theorem)</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
let-f-r-r-be-defined-as-f-x-0-sin-x-x-is-irrational-x-is-rational-then-which-of-the-following-is-true

slug:
let-f-r-r-be-defined-as-f-x-0-sin-x-x-is-irrational-x-is-rational-then-which-of-the-following-is-true-26581

question:
<div class="question">Let $f:R\rightarrow R$ be defined as
<br/>$f(x)=\left\{\begin{array}{lr}0, &amp; x \text { is irrational} \\ \sin |x|, &amp; x \text { is rational }\end{array}\right.$ Then, which of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">f is discontinuous for all x</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">f is continuous for all x</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">f is discontinuous at $x=k\pi$ where k is an integer</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">f is continuous at $x=k\pi$ where k is an integer</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">f is continuous at $x=k\pi$ where k is an integer</span> </div>

answer:
<div class="solution">We have, $f(x)=\left\{\begin{array}{ll}0, &amp; x \text { is irrational } \\ \sin |x|, x \text { is rational }\end{array}\right.$
<br/>If $f(x)$ is continuous, then $\sin |x|=0$ $\Rightarrow x=k \pi,$ where $k$ is an integer.</div>


title: let-f-r-r-be-defined-as-f-x-x-2-x-4-x-2-x-4-then-range-of-the-function-f-x-is
slug: let-f-r-r-be-defined-as-f-x-x-2-x-4-x-2-x-4-then-range-of-the-function-f-x-is-95962
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be defined as $f(x)=\frac{x^{2}-x+4}{x^{2}+x+4}$. Then, range of the function $f(x)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\frac{3}{5}, \frac{5}{3}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{3}{5}, \frac{5}{3}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(-\infty, \frac{3}{5}\right) \cup\left(\frac{5}{3}, \infty\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[-\frac{5}{3},-\frac{3}{5}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\frac{3}{5}, \frac{5}{3}\right]$</span> </div>

answer:
<div class="solution">Let $y=\frac{x^{2}-x+4}{x^{2}+x+4}$
<br/>$\Rightarrow \quad x^{2} y+x y+4 y=x^{2}-x+4$
<br/>$\Rightarrow(y-1) x^{2}+(y+1) x+4 y-4=0$
<br/>For $x$ to be real, discriminant of the above quadratic equation should be greater than equal to 0 
<br/>$\\ \Rightarrow \quad(y+1)^{2}-4(y-1)(4 y-4) \geq 0$
<br/>$\Rightarrow \quad(y+1)^{2}-16(y-1)^{2} \geq$
<br/>$\Rightarrow(y+1+4 y-4)(y+1-4 y+4) \geq$
<br/>$(5 y-3)(5-3 y) \geq 0$
<br/>$\therefore \quad y \in\left[\frac{3}{5}, \frac{5}{3}\right]$</div>


title: let-f-r-r-be-defined-by-f-x-x-2-1-x-2-x-2-for-all-x-r-then
slug: let-f-r-r-be-defined-by-f-x-x-2-1-x-2-x-2-for-all-x-r-then-81851
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be defined by $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$ for all $x \in R$. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$ is one-one but not onto mapping</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f$ is onto but not one-one mapping</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f$ is both one-one and onto</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$f$ is neither one-one nor onto</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ is neither one-one nor onto</span> </div>

answer:
<div class="solution">We have,
<br/>$$
<br/>\begin{aligned}
<br/>f(x) &amp;=x^{2}-\frac{x^{2}}{1+x^{2}} \\
<br/>f(-x) &amp;=(-x)^{2}-\frac{(-x)^{2}}{1+(-x)^{2}} \\
<br/>&amp;=x^{2}-\frac{x^{2}}{1+x^{2}}=f(x) \\
<br/>\therefore f(-x)=f(x) &amp;
<br/>\end{aligned}
<br/>$$
<br/>So, $f(x)$ is many one Again, $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$
<br/>$$
<br/>=\frac{x^{2}+x^{4}-x^{2}}{1+x^{2}}=\frac{x^{4}}{1+x^{2}}
<br/>$$
<br/>$\therefore$ Range of $f(x)$ will be $[0, \infty)$ But, co-domain of $f(x)=R$
<br/>$\therefore$ Range $\neq$ co-domain. So, $f(x)$ is onto.</div>


tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
let-f-r-r-be-differentiable-at-x-0-if-f00-and-f02-then-the-value-of-lim-x-0-x-1-f-x-f-2-x-f-3-x-f-2015-x-is

slug:
let-f-r-r-be-differentiable-at-x-0-if-f00-and-f02-then-the-value-of-lim-x-0-x-1-f-x-f-2-x-f-3-x-f-2015-x-is-75961

question:
<div class="question">Let $f: R \rightarrow R$ be differentiable at $x=0$. If f(0)=0 and f'(0)=2, then the value of $$
<br/>\left.\lim _{x \rightarrow 0} \frac{1}{x} \mid f(x)+f(2 x)+f(3 x)+\ldots+f(2015 x)\right]  \text{     is}
<br/>$$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2015</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2015 \times 2016$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2015 \times 2014$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2015 \times 2016$</span> </div>

answer:
<div class="solution">Given, $f(0)=0$ and $f^{\prime}(0)=2$
<br/>$\therefore \lim _{x \rightarrow 0} \frac{1}{x}[f(x)+f(2 x)+f(3 x)+\ldots+f(2015 x)]$
<br/>$=\lim _{x \rightarrow 0} \frac{\left[\begin{array}{c}f^{\prime}(x)+2 f^{\prime}(2 x)+3 f^{\prime}(3 x) \\ +\ldots+2015 f^{\prime}(2015 x)\end{array}\right]}{1}$
<br/>$=\frac{2+2 \times 2+3 \times 2+\ldots+2015 \times 2}{1}$
<br/>$=2[1+2+3+\ldots+2015]$
<br/>$=\frac{2(2015)(2015+1)}{2}=\frac{2 \times 2015 \times 2016}{2}$
<br/>$=2015 \times 2016$</div>


title: let-f-r-r-be-given-by-f-x-x-2-1-x-r-then
slug: let-f-r-r-be-given-by-f-x-x-2-1-x-r-then-67471
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be given by $f(x)=\left|x^{2}-1\right|, x \in R$. then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">f has a local minimum at $x=\pm 1$ but no local maximum</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">f has a local maximum at $x=0$ but no local minimum</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}$ has a local minima at $\mathrm{x}=\pm 1$ and a local maxima at $\mathrm{x}=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">f has neither a local maxima nor a local minima at any point</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}$ has a local minima at $\mathrm{x}=\pm 1$ and a local maxima at $\mathrm{x}=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/eymCZ7MlkJeZ4CNuVi3WDL6kfsXvX2_K-MzzGo_bIIs.original.fullsize.png"/><br/> <br/>$f$ has a local minima at $x=\pm 1$ and local maximum at $x=0$</div>


title: let-f-r-r-be-such-that-f-0-0-and-f-x-5-for-all-x-then-f-1-is-in
slug: let-f-r-r-be-such-that-f-0-0-and-f-x-5-for-all-x-then-f-1-is-in-22026
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be such that $f(0)=0$ and $\left|f^{\prime}(x)\right| \leq 5$ for all $x$. Then $f(1)$ is in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(5,6)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$[-5,5]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-\infty,-5) \cup(5, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$[-4,4]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[-5,5]$</span> </div>

answer:
<div class="solution">$\left|f^{\prime}(x)\right| \leq 5$
<br/>$\int_{0}^{1}-5 d x \leq \int_{0}^{1} f^{\prime}(x) d x \leq \int_{0}^{1} 5 d x$
<br/>$\Rightarrow-5 \leq f(1)-0 \leq 5$
<br/>$\Rightarrow-5 \leq f(1) \leq 5$
<br/>$\Rightarrow f(1) \in[-5,5]$</div>


title: let-f-r-r-be-such-that-f-2-x-1-f-x-for-all-x-r-if-f-is-continuous-at-x-1-an-f-1-1-then
slug: let-f-r-r-be-such-that-f-2-x-1-f-x-for-all-x-r-if-f-is-continuous-at-x-1-an-f-1-1-then-30877
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be such that $f(2 x-1)=f(x)$ for all $x \in R .$ If $f$ is continuous at $x=1$ an $f(1)=1,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(2)=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(2)=2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f$ is continuous only at $x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f$ is continuous at all points</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f$ is continuous only at $x=1$</span> </div>

answer:
<div class="solution">Given, $f: R \rightarrow R$
<br/>and $\quad f(2 x-1)=f(x), x \in R$
<br/>Hence, $f$ is continuous at $x=1$ and $f(1)=1$</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
let-f-r-r-be-such-that-f-is-injective-and-f-x-f-y-f-x-y-for-all-x-y-r-if-f-x-f-y-and-f-z-are-in-gp-then-x-y-and-z-are-in

slug:
let-f-r-r-be-such-that-f-is-injective-and-f-x-f-y-f-x-y-for-all-x-y-r-if-f-x-f-y-and-f-z-are-in-gp-then-x-y-and-z-are-in-60533

question:
<div class="question">Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x) f(y)=f(x+y) \quad$ for $\quad$ all $\quad x, y \in R, \quad$ if $f(x), f(y)$ and $f(z)$ are in $G P,$ then $x, y$ and $z$ are in</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">AP always</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">GP always</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">AP depending on the values of $x, y$ and $z$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">GP depending on the values of $x, y$ and $z$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">AP always</span> </div>

answer:
<div class="solution">Let the function, $f(x)=a^{kx}$ Which define in $f: R \rightarrow R$ and injective also.
<br/>Now, we have
<br/>$$
<br/>f(x) f(y)=f(x+y)
<br/>$$
<br/>$\Rightarrow$
<br/>$a^{k x} \cdot a^{k y}=a^{k(x+y)}$
<br/>$\Rightarrow$
<br/>$a^{k(x+y)}=a^{k(x+y)}$
<br/>$\because \quad f(x), f(y)$ and $f(z)$ are in GP
<br/>$\therefore$
<br/>$f(y)^{2}=f(x) \cdot f(z)$
<br/>$\Rightarrow$
<br/>$a^{2 k y}=a^{k x} \cdot a^{k z}$
<br/>$\Rightarrow$
<br/>$e^{2 k y}=e^{k(x+z)}$
<br/>On comparing, we get $2 k y=k(x+z) \Rightarrow 2 y=x+z$
<br/>$\Rightarrow x, y$ and $z$ are in $A P$.</div>


tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
let-f-r-r-be-such-that-f-is-injective-and-f-x-f-y-f-x-y-for-x-y-r-if-f-x-f-y-f-z-are-in-gp-then-x-y-z-are-in

slug:
let-f-r-r-be-such-that-f-is-injective-and-f-x-f-y-f-x-y-for-x-y-r-if-f-x-f-y-f-z-are-in-gp-then-x-y-z-are-in-82907

question:
<div class="question">Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x) f(y)=f(x+y)$ for $\forall x, y \in R .$ If $f(x)$
<br/>$f(y), f(z)$ are in G.P., then $x, y, z$ are in</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">AP always</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">GP always</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">AP depending on the value of $x, y, z$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">GP depending on the value of $x, y, z$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">AP always</span> </div>

answer:
<div class="solution">Since, $f: R \rightarrow R$ is injective and $f(x) f(y)=f(x+y), \forall x, y \in R$
<br/>$\therefore \quad f(x)=a^{x}$
<br/>Again, $f(x), f(y), f(z)$ are in GP. $\Rightarrow \quad(f(y))^{2}=f(x) \cdot f(y)$
<br/>$\Rightarrow \quad a^{2 y}=a^{x} \cdot a^{z}$
<br/>$\Rightarrow \quad a^{2 y}=a^{x+y}$
<br/>$\Rightarrow \quad 2 y=x+z$
<br/>$\therefore x, y, z$ are in $\mathrm{AP}$</div>


title: let-f-r-r-be-twice-continuously-differentiable-let-f-0-f-1-f-0-0-then
slug: let-f-r-r-be-twice-continuously-differentiable-let-f-0-f-1-f-0-0-then-10182
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f: R \rightarrow R$ be twice continuously differentiable. Let $f(0)=f(1)=f^{\prime}(0)=0 .$ Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f^{\prime \prime}(x) \neq 0$ for all $x$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f^{\prime \prime}(c)=0$ for some $c \in R$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f^{\prime \prime}(x) \neq 0$ if $x \neq 0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f^{\prime}(x)&gt;0$ for all $x$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f^{\prime \prime}(c)=0$ for some $c \in R$</span> </div>

answer:
<div class="solution">Let function $f(x)=x^{2}(x-1)$ 
<br/>$\Rightarrow \quad f^{\prime}(x)=3 x^{2}-2 x$
<br/>and $\quad f^{\prime \prime}(x)=6 x-2$
<br/>$\begin{array}{ll}\text { Now, } &amp; f(0)=f(1)=f^{\prime}(0)=0\end{array}$
<br/>Then, according to question, at $x=\frac{1}{3}, \quad f^{\prime \prime}(x)=0$
<br/>(i.e. $c=1 / 3$ for some $C \in R$ )</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
title:
let-f-x-0-for-all-x-and-f-x-exists-for-all-x-if-f-is-the-inverse-function-of-h-and-h-x-1-l-o-g-x-1-then-f-x-will-be

slug:
let-f-x-0-for-all-x-and-f-x-exists-for-all-x-if-f-is-the-inverse-function-of-h-and-h-x-1-l-o-g-x-1-then-f-x-will-be-41848

question:
<div class="question">Let $f(x)&gt;0$ for all $x$ and $f^{\prime}(x)$ exists for all $x$. If $f$ is the inverse function of $h$ and $h^{\prime}(x)=\frac{1}{1+\log x} \cdot$ Then, $f^{\prime}(x)$ will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1+\log (f(x))$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1+f(x)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1-\log (f(x))$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log f(x)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1+\log (f(x))$</span> </div>

answer:
<div class="solution">Given,
<br/>$$
<br/>h\{f(x)\}=x
<br/>$$
<br/>differentiating w.r.t. $x$, we get
<br/>$$
<br/>\begin{array}{r}
<br/>h^{\prime}\{f(x)\}, f^{\prime}(x)=1 \\
<br/>f^{\prime}(x)=\frac{1}{h^{\prime}\{f(x)\}}
<br/>\end{array}
<br/>$$
<br/>$\begin{aligned} \Rightarrow \quad f^{\prime}(x)=\frac{1}{\frac{1}{1+\log \{f(x)\}}} &amp; \\ \Rightarrow \quad\left[\because h^{\prime}(x)=\frac{1}{1+\log x}\right] \\ \Rightarrow &amp; f^{\prime}(x)=1+\log (f(x)) \end{aligned}$</div>


title: let-f-x-0-if-1-x-0-1-if-x-0-2-if-0-x-1-and-let-f-x-1-x-f-t-d-t-1-x-1-then
slug: let-f-x-0-if-1-x-0-1-if-x-0-2-if-0-x-1-and-let-f-x-1-x-f-t-d-t-1-x-1-then-37292
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">$\operatorname{Let} f(x)=\left\{\begin{array}{l}0, \text { if }-1 \leq x     &lt; 0 \\ 1, \text { if } x=0 \\ 2, \text { if } 0 &lt; x \leq 1\end{array}\right.$ and let $F(x)=\int_{-1}^{x} f(t) d t,-1 \leq x \leq 1$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{F}$ is continuous function in $[-1,1]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$F$ is discontinuous function in $[-1,1]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{F}^{\prime}(\mathrm{x})$ exists at $\mathrm{x}=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{F}^{\prime}(\mathrm{x})$ does not exists at $\mathrm{x}=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$\mathrm{F}$ is continuous function in $[-1,1]$, $\mathrm{F}^{\prime}(\mathrm{x})$ does not exists at $\mathrm{x}=0$</span> </div>

answer:
<div class="solution">$f(x)=\left\{\begin{array}{ll}0 &amp; -1 \leq x     &lt; 0 \\ 1 &amp; x=0 \\ 2 &amp; 0 &lt; x \leq 1\end{array}\right.$<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Sa3trje9i8W71FkBSIcLLeB3xroMcmTmJaoD3lKk9EM.original.fullsize.png"/><br/> <br/>$F(x)=\left\{\begin{array}{ll}0 &amp; -1 \leq x \leq 0 \\ 2 x &amp; 0 &lt; x \leq 1\end{array}\right.$</div>


title: let-f-x-0-x-1-t-d-t-x-2-1-x-0-x-1-then
slug: let-f-x-0-x-1-t-d-t-x-2-1-x-0-x-1-then-44882
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\left\{\begin{array}{ll}\int_{0}^{x}|1-t| d t, &amp; x&gt;0 \\ x-\frac{1}{2}, &amp; x \leq 1\end{array} .\right.$ Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(x)$ is continuous at $x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)$ is not continuous at $x=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)$ is differentiable at $x=1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$f(x)$ is not differentiable at $x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$f(x)$ is continuous at $x=1$, $f(x)$ is not differentiable at $x=1$</span> </div>

answer:
<div class="solution">Given. $f(x)=\left\{\begin{aligned} \int_{0}^{x}|1-t| d t, &amp; x&gt;1 \\ x-\frac{1}{2}, &amp; x \leq 1 \end{aligned}\right.$
<br/>Now, for $x&gt;1$, $\int_{0}^{x}|1-t| d t$
<br/>$$
<br/>\begin{aligned}
<br/>&amp;=\int_{0}^{1}(1-t) d t+\int_{1}^{x}(t-1) d t \\
<br/>&amp;=\left[t-\frac{t^{2}}{2}\right]_{0}^{1}+\left[\frac{t^{2}}{2}-t\right]_{1}^{x} \\
<br/>&amp;=\left[1-\frac{1}{2}-0\right]+\left[\frac{x^{2}}{2}-x-\frac{1}{2}+1\right] \\
<br/>&amp;=\frac{1}{2}+\frac{x^{2}}{2}-x-\frac{1}{2}+1 \\
<br/>\Rightarrow \quad \frac{x^{2}}{2}-x+1
<br/>\end{aligned}
<br/>$$
<br/>$\therefore$ Eq. (i) becomes, $f(x)=\left\{\begin{array}{ll}\frac{x^{2}}{2}-x+1, &amp; x&gt;1 \\ x-\frac{1}{2}, &amp; x \leq 1\end{array}\right.$
<br/>Continuity at $x=1$
<br/>and $\begin{aligned} \mathrm{LHL} &amp;=\lim (f(x)\\ &amp;=\lim _{x \rightarrow 1}\left(x-\frac{1}{2}\right)=1-\frac{1}{2} \\ \mathrm{RHL} &amp;=\lim _{x \rightarrow 1^{+}} f(x) \\ &amp;=\lim _{x \rightarrow 1}\left(\frac{x^{2}}{2}-x+1\right) \\ &amp;=\frac{1}{2}-1+1=\frac{1}{2} \\ f(1) &amp;=1-\frac{1}{2}=\frac{1}{2} \end{aligned}$
<br/>$\therefore \quad f(1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$
<br/>Hence, $f(x)$ is continuous at $x=1$ 
<br/>Differentiability at $x=1$
<br/>$\begin{aligned} L H D &amp;=\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h} \\ &amp;=\lim _{h \rightarrow 0} \frac{(1-h)-\frac{1}{2}-\left(1-\frac{1}{2}\right)}{-h}=\lim _{h \rightarrow 0} \frac{-h}{-h}=1 \end{aligned}$
<br/>$\begin{aligned} \mathrm{RHD} &amp;=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\frac{(1+h)^{2}}{2}-(1+h)+1-\left(1-\frac{1}{2}\right)}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\frac{1+h^{2}+2 h}{2}-1-h+1-\frac{1}{2}}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\frac{1}{2}+\frac{h^{2}}{2}+h-h-\frac{1}{2}}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{h^{2}}{2 h}=0 \end{aligned}$
<br/>$\mathrm{LHD} \neq \mathrm{RHD}$
<br/>Hence, $f(x)$ is not differentiable at $x=1$.</div>


title: let-f-x-1-x-2-where-the-square-root-is-to-be-taken-positive-then
slug: let-f-x-1-x-2-where-the-square-root-is-to-be-taken-positive-then-49383
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
answer: <div class="solution">Hint: <br/>$f(x)=1-|x|, f$ has maxima at $x=0$</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=1-\sqrt{\left(x^{2}\right)}$ where the square root is to be taken positive, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$ has no extrema at $x=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f$ has minima at $x=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$f$ has maxima at $x=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}^{\prime}$ exists at 0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ has maxima at $x=0$</span> </div>


title: let-f-x-2-sin-x-a-sin-x-b-cos-x-if-x-2-if-2-x-2-then-if-x-2
slug: let-f-x-2-sin-x-a-sin-x-b-cos-x-if-x-2-if-2-x-2-then-if-x-2-36313
tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\left\{\begin{array}{cc}-2 \sin x, &amp; \text { if } x \leq-\frac{\pi}{2} \\ A \sin x+B, &amp; \text { if }-\frac{\pi}{2}    &lt; x &lt; \frac{\pi}{2} \text { . Then, } \\ \cos x, &amp; \text { if } x \geq \frac{\pi}{2}\end{array}\right.$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$ is discontinuous for all $A$ and $B$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f$ is continuous for all $A=-1$ and $B=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f$ is continuous for all $A=1$ and $B=-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f$ is continuous for all real values of $A B$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ is continuous for all $A=-1$ and $B=1$</span> </div>

answer:
<div class="solution">At $\quad x=-\frac{\pi}{2}$
<br/>$$
<br/>\begin{array}{l}
<br/>\text { LHL }=-2 \\
<br/>\text { RHL }=-A+B \\
<br/>\text { For continuity, } \mathrm{LHL}=\mathrm{RHL}=f(-\pi / 2)
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad-A+B=2$
<br/>At $x=\pi / 2$
<br/>$$
<br/>\begin{aligned}
<br/>\mathrm{LHL} &amp;=A+B \\
<br/>\mathrm{RHL} &amp;=0 \\
<br/>\text { For continuity, } \mathrm{LHL} &amp;=\mathrm{RHL}=f(\pi / 2)
<br/>\end{aligned}
<br/>$$
<br/>$\Rightarrow \quad A+B=0$
<br/>On solving Eqs. (i) and (ii), we get
<br/>$$
<br/>A=-1 \text { and } B=1
<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
let-f-x-2-x-2-5-x-1-if-we-write-f-x-as-f-x-a-x-1-x-2-b-x-2-x-1-c-x-1-x-1-for-real-numbers-a-b-c-then

slug:
let-f-x-2-x-2-5-x-1-if-we-write-f-x-as-f-x-a-x-1-x-2-b-x-2-x-1-c-x-1-x-1-for-real-numbers-a-b-c-then-75576

question:
<div class="question">Let $f(x)=2 x^{2}+5 x+1$. If we write $f(x)$ as $f(x)=a(x+1)(x-2)+b(x-2)(x-1)+c(x-1)(x+1)$
<br/>for real numbers $a, b, c,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">there are infinite number of choices for $a, b, c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">only one choice for a but infinite number of choices for $b$ and $c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">exactly one choice for each of $a, b, c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">more than one but finite number of choices for $a, b, c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">exactly one choice for each of $a, b, c$</span> </div>

answer:
<div class="solution">$$
<br/>\begin{array}{l}
<br/>\text { Given, } f(x)=2 x^{2}+5 x+1 \\
<br/>\text { Also, } f(x)=a(x+1)(x-2)+b(x-2)(x-1) \\
<br/>+c(x-1)(x+1) \\
<br/>\qquad \begin{array}{r}
<br/>=a\left(x^{2}-x-2\right)+b\left(x^{2}-3 x+2\right) \\
<br/>+c\left(x^{2}-1\right) \\
<br/>f(x)=(a+b+c) x^{2}+(-a-3 b) x \\
<br/>+(-2 a+2 b-c)
<br/>\end{array}
<br/>\end{array}
<br/>$$
<br/>On equating the coefficients of $x^{2}$, $x$ and constant term  we get
<br/>$$
<br/>\begin{array}{l}
<br/>a+b+c=2,-a-3 b=5 \\
<br/>\text { and }-2 a+2 b-c=1
<br/>\end{array}
<br/>$$
<br/>On solving above equations, we get $a=-\frac{35}{4}, b=\frac{5}{4}$ and $c=\frac{38}{4}$
<br/>Hence, exactly one choice for each of b and c</div>


title: let-f-x-3-1-x-sin-x-1-cos-x-the-smallest-positive-interger-k-such-that-lim-x-0-x-k-f-x-0-is
slug: let-f-x-3-1-x-sin-x-1-cos-x-the-smallest-positive-interger-k-such-that-lim-x-0-x-k-f-x-0-is-57371
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answers are: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\frac{1}{3} x \sin x-(1-\cos x) .$ The smallest positive interger $k$ such that $\lim _{x \rightarrow 0} \frac{f(x)}{x^{k}} \neq 0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Hint: 
<br/>It $\left.\frac{x \sin x-3(1-\cos x)}{3 x^{k}}=\frac{1}{3} \operatorname{lt}_{x \rightarrow 0}\left(\frac{\sin x / 2}{x / 2}\right)_{x \rightarrow 0} \mid t_{x \rightarrow 0} \frac{2 x \cos x / 2-6 \sin x / 2}{2 x^{k-1}}\right)$
<br/>$k-1=1 \Rightarrow k=2$</div>


title: let-f-x-3-x-10-7-x-8-5-x-6-21-x-3-3-x-2-7-then-lim-h-0-h-3-3-h-f-1-h-f-1
slug: let-f-x-3-x-10-7-x-8-5-x-6-21-x-3-3-x-2-7-then-lim-h-0-h-3-3-h-f-1-h-f-1-43907
tags: ['Mathematics', 'Limits', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=3 x^{10}-7
<br/>x^{8}+5 x^{6}-21 x^{3}+3 x^{2}-7$
<br/>Then $\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">does not exist</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is $\frac{50}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is $\frac{53}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is $\frac{22}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">is $\frac{53}{3}$</span> </div>

answer:
<div class="solution">We have, $f(x)=3 x^{10}-7 x^{8}+5 x^{6}-21 x^{3}+3 x^{2}-7$
<br/>$\therefore \quad f(1-h)=3(1-h)^{10}-7(1-h)^{8}$
<br/>$+5(1-h)^{6}-21\left(1-h^{3}+3(1-h)^{2}-7\right.$
<br/>$=3\left(1-10 h+45 h^{2}-120 h^{3}+\ldots \ldots+h^{10}\right)$
<br/>$-7\left(1-8 h+28 h^{2}-56 h^{3}+\ldots . .+h^{8}\right)$
<br/>$+5\left(1-6 h+15 h^{2}-20 h^{3}+\ldots . .+h^{6}\right)$
<br/>$-21\left(1-3 h+3 h^{2}-h^{3}\right)$
<br/>$+3\left(1-2 h+h^{2}\right)-7$
<br/>$\Rightarrow f(1-h)=-24+53 h+h^{2}(-46)+h^{3}(-47)+\ldots$
<br/>and $\quad f(1)=-24$
<br/>$\therefore \quad \lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}$
<br/>$=\lim _{h \rightarrow 0} \frac{-24+53 h+h^{2}(-46)+h^{3}(-47)+\ldots-(-24)}{h\left(h^{2}+3\right)}$
<br/>$=\lim _{h \rightarrow 0} \frac{53 h+h^{2}(-46)+h^{3}(-47)+\ldots}{h\left(h^{2}+3\right)}$
<br/>$=\lim _{h \rightarrow 0} \frac{53+h(-46)+h^{2}(-47)+\ldots}{h^{2}+3}=\frac{53}{3}$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
let-f-x-a-0-a-1-x-a-2-x-2-a-3-x-3-where-a-0-a-1-a-2-a-3-are-real-constants-then-f-x-is-differentiable-at-x-0

slug:
let-f-x-a-0-a-1-x-a-2-x-2-a-3-x-3-where-a-0-a-1-a-2-a-3-are-real-constants-then-f-x-is-differentiable-at-x-0-74295

question:
<div class="question">Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are real constants. Then $f(x)$ is differentiable at $x=0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">whatever be $a_0, a_1, a_2, a_3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">for no values of $a_0, a_1, a_2, a_3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">only if $a_1=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">only if $a_1=0, a_3=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">only if $a_1=0$</span> </div>

answer:
<div class="solution">$f^{\prime}\left(0^{+}\right)=\lim _{x \rightarrow 0}\left(a_1+2 a_1 x+3 a_3 x^2\right)=a_1$<br/>$f^{\prime}\left(0^{-}\right)=\lim _{x \rightarrow 0}\left(-a_1+2 a_1 x-3 a_3 x^2\right)=-a_1$<br/>$f^{\prime}\left(0^{+}\right)=f^{\prime}\left(0^{-}\right) \quad \Rightarrow a_1=0$</div>


title: let-f-x-a-x-2-b-x-c-g-x-p-x-2-q-x-r-such-that-f-1-g-1-f-2-g-2-and-f-3-g-3-2-then-f-4-g-4-is
slug: let-f-x-a-x-2-b-x-c-g-x-p-x-2-q-x-r-such-that-f-1-g-1-f-2-g-2-and-f-3-g-3-2-then-f-4-g-4-is-92538
tags: ['Mathematics', 'Linear Programming', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=a x^{2}+b x+c, g(x)=p x^{2}+q x+r$
<br/>$\begin{array}{lll}\text { such that } &amp; f(1)=g(1), f(2)=g(2) &amp; \text { and }\end{array}$
<br/>$f(3)-g(3)=2 .$ Then, $f(4)-g(4)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7</span> </li> </ul>

answer:
<div class="solution">Given, $f(x)=e x^{2}+b x+c, g(x)=p x^{2}+q x+r$
<br/>since, $f(1)=g(1)$
<br/>$\Rightarrow \quad a+b+c=p+q+r$
<br/>$f(2)=8(2)$
<br/>$\Rightarrow \quad 4 a+2 b+c=4 p+2 q+r$
<br/>Subtracting Eq.
<br/>(tii) from Eq. (i), we get $3 a+b=3 p+q$
<br/>$f(3)-g(3)=2$
<br/>$\Rightarrow(9 a+3 b+c)-(9 p+3 q+r)=2$
<br/>$\Rightarrow \quad 3(3 a+b)+c-3(3 p+q)-r=2$
<br/>$\Rightarrow$
<br/>$c-r=2 \quad \ldots \text { (iv) }$
<br/>$(3 a+b=3 p+q)$
<br/>From Eq. (i), $(a-p)+(b-q)+(c-r)=0$
<br/>$\Rightarrow \quad(a-p)+(b-q)+2=0$
<br/>From Eq. (ii), $4(a-p)+2(b-q)+c-r=0$
<br/>$\Rightarrow \quad 2(a-p)+(b-q)+1=0$
<br/>Subtracting Eq.(v) from Eq. (vi), we get $(a-p)-1=0$
<br/>$a-p=1$
<br/>$\therefore$ From Eq. $(v), \quad b-q=-3$
<br/>Now, $f(4)-g(4)=(16 a+4 b+c)-(16 p+4 q+r)$
<br/>$=16(a-p)+4(b-q)+(c-r) \ldots(v$
<br/>Substituting the values of $(a-p),(b-q),(c-r)$ from above in Eq. (vii). we get $\begin{aligned} f(4)-g(4) &amp;=16 \times 1+4(-3)+2 \\ &amp;=16-12+2=6 \end{aligned}$</div>


title: let-f-x-be-a-continuous-periodic-function-with-period-t-let-i-a-a-t-f-x-d-x-then
slug: let-f-x-be-a-continuous-periodic-function-with-period-t-let-i-a-a-t-f-x-d-x-then-94345
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)$ be a continuous periodic function with period $T$. Let $I=\int_{a}^{a+T} f(x) d x$. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{I}$ is linear function in ' $\mathrm{a}$ '</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">I does not depend on ' $a$ '</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0     &lt; \mathrm{I} &lt; \mathrm{a}^{2}+1$ where $\mathrm{I}$ depends on ' $\mathrm{a}$ '</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{I}$ is quadratic function in ' $\mathrm{a}$ '</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I does not depend on ' $a$ '</span> </div>

answer:
<div class="solution">$I=\int_{a}^{a+T} f(x) d x=\int_{0}^{T} f(x) d x$
<br/>I does not depend on a.</div>


title: let-f-x-be-a-derivable-function-f-x-f-x-and-f-0-0-then
slug: let-f-x-be-a-derivable-function-f-x-f-x-and-f-0-0-then-84939
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">Let $f(x)$ be a derivable function, $f^{\prime}(x)&gt;f(x)$ and $f(0)=0 .$ Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(x)&gt;0$ for all $x&gt;0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)    &lt; 0$ for all $x&gt;0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">no sign of $f(x)$ can be ascertained</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(x)$ is a constant function</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)&gt;0$ for all $x&gt;0$</span> </div>

answer:
<div class="solution">Let $g(x)=e^{-x} f(x)$
<br/>$\therefore$ $g^{\prime}(x)=e^{-x} f^{\prime}(x)-e^{-x} f(x)$
<br/>$\quad=e^{-x}\left(f^{\prime}(x)-f(x)\right)$
<br/>Since, $f^{\prime}(x)&gt;f(x),$ so $f^{\prime}(x)-f(x)&gt;0$
<br/>$\therefore e^{-x}\left(f^{\prime}(x)-f(x)\right)&gt;0$
<br/>$\Rightarrow \quad g^{\prime}(x)&gt;0$
<br/>$\therefore g(x)$ is an increasing function. 
<br/>Now, $g(x)&gt;g(0) \forall x&gt;0$
<br/>$\begin{array}{lr}\Rightarrow e^{-x} f(x)&gt;e^{0} f(0) &amp; \\ \Rightarrow e^{-x} f(x)&gt;0 &amp; {[\because f(0)=0]}\end{array}$
<br/>$\Rightarrow \quad f(x)&gt;0, \forall x&gt;0$</div>


