Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

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.

4901.

The value of the definite integral \[\int_{0}^{1}{\frac{dx}{{{x}^{2}}+2x\cos \alpha +1}}\] for \[0

A.                 \[\sin \alpha \] 
B.                 \[{{\tan }^{-1}}(\sin \alpha )\]
C.                 \[\alpha \sin \alpha \]   
D.                 \[\frac{\alpha }{2}{{(\sin \alpha )}^{-1}}\]
Answer» E.
Discussion
4902.

\[\int_{0}^{\pi /2}{\frac{\sin x\cos x\,dx}{{{\cos }^{2}}x+3\cos x+2}}=\]                                 [MNR 1981]

A.                 \[\log \left( \frac{8}{9} \right)\]
B.                 \[\log \left( \frac{9}{8} \right)\]
C.                 \[\log (8\times 9)\]        
D.                 None of these
Answer» C.                 \[\log (8\times 9)\]        
Discussion
4903.

 \[\int_{0}^{\pi /6}{\frac{\sin x}{{{\cos }^{3}}x}\,dx=}\]                                  [SCRA 1979]

A.                 \[\frac{2}{3}\]   
B.                 \[\frac{1}{6}\]
C.                 2             
D.                 \[\frac{1}{3}\]
Answer» C.                 2             
Discussion
4904.

\[\int_{0}^{2}{\frac{{{x}^{3}}\,dx}{{{({{x}^{2}}+1)}^{\frac{3}{2}}}}}=\]

A.                 \[{{(\sqrt{2}-1)}^{2}}\]  
B.                 \[\frac{{{(\sqrt{2}-1)}^{2}}}{\sqrt{2}}\]
C.                 \[\frac{\sqrt{2}-1}{\sqrt{2}}\]    
D.                 None of these
Answer» E.
Discussion
4905.

\[\int_{0}^{\pi /4}{{{\tan }^{6}}x{{\sec }^{2}}x\,dx=}\]                                    [BIT Ranchi 1981]

A.                 \[\frac{1}{7}\]   
B.                 \[\frac{2}{7}\]
C.                 1             
D.                 None of these
Answer» B.                 \[\frac{2}{7}\]
Discussion
4906.

\[\int_{0}^{\pi /2}{\frac{\sin x\cos x}{1+{{\sin }^{4}}x}\,dx=}\]                                   [AISSE 1988]

A.                 \[\frac{\pi }{2}\]              
B.                 \[\frac{\pi }{4}\]
C.                 \[\frac{\pi }{6}\]              
D.                 \[\frac{\pi }{8}\]
Answer» E.
Discussion
4907.

\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,dx=}\]                                              [Roorkee 1978]

A.                 \[\frac{1}{2}({{e}^{\pi /2}}-1)\]
B.                 \[\frac{1}{2}({{e}^{\pi /2}}+1)\]
C.                 \[\frac{1}{2}(1-{{e}^{\pi /2}})\]
D.                 \[2({{e}^{\pi /2}}+1)\]
Answer» C.                 \[\frac{1}{2}(1-{{e}^{\pi /2}})\]
Discussion
4908.

\[\int_{0}^{\pi /6}{(2+3{{x}^{2}})\cos 3x\,dx=}\]                                               [DSSE 1985]

A.                 \[\frac{1}{36}(\pi +16)\]               
B.                 \[\frac{1}{36}(\pi -16)\]
C.                 \[\frac{1}{36}({{\pi }^{2}}-16)\] 
D.                 \[\frac{1}{36}({{\pi }^{2}}+16)\]
Answer» E.
Discussion
4909.

\[\int_{0}^{\pi /2}{\frac{\cos x}{1+\cos x+\sin x}}\,dx=\]                                               [Roorkee 1989]

A.                 \[\frac{\pi }{4}+\frac{1}{2}\log 2\]           
B.                 \[\frac{\pi }{4}+\log 2\]
C.                 \[\frac{\pi }{4}-\frac{1}{2}\log 2\]            
D.                 \[\frac{\pi }{4}-\log 2\]
Answer» D.                 \[\frac{\pi }{4}-\log 2\]
Discussion
4910.

\[\int_{0}^{1}{{{\cos }^{-1}}x\,dx=}\]                                      [DSSE 1988]

A.                 0             
B.                 1
C.                 2             
D.                 None of these
Answer» C.                 2             
Discussion
4911.

\[\int_{0}^{2\pi }{\sqrt{1+\sin \frac{x}{2}}\,dx=}\]            [MNR 1987; UPSEAT 2000]

A.                 0             
B.                 2
C.                 8             
D.                 4
Answer» D.                 4
Discussion
4912.

\[\int_{0}^{\pi /8}{\frac{{{\sec }^{2}}2x}{2}\,dx=}\]

A.                 \[\frac{1}{4}\]   
B.                 \[\frac{1}{3}\]
C.                 \[\frac{1}{2}\]   
D.                 None of these
Answer» B.                 \[\frac{1}{3}\]
Discussion
4913.

 \[\int_{0}^{\pi }{\frac{dx}{1+\sin x}}=\]                                                [CEE 1993]

A.                 0             
B.                 \[\frac{1}{2}\]
C.                 2             
D.                 \[\frac{3}{2}\]
Answer» D.                 \[\frac{3}{2}\]
Discussion
4914.

 \[\int_{0}^{2}{\sqrt{\frac{2+x}{2-x}}}\,dx=\]                                      [MNR 1984; CEE 1993]

A.                 \[\pi +2\]            
B.                 \[\pi +\frac{3}{2}\]
C.                 \[\pi +1\]            
D.                 None of these
Answer» B.                 \[\pi +\frac{3}{2}\]
Discussion
4915.

 \[\int_{\pi \text{/4}}^{\pi \text{/2}}{{{e}^{x}}(\log \sin x+\cot x)\,dx=}\]                                             [AI CBSE 1991]

A.                 \[{{e}^{\pi /4}}\log 2\]  
B.                 \[-{{e}^{\pi /4}}\log 2\]
C.                 \[\frac{1}{2}{{e}^{\pi /4}}\log 2\]             
D.                 \[-\frac{1}{2}{{e}^{\pi /4}}\log 2\]
Answer» D.                 \[-\frac{1}{2}{{e}^{\pi /4}}\log 2\]
Discussion
4916.

 \[\int_{0}^{\pi /4}{\frac{\sin x+\cos x}{9+16\sin 2x}\,dx=}\]                                        [IIT 1983]

A.                 \[\frac{1}{20}\log 3\]     
B.                 \[\log 3\]
C.                 \[\frac{1}{20}\log 5\]
D.                 None of these
Answer» B.                 \[\log 3\]
Discussion
4917.

\[\int_{0}^{\pi /2}{\frac{x+\sin x}{1+\cos x}\,dx=}\]                                              [MP PET 1989]

A.                 \[-\log 2\]
B.                 \[\log 2\]
C.                 \[\frac{\pi }{2}\]
D.                 0
Answer» D.                 0
Discussion
4918.

\[\int_{0}^{2\pi }{{{e}^{x/2}}.\sin \left( \frac{x}{2}+\frac{\pi }{4} \right)\,dx=}\]                                 [Roorkee 1982]

A.                 1             
B.                 \[2\sqrt{2}\]
C.                 0             
D.                 None of these
Answer» D.                 None of these
Discussion
4919.

\[\int_{0}^{a}{\frac{{{x}^{4}}\,dx}{{{({{a}^{2}}+{{x}^{2}})}^{4}}}}=\]

A.                 \[\frac{1}{16{{a}^{3}}}\left( \frac{\pi }{4}-\frac{1}{3} \right)\]     
B.                 \[\frac{1}{16{{a}^{3}}}\left( \frac{\pi }{4}+\frac{1}{3} \right)\]
C.                 \[\frac{1}{16}{{a}^{3}}\left( \frac{\pi }{4}-\frac{1}{3} \right)\]     
D.                 \[\frac{1}{16}{{a}^{3}}\left( \frac{\pi }{4}+\frac{1}{3} \right)\]
Answer» B.                 \[\frac{1}{16{{a}^{3}}}\left( \frac{\pi }{4}+\frac{1}{3} \right)\]
Discussion
4920.

The value of integral \[\int_{1/\pi }^{2/\pi }{\frac{\sin (1/x)}{{{x}^{2}}}}\,dx=\]  [IIT 1990]

A.                 2             
B.                 \[-1\]
C.                 0             
D.                 1
Answer» E.
Discussion
4921.

\[\int_{0}^{1}{\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}}\,dx=\]             [SCRA 1987; MNR 1990]

A.                 \[\frac{{{\pi }^{2}}}{8}\]
B.                 \[\frac{{{\pi }^{2}}}{16}\]
C.                 \[\frac{{{\pi }^{2}}}{4}\]
D.                 \[\frac{{{\pi }^{2}}}{32}\]
Answer» E.
Discussion
4922.

\[\int_{0}^{\pi /2}{\frac{dx}{2+\cos x}}=\]                                             [BIT Ranchi 1992]

A.                 \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]               
B.                 \[\sqrt{3}{{\tan }^{-1}}\left( \sqrt{3} \right)\]
C.                 \[\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]               
D.                 \[2\sqrt{3}{{\tan }^{-1}}\left( \sqrt{3} \right)\]
Answer» D.                 \[2\sqrt{3}{{\tan }^{-1}}\left( \sqrt{3} \right)\]
Discussion
4923.

The correct evaluation of \[\int_{0}^{\pi /2}{\sin x\,\sin 2x}\] is [MP PET 1993, 2003]

A.                 \[\frac{4}{3}\]   
B.                 \[\frac{1}{3}\]
C.                 \[\frac{3}{4}\]   
D.                 \[\frac{2}{3}\]
Answer» E.
Discussion
4924.

\[\int_{0}^{1/\sqrt{2}}{\frac{{{\sin }^{-1}}x}{{{(1-{{x}^{2}})}^{3/2}}}dx=}\]                                            [Roorkee 1984]

A.                 \[\frac{\pi }{4}+\frac{1}{2}\log 2\]           
B.                 \[\frac{\pi }{4}-\frac{1}{2}\log 2\]
C.                 \[\frac{\pi }{2}+\log 2\]
D.                 \[\frac{\pi }{2}-\log 2\]
Answer» C.                 \[\frac{\pi }{2}+\log 2\]
Discussion
4925.

\[\int_{0}^{\pi /4}{{{\tan }^{2}}x\,dx=}\]               [Roorkee 1983, Pb. CET 2000]

A.                 \[1-\frac{\pi }{4}\]
B.                 \[1+\frac{\pi }{4}\]
C.                 \[\frac{\pi }{4}-1\]
D.                 \[\frac{\pi }{4}\]
Answer» B.                 \[1+\frac{\pi }{4}\]
Discussion
4926.

\[\int_{\pi /4}^{\pi /2}{\cos \theta \,\text{cose}{{\text{c}}^{\text{2}}}\theta \,d\theta =}\]                                         [Roorkee 1978]

A.                 \[\sqrt{2}-1\]    
B.                 \[1-\sqrt{2}\]
C.                 \[\sqrt{2}+1\]   
D.                 None of these
Answer» B.                 \[1-\sqrt{2}\]
Discussion
4927.

If \[\int_{0}^{k}{\frac{dx}{2+8{{x}^{2}}}}=\frac{\pi }{16}\,,\] then \[k=\]

A.                 1             
B.                 \[\frac{1}{2}\]
C.                 \[\frac{1}{4}\]   
D.                 None of these
Answer» C.                 \[\frac{1}{4}\]   
Discussion
4928.

\[\int_{a}^{b}{\frac{\log x}{x}\,dx=}\]                                    [MP PET 1994]

A.                 \[\log \left( \frac{\log b}{\log a} \right)\]              
B.                 \[\log (a\,b)\log \,\left( \frac{b}{a} \right)\]
C.                 \[\frac{1}{2}\log (a\,b)\log \,\left( \frac{b}{a} \right)\]   
D.                 \[\frac{1}{2}\log (a\,b)\log \,\left( \frac{a}{b} \right)\]
Answer» D.                 \[\frac{1}{2}\log (a\,b)\log \,\left( \frac{a}{b} \right)\]
Discussion
4929.

\[\int_{0}^{1}{\frac{dx}{{{[ax+b(1-x)]}^{2}}}}=\]                                     [SCRA 1986]

A.                 \[\frac{a}{b}\]  
B.                 \[\frac{b}{a}\]
C.                 \[a\,b\]
D.                 \[\frac{1}{a\,b}\]
Answer» E.
Discussion
4930.

\[\int_{0}^{\pi /4}{{{\sec }^{7}}\theta {{\sin }^{3}}\theta }\,d\theta =\]

A.                 \[\frac{1}{12}\]
B.                 \[\frac{3}{12}\]
C.                 \[\frac{5}{12}\]
D.                 None of these
Answer» D.                 None of these
Discussion
4931.

\[\int_{0}^{\pi /2}{\sqrt{\cos \theta }{{\sin }^{3}}\theta }\,d\theta =\]

A.                 \[\frac{20}{21}\]              
B.                 \[\frac{8}{21}\]
C.                 \[\frac{-20}{21}\]            
D.                 \[\frac{-8}{21}\]
Answer» C.                 \[\frac{-20}{21}\]            
Discussion
4932.

\[\int_{\pi /6}^{\pi /4}{\text{cosec}\,2x\,dx=}\]                                                [MNR 1980]

A.                 \[\log 3\]            
B.                 \[\log \sqrt{3}\]
C.                 \[\log 9\]            
D.                 None of these
Answer» E.
Discussion
4933.

If \[g(1)=g(2)\], then \[\int_{1}^{2}{{{\left[ fg(x) \right]}^{-1}}}f'\{g(x)\}\ g'(x)\ dx\]is equal to                                                 [AMU 2005]

A.                 1             
B.                 2
C.                 0             
D.                 None of these
Answer» D.                 None of these
Discussion
4934.

If \[\int_{\log 2}^{x}{\frac{du}{{{({{e}^{u}}-1)}^{1/2}}}}=\frac{\pi }{6}\], then \[{{e}^{x}}=\]                        [Orissa JEE 2005]

A. 1
B. 2
C. 4             
D. -1
Answer» D. -1
Discussion
4935.

\[\int_{\pi /4}^{\pi /2}{\text{cose}{{\text{c}}^{2}}xdx=}\]                                            [Karnataka CET 2005]

A.                 \[-1\]    
B.                 1
C.                 0             
D.                 \[\frac{1}{2}\]
Answer» C.                 0             
Discussion
4936.

The value of \[\int_{1}^{{{e}^{2}}}{\frac{dx}{x{{(1+\ln x)}^{2}}}}\] is                        [J & K 2005]

A.                 \[2/3\] 
B.                 \[1/3\]
C.                 \[3/2\] 
D.                 \[\ln 2\]
Answer» B.                 \[1/3\]
Discussion
4937.

\[\int\limits_{\pi /4}^{3\pi /4}{\frac{dx}{1+\cos x}}\] is equal to                                [IIT 1999]

A.                 2             
B.                 \[-2\]
C.                 \[\frac{1}{2}\]   
D.                 \[-\frac{1}{2}\]
Answer» B.                 \[-2\]
Discussion
4938.

Let \[{{I}_{1}}=\int_{1}^{2}{\frac{dx}{\sqrt{1+{{x}^{2}}}}}\]and\[{{I}_{2}}=\int_{1}^{2}{\frac{dx}{x}}\] then          [Pb. CET 2004]

A.                 \[{{I}_{1}}>{{I}_{2}}\]    
B.                 \[{{I}_{2}}>{{I}_{1}}\]
C.                 \[{{I}_{1}}={{I}_{2}}\]    
D.                 \[{{I}_{1}}>2{{I}_{2}}\]
Answer» C.                 \[{{I}_{1}}={{I}_{2}}\]    
Discussion
4939.

The value of \[\int_{0}^{1}{{{x}^{2}}{{e}^{x}}dx}\]is equal to       [Pb. CET 2002]

A.                 \[e-2\] 
B.                 \[e+2\]
C.                 \[{{e}^{2}}-2\]  
D.                 \[{{e}^{2}}\]
Answer» B.                 \[e+2\]
Discussion
4940.

\[\int_{3}^{8}{\frac{2-3x}{x\sqrt{(1+x)}}\text{ }}dx\]is equal to                                  [Pb. CET 2001]

A.                 \[2\log \,\left( 3/2{{e}^{3}} \right)\]       
B.                 \[\log (3/{{e}^{3}})\]
C.                 \[4\log (3/{{e}^{3}})\]   
D.                 None of these
Answer» B.                 \[\log (3/{{e}^{3}})\]
Discussion
4941.

\[\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta d\theta }=\]                                        [Karnataka CET 2004]

A.                 \[\frac{2}{3}\]   
B.                 \[\frac{1}{4}\]
C.                 \[\frac{1}{3}\]   
D.                 \[\frac{1}{6}\]
Answer» E.
Discussion
4942.

\[\int_{1}^{e}{\frac{{{e}^{x}}}{x}(1+x\log x)\,dx}=\]

A.                 \[{{e}^{e}}\]
B.                 \[{{e}^{e}}-e\]
C.                 \[{{e}^{e}}+e\]
D.                 None of these
Answer» B.                 \[{{e}^{e}}-e\]
Discussion
4943.

\[\int_{0}^{1}{{{e}^{2\,\text{In}\,x}}\,dx}=\]                                                                                                         [MP PET 1990]

A.                 0
B.                 \[\frac{1}{2}\]
C.                 \[\frac{1}{3}\]
D.                 \[\frac{1}{4}\]
Answer» D.                 \[\frac{1}{4}\]
Discussion
4944.

Range of the function \[f(x)=9-7\sin x\] is

A.            (2, 16)
B.            [2, 16]
C.            [?1, 1]
D.            (2, 16]
Answer» C.            [?1, 1]
Discussion
4945.

Range of the function \[f(x)={{\sin }^{2}}({{x}^{4}})+{{\cos }^{2}}({{x}^{4}})\] is

A.                    \[(-\infty ,\ \infty )\]
B.             {1}
C.            (?1, 1)
D.            (0, 1)
Answer» C.            (?1, 1)
Discussion
4946.

The range of \[f(x)=\cos 2x-\sin 2x\] contains the set [IIT Screening]

A.                    [2, 4]
B.            [?1, 1]
C.            [?2, 2]
D.            [?4, 4]
Answer» C.            [?2, 2]
Discussion
4947.

If \[f:R\to R\], then the range of the function \[f(x)=\frac{{{x}^{2}}}{{{x}^{2}}+1}\] is [MP PET 1987]

A.                    \[{{R}^{-}}\]
B.            \[{{R}^{+}}\]
C.            R
D.            \[R\times R\]
Answer» C.            R
Discussion
4948.

The range of \[f(x)=\cos x-\sin x\] is            [MP PET 1995; Pb. CET 2001]           

A.            \[(-1,\ 1)\]
B.             \[[-1,\,\ 1)\]
C.            \[\left[ -\frac{\pi }{2},\ \frac{\pi }{2} \right]\]
D.            \[[-\sqrt{2},\ \sqrt{2}]\]
Answer» E.
Discussion
4949.

The range of the function \[f(x)=\frac{x+2}{|x+2|}\] is [RPET 2002]

A.            {0, 1}
B.             {?1, 1}
C.            R
D.            \[R-\{-2\}\]
Answer» C.            R
Discussion
4950.

The range of \[f(x)=\cos (x/3)\] is [RPET 2002]

A.                    \[(-1/3,\ 1/3)\]
B.             \[[-1,\ 1]\]
C.            \[(1/3,\ -1/3)\]
D.            \[(-3,\ 3)\]
Answer» C.            \[(1/3,\ -1/3)\]
Discussion