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.

3151.

The area of the region bounded by \[{{x}^{2}}+{{y}^{2}}-2x-3=0\] and \[y=\left| x \right|+1\]is

A. \[\frac{\pi }{2}-1\] sq. units
B. \[2\pi \]sq. units
C. \[4\pi \]sq. units 
D. \[\pi /2\]sq. units
Answer» B. \[2\pi \]sq. units
3152.

The area bounded by the two branches of curve \[{{(y-x)}^{2}}={{x}^{3}}\] and the straight line x = 1 is

A. 1/5 sq. units      
B. 3/5 sq. units
C. 4/5 sq. units      
D. 8/4 sq. units
Answer» D. 8/4 sq. units
3153.

The area of the closed figure bounded \[y=\frac{{{x}^{2}}}{2}-2x+2\] and the tangents to it at (1, 1/2) and (4, 2) is

A. 9/8 sq. units      
B. 3/8 sq. units
C. 3/2 sq. units      
D. 9/4 sq. units
Answer» B. 3/8 sq. units
3154.

The area bounded by the curve \[{{y}^{2}}(2-x)={{x}^{3}}\] and x = 2 is

A. \[\frac{\pi }{2}\]            
B. \[\pi \]
C. \[2\pi \] 
D. \[3\pi \]
Answer» E.
3155.

The area enclosed by the curves \[y=\text{sin}\,x+\text{cos}\,x\]and \[y=\left| \text{cos }x-\text{sin }x \right|\] over the interval [0,\[\pi /2\]] is

A. \[4(\sqrt{2}-1)\] 
B. \[2\sqrt{2}(\sqrt{2}-1)\]
C. \[2(\sqrt{2}+1)\]
D. \[2\sqrt{2}(\sqrt{2}+1)\]
Answer» C. \[2(\sqrt{2}+1)\]
3156.

The area enclosed by \[y={{x}^{2}}+\cos x\] and its normal at \[x=\frac{\pi }{2}\] in the first quadrant is

A. \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{64}+\frac{{{\pi }^{3}}}{32}+1\]
B. \[\frac{{{\pi }^{5}}}{16}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{24}-1\]
C. \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{16}\]
D. \[\frac{{{\pi }^{5}}}{32}-\frac{{{\pi }^{4}}}{32}+\frac{{{\pi }^{3}}}{24}+1\]
Answer» E.
3157.

The area bounded by the curve y = sin x and the line x = 0, \[\left| y \right|=\frac{\pi }{2}\] is

A. 1         
B. 2
C. \[\pi \]   
D. 2\[\pi \]
Answer» C. \[\pi \]   
3158.

The area bounded by the curves \[x={{y}^{2}}\] and \[x=\frac{2}{1+{{y}^{2}}}\] is

A. \[\pi -\frac{2}{3}\]
B. \[\pi +\frac{2}{3}\]
C. \[-\pi -\frac{2}{3}\]        
D. none of these
Answer» B. \[\pi +\frac{2}{3}\]
3159.

Area lying in the first quadrant and bounded by the curve \[y={{x}^{3}}\] and the line y = 4x is

A. 2                     
B. 3
C. 4         
D. 5
Answer» D. 5
3160.

The area bounded by the curve y = x \[\left| x \right|\], x-axis and the ordinates x = 1, \[x=-1\] is given by

A. 0         
B. 44256
C. 2/3                   
D. None of these
Answer» D. None of these
3161.

The area enclosed between the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\] and the line x = 2 above the x-axis is

A. \[\pi {{a}^{2}}\]sq. units          
B. \[\frac{3\pi \,{{a}^{2}}}{2}\] sq. units
C. \[2\pi \,{{a}^{2}}\]sq. units
D. \[3\pi \,{{a}^{2}}\] sq. units
Answer» C. \[2\pi \,{{a}^{2}}\]sq. units
3162.

The area of the closed figure bounded by \[x=-1,\]\[y=0,\] \[y={{x}^{2}}+x+1\], and the tangent to the curve \[y={{x}^{2}}+x+1\] at A(1,3) is

A. 4/3 sq. units      
B. 7/3 sq. units
C. 7/6 sq. units      
D. None of these
Answer» D. None of these
3163.

Let f(x) and g(x) be differentiable for \[0\le x\le 1\]such that f(0)=0, g(0)=0, f(1)=6. Let there exists a real number c in (0, 1) such that f?= 2g?. then the value of g(1) must be

A. 1
B. 3
C. \[-\,2\]              
D. \[-\,1\]
Answer» C. \[-\,2\]              
3164.

A curve is represented by the equations \[x={{\sec }^{2}}\]t and\[y=\cot \,t\], where t is a parameter. If the tangent at the point P on the curve where \[t=\pi /4\] meets the curve again at the point Q, Then \[\left| PQ \right|\] is equal to

A. \[\frac{5\sqrt{3}}{2}\]   
B. \[\frac{5\sqrt{5}}{2}\]
C. \[\frac{2\sqrt{5}}{3}\]   
D. \[\frac{3\sqrt{5}}{2}\]
Answer» E.
3165.

Let \[f\,\left( 1 \right)=-\,2\] and \[f'(x)\ge 4.2\] for \[1\le x\le 6\]. The smallest possible value of f(6) is

A. 9                     
B. 12
C. 15                    
D. 19
Answer» E.
3166.

If a variable tangent to the curve \[{{x}^{2}}y={{c}^{3}}\]makes intercepts a and b on x-and y-axis, respectively, then the value of \[{{a}^{2}}b\]is

A. 27\[{{c}^{3}}\]           
B. \[\frac{4}{27}{{c}^{3}}\]
C. \[\frac{27}{4}{{c}^{3}}\]        
D. \[\frac{4}{9}{{c}^{3}}\]
Answer» D. \[\frac{4}{9}{{c}^{3}}\]
3167.

The normal to the curve \[2{{x}^{2}}+{{y}^{2}}=12\] at the point (2, 2) cuts the curve again at

A. \[\left( -\frac{22}{9},-\frac{2}{9} \right)\] 
B. \[\left( \frac{22}{9},\frac{2}{9} \right)\]
C. \[\left( -\,2,\,\,-2 \right)\]  
D. none of these
Answer» B. \[\left( \frac{22}{9},\frac{2}{9} \right)\]
3168.

If \[x+4y=14\] is a normal to the curve \[{{y}^{2}}=a{{x}^{3}}-\beta \] at  (2, 3), then the value of \[\alpha +\beta \]is

A. 9                     
B. \[-\,5\]
C. 7                     
D. \[-\,7\]
Answer» B. \[-\,5\]
3169.

The equation of the tangent to the curve \[y=b{{e}^{-x/a}}\] at the point where it crosses the y-axis is

A. \[\frac{x}{a}-\frac{y}{b}=1\]
B. \[ax+by=1\]
C. \[ax-by=1\]       
D. \[\frac{x}{a}+\frac{y}{b}=1\]
Answer» E.
3170.

The maximum distance from the origin of a point on the curve \[x=a\text{ }sin\text{ }t-b\]\[\sin \left( \frac{at}{b} \right),\] \[y=a\,\cos \,\,t-b\cos \left( \frac{at}{b} \right),\] both a, b>0, is

A. a - b                
B. a+b
C. \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]        
D. \[\sqrt{{{a}^{2}}-{{b}^{2}}}\]
Answer» B. a+b
3171.

If \[2a+3b+6c=0,\] then at least one root of the equation \[a{{x}^{2}}+bx+c=0\]lies in the interval

A. (0, 1)    
B. (1, 3)
C. (2, 3)    
D. (1, 3)
Answer» B. (1, 3)
3172.

  A function\[y=f\left( x \right)\]has a second order derivative\[f''\left( x \right)=6\left( x-1 \right)\]. If its graph passes through the point (2, 1) and at that point the tangent to the graph is \[y=3x-5,\]then the function is

A. \[{{(x-1)}^{2}}\]         
B. \[{{(x-1)}^{3}}\]
C. \[{{(x+1)}^{3}}\]       
D. \[{{(x+1)}^{2}}\]
Answer» C. \[{{(x+1)}^{3}}\]       
3173.

If f(x) and g(x) are differentiable functions for \[0\le x\le 1\] such that f(0)=10, g(0)=2,  f(1)=2, g(1)=4, then in the interval (0, 1),

A. f'(x) =0 for all x
B. f'(x)+4g'(x)=0 for at least one x
C. f'(x)=2g'(x) for at most done x
D. none of these
Answer» C. f'(x)=2g'(x) for at most done x
3174.

In which of the following function is rolle's theorem applicable?

A. \[f(x)=\left\{ \begin{matrix}    x,\,  \\    0,\,  \\ \end{matrix} \right.\,\begin{matrix}    \,\,\,\,\,0\le x<1  \\    x=1  \\ \end{matrix}\] on [0, 1]
B. \[f(x)=\left\{ \begin{matrix}    \frac{\sin x}{x},\,  \\    1,\,  \\ \end{matrix} \right.\,\,\,\,\begin{matrix}    -\pi \le x<0  \\    x=0  \\ \end{matrix}\] on [-\[\pi \],0]
C. \[f(x)=\frac{{{x}^{2}}-x-6}{x-1}\] on [-2, 3]
D. \[f(x)=\left\{ \begin{matrix}    \frac{{{x}^{3}}-2{{x}^{2}}-5x+6}{x-1},\,\,\,if\,\,x\ne 1,  \\    -\,6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,x=1  \\ \end{matrix} \right.\]on [-2, 3]
Answer» E.
3175.

A lamp of negligible height is placed on the ground \[{{\ell }_{1}}\] away from a wall. A man \[{{\ell }_{2}}\]m tall is walking at a speed of \[\frac{{{\ell }_{1}}}{10}\]m/s form the lamp to the nearest point on the wall. When he is midway between the lamp and the wall, the rate of change in the length of this shadow on the all is

A. \[-\frac{5{{\ell }_{2}}}{2}m/s\] 
B. \[-\frac{2{{\ell }_{2}}}{5}m/s\]
C. \[-\frac{{{\ell }_{2}}}{2}m/s\]   
D. \[-\frac{{{\ell }_{2}}}{5}m/s\]
Answer» C. \[-\frac{{{\ell }_{2}}}{2}m/s\]   
3176.

Let f(x) be a twice differentiable function for all real values of x and satisfies f(1)=1, f(2)=4, f(3)=9. Then which of the following is definitely true?

A. \[f''(x)=2\forall x\in (1,3)\]
B. \[f''(x)=f'(x)=5\] for some \[x\in (2,3)\]
C. \[f''(x)=3\,\forall \,x\in (2,\,\,3)\]
D. \[f''(x)=2\] for some \[x\in (1,3)\]
Answer» E.
3177.

The lines tangent to the curves \[{{y}^{3}}-{{x}^{2}}y+5y-2x=0\]and \[{{x}^{4}}-{{x}^{3}}{{y}^{2}}+5x+2y=0\]at the origin intersect at an angles \[\theta \]equal to

A. \[\frac{\pi }{6}\]            
B. \[\frac{\pi }{4}\]
C. \[\frac{\pi }{3}\]            
D. \[\frac{\pi }{2}\]
Answer» E.
3178.

The slope of the tangent to the curve \[y=\sqrt{4-{{x}^{2}}}\] at the point where the ordinate and abscissa equal is

A. -1
B. 1
C. 0                     
D. none of these
Answer» B. 1
3179.

If \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+\sin x=0,\] then solution of the differential equation is.                                                                [Pb. CET 2001]

A.            \[\sin x+{{c}_{1}}x+{{c}_{2}}\]
B.            \[\cos x+{{c}_{1}}x+{{c}_{2}}\]
C.            \[\tan x+{{c}_{1}}x+{{c}_{2}}\]                                           
D.            \[\log \sin x+{{c}_{1}}x+{{c}_{2}}\]
Answer» B.            \[\cos x+{{c}_{1}}x+{{c}_{2}}\]
3180.

The solution of the equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{e}^{-2x}}\] is               [AIEEE 2002]

A.            \[\frac{1}{4}{{e}^{-2x}}\]  
B.            \[\frac{1}{4}{{e}^{-2x}}+cx+d\]           
C.            \[\frac{1}{4}{{e}^{-2x}}+c{{x}^{2}}+d\]                       
D.  \[\frac{1}{4}{{e}^{-2x}}+c+d\]
Answer» C.            \[\frac{1}{4}{{e}^{-2x}}+c{{x}^{2}}+d\]                       
3181.

If \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=0,\] then  [UPSEAT 1999]                             

A.            \[y=ax+b\]                              
B.            \[{{y}^{2}}=ax+b\]
C.            \[y=\log x\]                               
D.  \[y={{e}^{x}}+c\]
Answer» B.            \[{{y}^{2}}=ax+b\]
3182.

The solution of  \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\sec }^{2}}x+x{{e}^{x}}\]is        [DSSE 1985]

A.                    \[y=\log (\sec x)+(x-2){{e}^{x}}+{{c}_{1}}x+{{c}_{2}}\]
B.                    \[y=\log (\sec x)+(x+2){{e}^{x}}+{{c}_{1}}x+{{c}_{2}}\]
C.            \[y=\log (\sec x)-(x+2){{e}^{x}}+{{c}_{1}}x+{{c}_{2}}\]
D.            None of these
Answer» B.                    \[y=\log (\sec x)+(x+2){{e}^{x}}+{{c}_{1}}x+{{c}_{2}}\]
3183.

The solution of the differential equation \[{{\cos }^{2}}x\frac{{{d}^{2}}y}{d{{x}^{2}}}=1\] is

A.            \[y=\log \cos x+cx\]                
B.            \[y=\log \sec x+{{c}_{1}}x+{{c}_{2}}\]
C.            \[y=\log \sec x-{{c}_{1}}x+{{c}_{2}}\]                                
D.            None of these
Answer» C.            \[y=\log \sec x-{{c}_{1}}x+{{c}_{2}}\]                                
3184.

The solution of the differential equation \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}=1\], given that \[y=1,\ \frac{dy}{dx}=0\]when \[x=1\], is

A.            \[y=x\log x+x+2\]                  
B.            \[y=x\log x-x+2\]
C.            \[y=x\log x+x\]                       
D.            \[y=x\log x-x\]
Answer» C.            \[y=x\log x+x\]                       
3185.

If \[A=\left| \,\begin{matrix}    5 & 6 & 3  \\    -4 & 3 & 2  \\    -4 & -7 & 3  \\ \end{matrix}\, \right|\,,\]then cofactors of the elements of 2nd row are [RPET 2002]

A. \[39,\,-3,\,11\]
B. \[-39,\,3,\,11\]
C. \[-39,\,27,\,11\]
D. \[-39,\,-3,\,11\]
Answer» D. \[-39,\,-3,\,11\]
3186.

Let \[A={{[{{a}_{ij}}]}_{n\times n}}\]be a square matrix and let \[{{c}_{ij}}\]be cofactor of \[{{a}_{ij}}\]in A. If \[C=[{{c}_{ij}}]\],then

A. \[|C|\,=\,|A|\]
B.   \[|C|\,=\,|A{{|}^{n-1}}\]
C. \[|C|\,=\,|A{{|}^{n-2}}\]
D. None of these
Answer» C. \[|C|\,=\,|A{{|}^{n-2}}\]
3187.

If \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\].... are respectively the co-factors of the elements \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\],...... of the determinant \[\Delta =\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\], then \[\left| \begin{matrix}    {{B}_{2}} & {{C}_{2}}  \\    {{B}_{3}} & {{C}_{3}}  \\ \end{matrix} \right|=\]

A. \[{{a}_{1}}\Delta \]
B. \[{{a}_{1}}{{a}_{3}}\Delta \]
C. \[({{a}_{1}}+{{b}_{1}})\Delta \]
D. None of these
Answer» B. \[{{a}_{1}}{{a}_{3}}\Delta \]
3188.

If \[{{\Delta }_{1}}=\left| \,\begin{matrix}    1 & 0  \\    a & b  \\ \end{matrix}\, \right|\] and  \[{{\Delta }_{2}}=\left| \begin{matrix}    1 & 0  \\    c & d  \\ \end{matrix} \right|\], then \[{{\Delta }_{2}}{{\Delta }_{1}}\]is equal to [RPET 1984]

A. ac
B. bd
C. \[(b-a)(d-c)\]
D. None of these
Answer» C. \[(b-a)(d-c)\]
3189.

If \[\omega \] is a cube root of unity and \[\Delta =\left| \begin{matrix}    1 & 2\omega   \\    \omega  & {{\omega }^{2}}  \\ \end{matrix} \right|\], then \[{{\Delta }^{2}}\]is equal to [RPET 1984]

A. \[-\omega \]
B. \[\omega \]
C. 1
D. \[{{\omega }^{2}}\]
Answer» C. 1
3190.

If in the determinant \[\Delta =\left| \begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\], \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\] etc. be the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\]etc., then which of the following relations is incorrect

A. \[{{a}_{1}}{{A}_{1}}+{{b}_{1}}{{B}_{1}}+{{c}_{1}}{{C}_{1}}=\Delta \]
B. \[{{a}_{2}}{{A}_{2}}+{{b}_{2}}{{B}_{2}}+{{c}_{2}}{{C}_{2}}=\Delta \]
C. \[{{a}_{3}}{{A}_{3}}+{{b}_{3}}{{B}_{3}}+{{c}_{3}}{{C}_{3}}=\Delta \]
D. \[{{a}_{1}}{{A}_{2}}+{{b}_{1}}{{B}_{2}}+{{c}_{1}}{{C}_{2}}=\Delta \]
Answer» E.
3191.

If \[\Delta =\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\] and \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\]denote the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\] respectively, then the value of the determinant \[\left| \begin{matrix}    {{A}_{1}} & {{B}_{1}} & {{C}_{1}}  \\    {{A}_{2}} & {{B}_{2}} & {{C}_{2}}  \\    {{A}_{3}} & {{B}_{3}} & {{C}_{3}}  \\ \end{matrix} \right|\] is [MP PET 1989]

A. \[\Delta \]
B. \[{{\Delta }^{2}}\]
C. \[{{\Delta }^{3}}\]
D. 0
Answer» C. \[{{\Delta }^{3}}\]
3192.

The minors of - 4 and 9 and the co-factors of - 4 and 9 in determinant \[\,\left| \,\begin{matrix}    -1 & -2 & 3  \\    -4 & -5 & -6  \\    -7 & 8 & 9  \\ \end{matrix}\, \right|\] are respectively [J & K 2005]

A. 42, 3; - 42, 3
B. -42, -3; 42, -3
C. 42, 3; - 42, - 3
D. 42, 3; 42, 3
Answer» C. 42, 3; - 42, - 3
3193.

The cofactor of the element '4' in the determinant \[\left| \,\begin{matrix}    1 & 3 & 5 & 1  \\    2 & 3 & 4 & 2  \\    8 & 0 & 1 & 1  \\    0 & 2 & 1 & 1  \\ \end{matrix}\, \right|\] is [MP PET 1987]

A. 4
B. 10
C. -10
D. -4
Answer» C. -10
3194.

The upper quartile for the following distribution Size of items 1 2 3 4 5 6 7 Frequency 2 4 5 8 7 3 2                 is given by the size of

A.                 \[\left( \frac{31+1}{4} \right)\]th item
B.                 \[\left[ 2\left( \frac{31+1}{4} \right) \right]\]th item         
C.                 \[\left[ 3\left( \frac{31+1}{4} \right) \right]\]th item          
D.                  \[\left[ 4\left( \frac{31+1}{4} \right) \right]\]th item
Answer» D.                  \[\left[ 4\left( \frac{31+1}{4} \right) \right]\]th item
3195.

For a symmetrical distribution \[{{Q}_{1}}=25\] and \[{{Q}_{3}}=45\], the median is

A.                 20           
B.                 25
C.                 35           
D.                  None of these
Answer» D.                  None of these
3196.

The relation between the median M, the second quartile \[{{Q}_{2}}\], the fifth decile \[{{D}_{5}}\] and the 50th percentile \[{{P}_{50}}\], of a set of observations is                                   [AMU 1990]

A.                 \[M={{Q}_{2}}={{D}_{5}}={{P}_{50}}\]  
B.                 \[M<{{Q}_{2}}<{{D}_{5}}<{{P}_{50}}\]   
C.                 \[M>{{Q}_{2}}>{{D}_{5}}>{{P}_{50}}\]   
D.                  None of these
Answer» B.                 \[M<{{Q}_{2}}<{{D}_{5}}<{{P}_{50}}\]   
3197.

The median of 10, 14, 11, 9, 8, 12, 6 is  [Kurukshetra CEE 1997]

A.                 10
B.                 12           
C.                 14           
D.                  11
Answer» B.                 12           
3198.

Which of the following, in case of a discrete data, is not equal to the median

A.                 50th percentile      
B.                 5th decile
C.                 2nd quartile           
D.                 Lower quartile
Answer» E.
3199.

The central value of the set of observations is called

A.                 Mean      
B.                 Median
C.                 Mode      
D.                  G.M.
Answer» C.                 Mode      
3200.

Consider the following statements                        [AIEEE 2004]                 (1) Mode can be computed from histogram                 (2) Median is not independent of change of scale                 (3) Variance is independent of change of origin and scale                 Which of these is/are correct

A.                 (1), (2) and (3)     
B.                 Only (2)
C.                 Only (1) and (2)   
D.                 Only (1)
Answer» E.