MCQOPTIONS
Saved Bookmarks
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.
| 5901. |
If \[y=\sqrt{\sin x+y},\] then \[\frac{dy}{dx}\] equals to [RPET 2001] |
| A. | \[\frac{\sin x}{2y-1}\] |
| B. | \[\frac{\cos x}{2y-1}\] |
| C. | \[\frac{\sin x}{2y+1}\] |
| D. | \[\frac{\cos x}{2y+1}\] |
| Answer» C. \[\frac{\sin x}{2y+1}\] | |
| 5902. |
If \[x=y\sqrt{1-{{y}^{2}},}\]then \[\frac{dy}{dx}=\] [MP PET 2001] |
| A. | 0 |
| B. | x |
| C. | \[\frac{\sqrt{1-{{y}^{2}}}}{1-2{{y}^{2}}}\] |
| D. | \[\frac{\sqrt{1-{{y}^{2}}}}{1+2{{y}^{2}}}\] |
| Answer» D. \[\frac{\sqrt{1-{{y}^{2}}}}{1+2{{y}^{2}}}\] | |
| 5903. |
The derivative of \[f(x)=|x{{|}^{3}}\] at \[x=0\] is [RPET 2001; Kurukshetra CEE 2002] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | Not defined |
| Answer» B. 1 | |
| 5904. |
Given that \[\frac{d}{dx}f(x)=f\,'(x)\]. The relationship \[f\,'(a+b)=f\,'(a)+f\,'(b)\] is valid if \[f(x)\] is equal to [AMU 2000] |
| A. | \[x\] |
| B. | \[{{x}^{2}}\] |
| C. | \[{{x}^{3}}\] |
| D. | \[{{x}^{4}}\] |
| Answer» C. \[{{x}^{3}}\] | |
| 5905. |
If \[f(x)\] has a derivative at \[x=a,\]then \[\underset{x\to a}{\mathop{\lim }}\,\frac{xf(a)-af(x)}{x-a}\] is equal to [AMU 2000] |
| A. | \[f(a)-a\,f\,'(a)\] |
| B. | \[a\,f(a)-f\,'(a)\] |
| C. | \[f(a)+f'(a)\] |
| D. | \[a\,f(a)+f\,'(a)\] |
| Answer» B. \[a\,f(a)-f\,'(a)\] | |
| 5906. |
If \[y={{\tan }^{-1}}\left( \frac{\sqrt{x}-x}{1+{{x}^{3/2}}} \right),\]then \[y'(1)\] is [AMU 2000] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | ? 1 |
| D. | \[-\frac{1}{4}\] |
| Answer» E. | |
| 5907. |
\[\frac{d}{dx}\left[ \left( \frac{{{\tan }^{2}}2x-{{\tan }^{2}}x}{1-{{\tan }^{2}}2x{{\tan }^{2}}x} \right)\cot 3x \right]\] [AMU 2000] |
| A. | \[\tan 2x\,\tan x\] |
| B. | \[\tan 3x\tan x\] |
| C. | \[{{\sec }^{2}}x\] |
| D. | \[\sec x\tan x\] |
| Answer» D. \[\sec x\tan x\] | |
| 5908. |
\[\frac{d}{dx}{{\log }_{\sqrt{x}}}(1/x)\]is equal to [AMU 1999] |
| A. | \[-\frac{1}{2\sqrt{x}}\] |
| B. | ? 2 |
| C. | \[-\frac{1}{{{x}^{2}}\sqrt{x}}\] |
| D. | 0 |
| Answer» E. | |
| 5909. |
If \[f(1)=3,\,{f}'(1)=2,\]then \[\frac{d}{dx}\{\log f\,({{e}^{x}}+2x)\}\] at \[x=0\] is [AMU 1999] |
| A. | 2 / 3 |
| B. | 3 / 2 |
| C. | 2 |
| D. | 0 |
| Answer» D. 0 | |
| 5910. |
The derivative of \[f(x)=\,|{{x}^{2}}-x|\] at x = 2 is [AMU 1999] |
| A. | ? 3 |
| B. | 0 |
| C. | 3 |
| D. | Not defined |
| Answer» D. Not defined | |
| 5911. |
If \[f(x)={{\cos }^{-1}}\left[ \frac{1-{{(\log x)}^{2}}}{1+{{(\log x)}^{2}}} \right]\,,\]then the value of \[f'(e)=\] [Karnataka CET 1999; Pb. CET 2000] |
| A. | 1 |
| B. | 1/e |
| C. | 2/e |
| D. | \[\frac{2}{{{e}^{2}}}\] |
| Answer» C. 2/e | |
| 5912. |
The derivative of \[\sqrt{\sqrt{x}+1}\]is [SCRA 1996] |
| A. | \[\frac{1}{\sqrt{x}(\sqrt{x}+1)}\] |
| B. | \[\frac{1}{\sqrt{x}\sqrt{x+1}}\] |
| C. | \[\frac{4}{\sqrt{x(\sqrt{x}+1)}}\] |
| D. | \[\frac{1}{4\sqrt{x(\sqrt{x}+1)}}\] |
| Answer» E. | |
| 5913. |
The derivative of \[y=1-|x|\]at \[x=0\]is [SCRA 1996] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | Does not exist |
| Answer» E. | |
| 5914. |
If \[y=\cos (\sin {{x}^{2}}),\]then at \[x=\sqrt{\frac{\pi }{2}},\frac{dy}{dx}\]= |
| A. | ? 2 |
| B. | 2 |
| C. | \[-2\sqrt{\frac{\pi }{2}}\] |
| D. | 0 |
| Answer» E. | |
| 5915. |
The derivative of \[f(x)=x|x|\] is [SCRA 1996] |
| A. | \[2x\] |
| B. | ? 2x |
| C. | \[2{{x}^{2}}\] |
| D. | \[2|x|\] |
| Answer» E. | |
| 5916. |
If \[f(x)=\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}+\sqrt{{{x}^{2}}+{{b}^{2}}}}\], then \[{f}'(x)\] is equal to [Kurukshetra CEE 1998] |
| A. | \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] |
| B. | \[\frac{x}{({{a}^{2}}+{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] |
| C. | \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}+\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] |
| D. | \[({{a}^{2}}+{{b}^{2}})\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] |
| Answer» B. \[\frac{x}{({{a}^{2}}+{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] | |
| 5917. |
\[\frac{d}{dx}\cos \,{{\text{h}}^{-1}}(\sec x)=\] [RPET 1997] |
| A. | sec x |
| B. | sin x |
| C. | tan x |
| D. | cosec x |
| Answer» B. sin x | |
| 5918. |
\[\frac{d}{dx}(\sin 2{{x}^{2}})\] equals [RPET 1996] |
| A. | \[4x\cos \,(2{{x}^{2}})\] |
| B. | \[2\sin {{x}^{2}}\cos {{x}^{2}}\] |
| C. | \[4x\sin ({{x}^{2}})\] |
| D. | \[4x\sin ({{x}^{2}})\cos ({{x}^{2}})\] |
| Answer» B. \[2\sin {{x}^{2}}\cos {{x}^{2}}\] | |
| 5919. |
If \[y=\frac{{{(1-x)}^{2}}}{{{x}^{2}}}\], then \[\frac{dy}{dx}\]is [MP PET 1999] |
| A. | \[\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] |
| B. | \[-\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] |
| C. | \[-\frac{2}{{{x}^{2}}}-\frac{2}{{{x}^{3}}}\] |
| D. | \[-\frac{2}{{{x}^{3}}}+\frac{2}{{{x}^{2}}}\] |
| Answer» E. | |
| 5920. |
If \[pv=81\], then \[\frac{dp}{dv}\] is at v = 9 equal to [MP PET 1999] |
| A. | 1 |
| B. | ?1 |
| C. | 2 |
| D. | None of these |
| Answer» C. 2 | |
| 5921. |
The values of x, at which the first derivative of the function \[{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}\]w.r.t. x is \[\frac{3}{4}\], are [MP PET 1998] |
| A. | \[\pm \,2\] |
| B. | \[\pm \frac{1}{2}\] |
| C. | \[\pm \frac{\sqrt{3}}{2}\] |
| D. | \[\pm \frac{2}{\sqrt{3}}\] |
| Answer» B. \[\pm \frac{1}{2}\] | |
| 5922. |
The first derivative of the function \[(\sin 2x\cos 2x\cos 3x+{{\log }_{2}}{{2}^{x+3}})\] with respect to x at \[x=\pi \]is [MP PET 1998] |
| A. | 2 |
| B. | ?1 |
| C. | \[-2+{{2}^{\pi }}{{\log }_{e}}2\] |
| D. | \[-2+{{\log }_{e}}2\] |
| Answer» C. \[-2+{{2}^{\pi }}{{\log }_{e}}2\] | |
| 5923. |
If\[y={{\log }_{10}}{{x}^{2}}\], then \[\frac{dy}{dx}\] is equal to |
| A. | \[\frac{2}{x}\] |
| B. | \[\frac{2}{x{{\log }_{e}}10}\] |
| C. | \[\frac{1}{x{{\log }_{e}}10}\] |
| D. | \[\frac{1}{10x}\] |
| Answer» C. \[\frac{1}{x{{\log }_{e}}10}\] | |
| 5924. |
If \[y={{3}^{{{x}^{2}}}}\], then \[\frac{dy}{dx}\] is equal to |
| A. | \[({{x}^{2}}){{3}^{{{x}^{2}}-1}}\] |
| B. | \[3{{x}^{2}}.2x\] |
| C. | \[{{3}^{{{x}^{2}}}}.2x.\log 3\] |
| D. | \[({{x}^{2}}-1).3\] |
| Answer» D. \[({{x}^{2}}-1).3\] | |
| 5925. |
\[\frac{d}{dx}\left( {{x}^{2}}\sin \frac{1}{x} \right)=\] |
| A. | \[\cos \,\left( \frac{1}{x} \right)+2x\sin \left( \frac{1}{x} \right)\] |
| B. | \[2x\sin \left( \frac{1}{x} \right)-\cos \left( \frac{1}{x} \right)\] |
| C. | \[\cos \left( \frac{1}{x} \right)-2x\sin \left( \frac{1}{x} \right)\] |
| D. | None of these |
| Answer» C. \[\cos \left( \frac{1}{x} \right)-2x\sin \left( \frac{1}{x} \right)\] | |
| 5926. |
If \[y=\sqrt{\sin \sqrt{x}}\], then \[\frac{dy}{dx}=\] [MP PET 1997] |
| A. | \[\frac{1}{2\sqrt{\cos \sqrt{x}}}\] |
| B. | \[\frac{\sqrt{\cos \sqrt{x}}}{2x}\] |
| C. | \[\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}\] |
| D. | \[\frac{1}{2\sqrt{\sin x}}\] |
| Answer» D. \[\frac{1}{2\sqrt{\sin x}}\] | |
| 5927. |
If \[y=\sec {{x}^{0}}\], then \[\frac{dy}{dx}=\] [MP PET 1997] |
| A. | \[\sec x\tan x\] |
| B. | \[\sec {{x}^{o}}\tan {{x}^{o}}\] |
| C. | \[\frac{\pi }{180}\sec {{x}^{o}}\tan {{x}^{o}}\] |
| D. | \[\frac{180}{\pi }\sec {{x}^{o}}\tan {{x}^{o}}\] |
| Answer» D. \[\frac{180}{\pi }\sec {{x}^{o}}\tan {{x}^{o}}\] | |
| 5928. |
If \[f(x)={{e}^{x}}g(x),g(0)=2,g'(0)=1\], then \[f'(0)\]is |
| A. | 1 |
| B. | 3 |
| C. | 2 |
| D. | 0 |
| Answer» C. 2 | |
| 5929. |
If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], then \[\frac{dy}{dx}=\] |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{2}{3}\] |
| C. | 3 |
| D. | 1 |
| Answer» B. \[\frac{2}{3}\] | |
| 5930. |
If \[y={{e}^{x}}\log x\], then \[\frac{dy}{dx}\]is [SCRA 1996] |
| A. | \[\frac{{{e}^{x}}}{x}\] |
| B. | \[{{e}^{x}}\left( \frac{1}{x}+x\log x \right)\] |
| C. | \[{{e}^{x}}\left( \frac{1}{x}+\log x \right)\] |
| D. | \[\frac{{{e}^{x}}}{\log x}\] |
| Answer» D. \[\frac{{{e}^{x}}}{\log x}\] | |
| 5931. |
For the function \[f(x)={{x}^{2}}-6x+8,2\le x\le 4\], the value of x for which \[f'(x)\] vanishes, is [MP PET 1996] |
| A. | \[\frac{9}{4}\] |
| B. | \[\frac{5}{2}\] |
| C. | 3 |
| D. | \[\frac{7}{2}\] |
| Answer» D. \[\frac{7}{2}\] | |
| 5932. |
If \[y={{e}^{(1+{{\log }_{e}}x)}}\], then the value of \[\frac{dy}{dx}=\] [MP PET 1996; Pb. CET 2001] |
| A. | e |
| B. | 1 |
| C. | 0 |
| D. | \[{{\log }_{e}}x\,\,{{e}^{{{\log }_{e}}ex}}\] |
| Answer» B. 1 | |
| 5933. |
Differential coefficient of \[\sqrt{\sec \sqrt{x}}\]is [MP PET 1996] |
| A. | \[\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] |
| B. | \[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\] |
| C. | \[\frac{1}{2}\sqrt{x}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] |
| D. | \[\frac{1}{2}\sqrt{x}\sec \sqrt{x}\sin \sqrt{x}\] |
| Answer» B. \[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\] | |
| 5934. |
\[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x+2}{x-2} \right)}^{3/4}} \right\} \right]\] equals |
| A. | \[\frac{{{x}^{2}}-7}{{{x}^{2}}-4}\] |
| B. | 1 |
| C. | \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] |
| D. | \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] |
| Answer» B. 1 | |
| 5935. |
\[\frac{d}{dx}[\cos {{(1-{{x}^{2}})}^{2}}]\]= [AISSE 1981; AI CBSE 1979] |
| A. | \[-2x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] |
| B. | \[-4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] |
| C. | \[4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] |
| D. | \[-2(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] |
| Answer» D. \[-2(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] | |
| 5936. |
If \[y={{\log }_{\cos x}}\sin x\], then \[\frac{dy}{dx}\]is equal to |
| A. | \[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \cos x)}^{2}}}\] |
| B. | \[\frac{\tan x\log \cos x+\cot x\log \sin x}{{{(\log \cos x)}^{2}}}\] |
| C. | \[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \sin x)}^{2}}}\] |
| D. | None of these |
| Answer» B. \[\frac{\tan x\log \cos x+\cot x\log \sin x}{{{(\log \cos x)}^{2}}}\] | |
| 5937. |
The derivative of tanx ? x with respect to x is [SCRA 1996] |
| A. | \[1-{{\tan }^{2}}x\] |
| B. | tan x |
| C. | \[-{{\tan }^{2}}x\] |
| D. | \[{{\tan }^{2}}x\] |
| Answer» E. | |
| 5938. |
If \[y\sqrt{{{x}^{2}}+1}=\log \left\{ \sqrt{{{x}^{2}}+1}-x \right\}\], then\[({{x}^{2}}+1)\frac{dy}{dx}+xy+1=\] [Roorkee 1978; Kurukshetra CEE 1998] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | None of these |
| Answer» B. 1 | |
| 5939. |
If \[y={{x}^{n}}\log x+x{{(\log x)}^{n}}\], then \[\frac{dy}{dx}=\] |
| A. | \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] |
| B. | \[{{x}^{n-2}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] |
| C. | \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n-\log x]\] |
| D. | None of these |
| Answer» B. \[{{x}^{n-2}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] | |
| 5940. |
If \[y={{\sin }^{-1}}\sqrt{x}\], then \[\frac{dy}{dx}=\] [MP PET 1995] |
| A. | \[\frac{2}{\sqrt{x}\sqrt{1-x}}\] |
| B. | \[\frac{-2}{\sqrt{x}\sqrt{1-x}}\] |
| C. | \[\frac{1}{2\sqrt{x}\sqrt{1-x}}\] |
| D. | \[\frac{1}{\sqrt{1-x}}\] |
| Answer» D. \[\frac{1}{\sqrt{1-x}}\] | |
| 5941. |
\[\frac{d}{dx}\left[ \log \left( x+\frac{1}{x} \right) \right]=\] [MP PET 1995] |
| A. | \[\left( x+\frac{1}{x} \right)\] |
| B. | \[\frac{\left( 1+\frac{1}{{{x}^{2}}} \right)}{\left( 1+\frac{1}{x} \right)}\] |
| C. | \[\frac{\left( 1-\frac{1}{{{x}^{2}}} \right)}{\left( x+\frac{1}{x} \right)}\] |
| D. | \[\left( 1+\frac{1}{x} \right)\] |
| Answer» D. \[\left( 1+\frac{1}{x} \right)\] | |
| 5942. |
If \[f(2)=4\], \[f'(2)=1\]then \[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}=\] [RPET 1995, 2000] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | ?2 |
| Answer» C. 3 | |
| 5943. |
If \[y=\frac{{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\], then \[(1-{{x}^{2}})\frac{dy}{dx}\] is equal to [RPET 1995] |
| A. | \[x+y\] |
| B. | \[1+xy\] |
| C. | 1? xy |
| D. | \[xy-2\] |
| Answer» C. 1? xy | |
| 5944. |
If \[y={{\tan }^{-1}}\sqrt{\frac{1+\cos x}{1-\cos x}}\], then \[\frac{dy}{dx}\] is equal to [Roorkee 1995] |
| A. | 0 |
| B. | \[-\frac{1}{2}\] |
| C. | ½ |
| D. | 1 |
| Answer» C. ½ | |
| 5945. |
\[\frac{d}{dx}({{\sin }^{-1}}x)\] is equal to [RPET 1995] |
| A. | \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] |
| B. | \[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] |
| C. | \[\frac{1}{\sqrt{1+{{x}^{2}}}}\] |
| D. | \[\frac{-1}{\sqrt{1+{{x}^{2}}}}\] |
| Answer» B. \[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] | |
| 5946. |
\[\frac{d}{dx}({{e}^{{{x}^{3}}}})\] is equal to [RPET 1995] |
| A. | \[3x{{e}^{{{x}^{3}}}}\] |
| B. | \[3{{x}^{2}}{{e}^{{{x}^{3}}}}\] |
| C. | \[3x{{\left( {{e}^{{{x}^{3}}}} \right)}^{2}}\] |
| D. | \[2{{x}^{2}}{{e}^{{{x}^{3}}}}\] |
| Answer» C. \[3x{{\left( {{e}^{{{x}^{3}}}} \right)}^{2}}\] | |
| 5947. |
If\[y={{\log }_{\sin x}}(\tan x),\] then \[{{\left( \frac{dy}{dx} \right)}_{\pi /4}}=\] |
| A. | \[\frac{4}{\log 2}\] |
| B. | \[-4\log 2\] |
| C. | \[\frac{-4}{\log 2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5948. |
If \[f(x)=(x-{{x}_{0}})g(x)\], where \[g(x)\] is continuous at \[{{x}_{0}}\], then \[f'({{x}_{0}})\] is equal to |
| A. | 0 |
| B. | \[{{x}_{0}}\] |
| C. | \[g({{x}_{0}})\] |
| D. | None of these |
| Answer» D. None of these | |
| 5949. |
If \[y=\frac{{{a}^{{{\cos }^{-1}}x}}}{1+{{a}^{{{\cos }^{-1}}x}}}\]and \[z={{a}^{{{\cos }^{-1}}x}}\], then \[\frac{dy}{dx}\]= [MP PET 1994] |
| A. | \[\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] |
| B. | \[-\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] |
| C. | \[\frac{1}{{{(1+{{a}^{{{\cos }^{-1}}x}})}^{2}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5950. |
\[\frac{d}{dx}(\log \tan x)=\] [MNR 1986] |
| A. | \[2\sec 2x\] |
| B. | \[2\,\text{cosec }2x\] |
| C. | \[\sec 2x\] |
| D. | \[\text{cosec}\,2x\] |
| Answer» C. \[\sec 2x\] | |