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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 5851. |
If \[y={{\sec }^{-1}}\left( \frac{\sqrt{x}+1}{\sqrt{x}-1} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{x}-1}{\sqrt{x}+1} \right)\], then \[\frac{dy}{dx}=\] [UPSEAT 1999; AMU 2002; Kerala (Engg.) 2005] |
| A. | 0 |
| B. | \[\frac{1}{\sqrt{x}+1}\] |
| C. | 1 |
| D. | None of these |
| Answer» B. \[\frac{1}{\sqrt{x}+1}\] | |
| 5852. |
\[\frac{d}{dx}{{\tan }^{-1}}\frac{4\sqrt{x}}{1-4x}=\] |
| A. | \[\frac{1}{\sqrt{x}(1+4x)}\] |
| B. | \[\frac{2}{\sqrt{x}(1+4x)}\] |
| C. | \[\frac{4}{\sqrt{x}(1+4x)}\] |
| D. | None of these |
| Answer» C. \[\frac{4}{\sqrt{x}(1+4x)}\] | |
| 5853. |
If \[y=\sin [\cos (\sin x)],\]then \[dy/dx=\] |
| A. | \[-\cos [\cos (\sin x)]\sin (\cos x).\cos x\] |
| B. | \[-\cos [\cos (\sin x)]\sin (\sin x).\cos x\] |
| C. | \[\cos [\cos (\sin x)]\sin (\cos x).\cos x\] |
| D. | \[\cos [\cos (\sin x)]\sin (\sin x).\cos x\] |
| Answer» C. \[\cos [\cos (\sin x)]\sin (\cos x).\cos x\] | |
| 5854. |
\[\frac{d}{dx}({{e}^{x}}\log \sin 2x)=\] [AI CBSE 1985] |
| A. | \[{{e}^{x}}(\log \sin 2x+2\cot 2x)\] |
| B. | \[{{e}^{x}}(\log \cos 2x+2\cot 2x)\] |
| C. | \[{{e}^{x}}(\log \cos 2x+\cot 2x)\] |
| D. | None of these |
| Answer» B. \[{{e}^{x}}(\log \cos 2x+2\cot 2x)\] | |
| 5855. |
\[\frac{d}{dx}{{\tan }^{-1}}(\sec x+\tan x)=\] [AISSE 1985, 87; DSSE 1982, 84] |
| A. | 1 |
| B. | 1/2 |
| C. | \[\cos x\] |
| D. | \[\sec x\] |
| Answer» C. \[\cos x\] | |
| 5856. |
\[\frac{d}{dx}\log (\sqrt{x-a}+\sqrt{x-b})=\] |
| A. | \[\frac{1}{2[\sqrt{(x-a)}+\sqrt{(x-b)}]}\] |
| B. | \[\frac{1}{2\sqrt{(x-a)(x-b)}}\] |
| C. | \[\frac{1}{\sqrt{(x-a)(x-b)}}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{\sqrt{(x-a)(x-b)}}\] | |
| 5857. |
\[\frac{d}{dx}\sqrt{\frac{1+\cos 2x}{1-\cos 2x}}=\] |
| A. | \[{{\sec }^{2}}x\] |
| B. | \[-\text{cose}{{\text{c}}^{2}}x\] |
| C. | \[2\,{{\sec }^{2}}\frac{x}{2}\] |
| D. | \[-2\text{cose}{{\text{c}}^{2}}\frac{x}{2}\] |
| Answer» C. \[2\,{{\sec }^{2}}\frac{x}{2}\] | |
| 5858. |
\[\frac{d}{dx}\log \tan \left( \frac{\pi }{4}+\frac{x}{2} \right)=\] |
| A. | \[\cos \text{ec}\,x\] |
| B. | \[-\cos \text{ec}\,x\] |
| C. | \[\sec x\] |
| D. | \[-\sec x\] |
| Answer» D. \[-\sec x\] | |
| 5859. |
If \[y=\log \frac{1+\sqrt{x}}{1-\sqrt{x}},\]then \[\frac{dy}{dx}=\] |
| A. | \[\frac{\sqrt{x}}{1-x}\] |
| B. | \[\frac{1}{\sqrt{x}(1-x)}\] |
| C. | \[\frac{\sqrt{x}}{1+x}\] |
| D. | \[\frac{1}{\sqrt{x}(1+x)}\] |
| Answer» C. \[\frac{\sqrt{x}}{1+x}\] | |
| 5860. |
If \[f(x)=\sqrt{1+{{\cos }^{2}}({{x}^{2}})}\], then \[f'\left( \frac{\sqrt{\pi }}{2} \right)\] is [Orissa JEE 2004] |
| A. | \[\sqrt{\pi }/6\] |
| B. | \[-\,\sqrt{(\pi /6)}\] |
| C. | \[1/\sqrt{6}\] |
| D. | \[\pi /\sqrt{6}\] |
| Answer» C. \[1/\sqrt{6}\] | |
| 5861. |
\[\frac{d}{dx}[(1+{{x}^{2}}){{\tan }^{-1}}x]=\] |
| A. | \[x{{\tan }^{-1}}x\] |
| B. | \[2{{\tan }^{-1}}x\] |
| C. | \[2x{{\tan }^{-1}}x+1\] |
| D. | \[x{{\tan }^{-1}}x+1\] |
| Answer» D. \[x{{\tan }^{-1}}x+1\] | |
| 5862. |
\[\frac{d}{dx}{{\log }_{7}}({{\log }_{7}}x)\]= |
| A. | \[\frac{1}{x{{\log }_{e}}x}\] |
| B. | \[\frac{{{\log }_{e}}7}{x{{\log }_{e}}x}\] |
| C. | \[\frac{{{\log }_{7}}e}{x{{\log }_{e}}x}\] |
| D. | \[\frac{{{\log }_{7}}e}{x{{\log }_{7}}x}\] |
| Answer» D. \[\frac{{{\log }_{7}}e}{x{{\log }_{7}}x}\] | |
| 5863. |
If \[y={{\tan }^{-1}}\frac{4x}{1+5{{x}^{2}}}+{{\tan }^{-1}}\frac{2+3x}{3-2x}\], then \[\frac{dy}{dx}=\] |
| A. | \[\frac{1}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] |
| B. | \[\frac{5}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] |
| C. | \[\frac{5}{1+25{{x}^{2}}}\] |
| D. | \[\frac{1}{1+25{{x}^{2}}}\] |
| Answer» D. \[\frac{1}{1+25{{x}^{2}}}\] | |
| 5864. |
If \[y=\log {{\left( \frac{1+x}{1-x} \right)}^{1/4}}-\frac{1}{2}{{\tan }^{-1}}x,\]then \[\frac{dy}{dx}=\] |
| A. | \[\frac{{{x}^{2}}}{1-{{x}^{4}}}\] |
| B. | \[\frac{2{{x}^{2}}}{1-{{x}^{4}}}\] |
| C. | \[\frac{{{x}^{2}}}{2\,\,(1-{{x}^{4}})}\] |
| D. | None of these |
| Answer» B. \[\frac{2{{x}^{2}}}{1-{{x}^{4}}}\] | |
| 5865. |
\[\frac{d}{dx}[{{\sin }^{n}}x\cos \,nx]=\] |
| A. | \[n{{\sin }^{n-1}}x\cos (n+1)x\] |
| B. | \[n{{\sin }^{n-1}}x\cos \,nx\] |
| C. | \[n{{\sin }^{n-1}}x\cos (n-1)x\] |
| D. | \[n{{\sin }^{n-1}}x\sin (n+1)x\] |
| Answer» B. \[n{{\sin }^{n-1}}x\cos \,nx\] | |
| 5866. |
If \[y=b\cos \log {{\left( \frac{x}{n} \right)}^{n}}\], then \[\frac{dy}{dx}=\] |
| A. | \[-n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] |
| B. | \[n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] |
| C. | \[-n\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5867. |
\[\frac{d}{dx}{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}=\] [AI CBSE 1980] |
| A. | \[1-\frac{1}{{{x}^{2}}}\] |
| B. | \[1+\frac{1}{{{x}^{2}}}\] |
| C. | \[1-\frac{1}{2x}\] |
| D. | None of these |
| Answer» B. \[1+\frac{1}{{{x}^{2}}}\] | |
| 5868. |
If \[y=\sqrt{(1-x)(1+x)}\], then |
| A. | \[(1-{{x}^{2}})\frac{dy}{dx}-xy=0\] |
| B. | \[(1-{{x}^{2}})\frac{dy}{dx}+xy=0\] |
| C. | \[(1-{{x}^{2}})\frac{dy}{dx}-2xy=0\] |
| D. | \[(1-{{x}^{2}})\frac{dy}{dx}+2xy=0\] |
| Answer» C. \[(1-{{x}^{2}})\frac{dy}{dx}-2xy=0\] | |
| 5869. |
\[\frac{d}{dx}\left( \frac{{{\cot }^{2}}x-1}{{{\cot }^{2}}x+1} \right)=\] |
| A. | \[-\sin 2x\] |
| B. | \[2\sin 2x\] |
| C. | \[2\cos 2x\] |
| D. | \[-2\sin 2x\] |
| Answer» E. | |
| 5870. |
\[\frac{d}{dx}\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}=\] [AISSE 1985; DSSE 1986] |
| A. | \[{{\sec }^{2}}x\] |
| B. | \[-{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] |
| C. | \[{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] |
| D. | \[{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] |
| Answer» C. \[{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] | |
| 5871. |
The differential coefficient of \[{{a}^{x}}+\log x.\sin x\]is |
| A. | \[{{a}^{x}}{{\log }_{e}}a+\frac{\sin x}{x}+\log x.\cos x\] |
| B. | \[{{a}^{x}}+\frac{\sin x}{x}+\cos x.\log x\] |
| C. | \[{{a}^{x}}\log a+\frac{\cos x}{x}+\sin x.\log x.\] |
| D. | None of these |
| Answer» B. \[{{a}^{x}}+\frac{\sin x}{x}+\cos x.\log x\] | |
| 5872. |
If \[y=x\text{ }\left[ \left( \cos \frac{x}{2}+\sin \frac{x}{2} \right)\text{ }\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)+\sin x \right]+\frac{1}{2\sqrt{x}}\], then \[\frac{dy}{dx}=\] |
| A. | \[(1+x)\cos x+(1-x)\sin x-\frac{1}{4x\sqrt{x}}\] |
| B. | \[(1-x)\cos x+(1+x)\sin x+\frac{1}{4x\sqrt{x}}\] |
| C. | \[(1+x)\cos x+(1+x)\sin x-\frac{1}{4x\sqrt{x}}\] |
| D. | None of these |
| Answer» B. \[(1-x)\cos x+(1+x)\sin x+\frac{1}{4x\sqrt{x}}\] | |
| 5873. |
If \[y=3{{x}^{5}}+4{{x}^{4}}+2x+3\], then |
| A. | \[{{y}_{4}}=0\] |
| B. | \[{{y}_{5}}=0\] |
| C. | \[{{y}_{6}}=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 5874. |
\[\frac{d}{dx}\log (\log x)\]= [IIT 1985] |
| A. | \[\frac{x}{\log x}\] |
| B. | \[\frac{\log x}{x}\] |
| C. | \[{{(x\log x)}^{-1}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5875. |
If \[f(x)=mx+c,f(0)=f'(0)=1\]then \[f(2)=\] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | ? 3 |
| Answer» D. ? 3 | |
| 5876. |
If \[y={{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right)\] then \[\frac{dy}{dx}=\] [Kerala (Engg.) 2005] |
| A. | 2 |
| B. | ? 1 |
| C. | \[\frac{a}{b}\] |
| D. | 0 |
| E. | \[\frac{b}{a}\] |
| Answer» C. \[\frac{a}{b}\] | |
| 5877. |
The derivative of \[y=(1-x)\,(2-x)....(n-x)\] at \[x=1\] is equal to [Kerala (Engg.) 2005] |
| A. | 0 |
| B. | \[(-1)\,(n-1)\,!\] |
| C. | \[n!\,-\,1\] |
| D. | \[{{(-1)}^{n-1}}(n-1)\,!\] |
| E. | \[{{(-1)}^{n}}\,(n-1)\,!\] |
| Answer» C. \[n!\,-\,1\] | |
| 5878. |
If \[f(x)=\cos x\cos 2x\cos 4x\cos 8x\cos 16x\], then \[{f}'\left( \frac{\pi }{4} \right)\] is [AMU 2005] |
| A. | \[\sqrt{2}\] |
| B. | \[\frac{1}{\sqrt{2}}\] |
| C. | 1 |
| D. | \[\frac{\sqrt{3}}{2}\] |
| Answer» B. \[\frac{1}{\sqrt{2}}\] | |
| 5879. |
\[\frac{d}{dx}\,\,\left[ {{\tan }^{-1}}\left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \right]\]= [Kerala (Engg.) 2005] |
| A. | \[\frac{1}{2(1+x)\,\sqrt{x}}\] |
| B. | \[\frac{3}{(1+x)\,\sqrt{x}}\] |
| C. | \[\frac{2}{(1+x)\,\sqrt{x}}\] |
| D. | \[\frac{2\sqrt{2}y-3}{2\sqrt{2}}=\frac{-3\sqrt{2}\times +3}{2\sqrt{2}}\] |
| E. | \[\frac{3}{2(1+x)\sqrt{x}}\] |
| Answer» F. | |
| 5880. |
\[f(x)={{x}^{2}}-3x\], then the points at which \[f(x)=f'(x)\]are |
| A. | 1, 3 |
| B. | 1, ? 3 |
| C. | ? 1, 3 |
| D. | None of these |
| Answer» E. | |
| 5881. |
Let \[f(x)\] be a polynomial function of the second degree. If \[f(1)=f(-1)\] and \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] are in A.P. then \[{f}'({{a}_{1}})\], \[{f}'({{a}_{2}})\], \[{f}'({{a}_{3}})\] are in [AMU 2005] |
| A. | A.P |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 5882. |
The derivative of function \[f(x)\] is \[{{\tan }^{4}}x\]. If \[f(0)=0\] then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f(x)}{x}\]is equal to [J & K 2005] |
| A. | 1 |
| B. | 0 |
| C. | ?1 |
| D. | None of these |
| Answer» C. ?1 | |
| 5883. |
If \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}}\]then \[{{\left. \frac{dy}{dx} \right|}_{x=1,y=2}}\] is equal to [J & K 2005] |
| A. | ½ |
| B. | 2 |
| C. | 2m/n |
| D. | m/ 2n |
| Answer» C. 2m/n | |
| 5884. |
If \[y={{\cos }^{-1}}\cos (|x|-f(x)),\] where \[\] \[f(x)\left\{ \begin{align} & =1\,,\,\text{if}\,\,\,x>0 \\ & =-1\,,\,\text{if}\,\,\,x |
| A. | ? 1 |
| B. | 1 |
| C. | 0 |
| D. | Indeterminate |
| Answer» C. 0 | |
| 5885. |
If \[y={{\tan }^{-1}}(\sec x-\tan x)\]then \[\frac{dy}{dx}=\] [Karnataka CET 2004] |
| A. | 2 |
| B. | ?2 |
| C. | ½ |
| D. | ?1/2 |
| Answer» C. ½ | |
| 5886. |
If \[y={{(\cos {{x}^{2}})}^{2}}\]then \[\frac{dy}{dx}\]is equal to [Pb. CET 2004] |
| A. | \[-4x\sin 2{{x}^{2}}\] |
| B. | \[-x\sin {{x}^{2}}\] |
| C. | \[-2x\sin 2{{x}^{2}}\] |
| D. | \[-x\cos 2{{x}^{2}}\] |
| Answer» D. \[-x\cos 2{{x}^{2}}\] | |
| 5887. |
If \[y={{\cot }^{-1}}({{x}^{2}})\], then \[\frac{dy}{dx}\] is equal to [Pb. CET 2002] |
| A. | \[\frac{2x}{1+{{x}^{4}}}\] |
| B. | \[\frac{2x}{\sqrt{1+4x}}\] |
| C. | \[\frac{-2x}{1+{{x}^{4}}}\] |
| D. | \[\frac{-2x}{\sqrt{1+{{x}^{2}}}}\] |
| Answer» D. \[\frac{-2x}{\sqrt{1+{{x}^{2}}}}\] | |
| 5888. |
If \[y=a\sin x+b\cos x,\]then \[{{y}^{2}}+{{\left( \frac{dy}{dx} \right)}^{2}}\]is a |
| A. | Function of x |
| B. | Function of y |
| C. | Function of x and y |
| D. | Constant |
| Answer» E. | |
| 5889. |
If \[\sin y+{{e}^{-x\,\cos y}}=e,\]then \[\frac{dy}{dx}\] at \[(1,\pi )\] is [Kerala (Engg.) 2002] |
| A. | \[\sin y\] |
| B. | \[-x\cos y\] |
| C. | \[e\] |
| D. | \[\sin y-x\,\cos y\] |
| Answer» D. \[\sin y-x\,\cos y\] | |
| 5890. |
Derivative of the function \[f(x)={{\log }_{5}}({{\log }_{7}}x)\], \[x>7\] is [Orissa JEE 2002] |
| A. | \[\frac{1}{x(\text{In}\,\text{5)(In}\,\text{7)(lo}{{\text{g}}_{\text{7}}}x)}\] |
| B. | \[\frac{1}{x(\text{ln}\,\text{5)(ln}\,\text{7)}}\] |
| C. | \[\frac{1}{x(In\,x)}\] |
| D. | None of these |
| Answer» B. \[\frac{1}{x(\text{ln}\,\text{5)(ln}\,\text{7)}}\] | |
| 5891. |
If \[f(x)=\sqrt{ax}+\frac{{{a}^{2}}}{\sqrt{ax}},\]then \[f'(a)=\] [EAMCET 2002] |
| A. | ? 1 |
| B. | 1 |
| C. | 0 |
| D. | a |
| Answer» D. a | |
| 5892. |
Derivative of \[{{x}^{6}}+{{6}^{x}}\]with respect to x is [Kerala (Engg.) 2002] |
| A. | \[12x\] |
| B. | \[x+4\] |
| C. | \[6{{x}^{5}}+{{6}^{x}}\,\,\log 6\] |
| D. | \[6{{x}^{5}}+x{{6}^{x-1}}\] |
| Answer» D. \[6{{x}^{5}}+x{{6}^{x-1}}\] | |
| 5893. |
If \[x=\exp \left\{ {{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right) \right\}\,\,\], then \[\frac{dy}{dx}\]equals [MP PET 2002] |
| A. | \[2x\,[1+\tan \,(\log x)]+x{{\sec }^{2}}(\log x)\] |
| B. | \[x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] |
| C. | \[2x\,[1+\tan \,(\log x)]+{{x}^{2}}\,\,{{\sec }^{2}}(\log x)\] |
| D. | \[2x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] |
| Answer» B. \[x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] | |
| 5894. |
If \[y=\sec ({{\tan }^{-1}}x),\]then \[\frac{dy}{dx}\] is [DCE 2002; Kurukshetra CEE 2001] |
| A. | \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] |
| B. | \[\frac{-x}{\sqrt{1+{{x}^{2}}}}\] |
| C. | \[\frac{x}{\sqrt{1-{{x}^{2}}}}\] |
| D. | None of these |
| Answer» B. \[\frac{-x}{\sqrt{1+{{x}^{2}}}}\] | |
| 5895. |
The differential coefficient of the function \[|x-1|+|x-3|\] at the point \[x=2\] is [RPET 2002; Pb. CET 2000, 04] |
| A. | ? 2 |
| B. | 0 |
| C. | 2 |
| D. | Undefined |
| Answer» C. 2 | |
| 5896. |
\[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]\] equals to [RPET 2001] |
| A. | 1 |
| B. | \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] |
| C. | \[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] |
| D. | \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] |
| Answer» D. \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] | |
| 5897. |
\[\frac{d}{dx}\log |x|\text{ }=......,(x\ne 0)\] |
| A. | \[\frac{1}{x}\] |
| B. | \[-\frac{1}{x}\] |
| C. | x |
| D. | \[-x\] |
| Answer» B. \[-\frac{1}{x}\] | |
| 5898. |
\[\frac{d}{dx}\left[ {{\tan }^{-1}}\left( \frac{a-x}{1+ax} \right) \right]=\] [Karnataka CET 2001; Pb. CET 2001] |
| A. | \[-\frac{1}{1+{{x}^{2}}}\] |
| B. | \[\frac{1}{1+{{a}^{2}}}-\frac{1}{1+{{x}^{2}}}\] |
| C. | \[\frac{1}{1+{{\left( \frac{a-x}{1+ax} \right)}^{2}}}\] |
| D. | \[\frac{-1}{\sqrt{1-{{\left( \frac{a-x}{1+ax} \right)}^{2}}}}\] |
| Answer» B. \[\frac{1}{1+{{a}^{2}}}-\frac{1}{1+{{x}^{2}}}\] | |
| 5899. |
If \[y=\frac{a+b{{x}^{3/2}}}{{{x}^{5/4}}}\] and \[{y}'=0\] at \[x=5\], then the ratio \[a:b\] is equal to [AMU 2001] |
| A. | \[\sqrt{5}:1\] |
| B. | 5 : 2 |
| C. | 3 : 5 |
| D. | 1 : 2 |
| Answer» B. 5 : 2 | |
| 5900. |
If \[y={{\tan }^{-1}}\left[ \frac{\sin x+\cos x}{\cos x-\sin x} \right]\,,\] then \[\frac{dy}{dx}\] is [UPSEAT 2001] |
| A. | \[1/2\] |
| B. | \[\pi /4\] |
| C. | 0 |
| D. | 1 |
| Answer» E. | |