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.

8351.

If the value of the determinant \[\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{matrix} \right|\] is positive, then (a, b, c > 0)

A. \[abc>1\]
B. \[abc>-\,8\]
C. \[abc<-\,8\]
D. \[abc>-\,2\]
Answer» C. \[abc<-\,8\]
8352.

If , then y =\[f(x)\]represents

A. a straight line parallel to x-axis
B. a straight line parallel to y-axis
C. parabola
D. a straight line with negative slope
Answer» C. parabola
8353.

If \[p+q+r=0=a+b+c\], then the value of the determinant is

A. 0
B. \[pa+qb+rc\]
C. 1
D. none of these
Answer» B. \[pa+qb+rc\]
8354.

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.
8355.

If \[\Delta =\left| \,\begin{matrix} a & b & c \\ x & y & z \\ p & q & r \\ \end{matrix}\, \right|\], then \[\left| \,\begin{matrix} ka & kb & kc \\ kx & ky & kz \\ kp & kq & kr \\ \end{matrix}\, \right|\]= [RPET 1986]

A. \[\Delta \]
B. \[k\Delta \]
C. \[3k\Delta \]
D. \[{{k}^{3}}\Delta \]
Answer» E.
8356.

\[\left| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right|=\] [MP PET 1991]

A. \[3abc+{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
B. \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
C. \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
D. \[abc+{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
Answer» C. \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
8357.

If \[a,b,c\] are different and \[\left| \,\begin{matrix} a & {{a}^{2}} & {{a}^{3}}-1 \\ b & {{b}^{2}} & {{b}^{3}}-1 \\ c & {{c}^{2}} & {{c}^{3}}-1 \\ \end{matrix}\, \right|=0\], then [EAMCET 1989]

A. \[a+b+c=0\]
B. \[abc=1\]
C. \[a+b+c=1\]
D. \[ab+bc+ca=0\]
Answer» C. \[a+b+c=1\]
8358.

\[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\frac{k}{{{n}^{2}}+{{k}^{2}}}}\]is equals to [Roorkee 1999]

A. \[\frac{1}{2}\log 2\]
B. \[x=\frac{3\pi }{4}\]
C. \[\pi /4\]
D. \[\pi /2\]
Answer» B. \[x=\frac{3\pi }{4}\]
8359.

\[\underset{n\to \infty }{\mathop{\text{lim}\,}}\,\left[ \frac{1}{{{n}^{2}}}{{\sec }^{2}}\frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}{{\sec }^{2}}\frac{4}{{{n}^{2}}}+.....+\frac{1}{n}{{\sec }^{2}}1 \right]\] equals [AIEEE 2005]

A. \[\tan 1\]
B. \[\frac{1}{2}\tan 1\]
C. \[\frac{1}{2}\sec 1\]
D. \[\frac{1}{2}\text{cosec}1\]
Answer» C. \[\frac{1}{2}\sec 1\]
8360.

\[\int_{\,-\pi /2}^{\,\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx=}\] [EAMCET 2002]

A. \[\frac{3\pi }{64}\]
B. \[\frac{3\pi }{572}\]
C. \[\frac{3\pi }{256}\]
D. \[\frac{3\pi }{128}\]
Answer» D. \[\frac{3\pi }{128}\]
8361.

\[\int_{\,0}^{\,\infty }{\,\log \left( x+\frac{1}{x} \right)\frac{dx}{1+{{x}^{2}}}}\] is equal to [RPET 2000, 02]

A. \[\pi \log 2\]
B. \[-\pi \log 2\]
C. \[(\pi /2)\log 2\]
D. \[-(\pi /2)\log 2\]
Answer» B. \[-\pi \log 2\]
8362.

\[\int_{0}^{\pi }{{{\sin }^{5}}\left( \frac{x}{2} \right)\,dx}\] equals [Kurukshetra CEE 1996]

A. \[\frac{16}{15}\]
B. \[\frac{32}{15}\]
C. \[\frac{8}{15}\]
D. \[\frac{5}{6}\]
Answer» B. \[\frac{32}{15}\]
8363.

If \[f(x)=\int_{{{x}^{2}}}^{{{x}^{2}}+1}{{{e}^{-{{t}^{2}}}}}dt,\] then \[f(x)\] increases in [IIT Screening 2003]

A. \[(2,\,\,2)\]
B. No value of \[x\]
C. \[(0,\,\,\infty )\]
D. \[(-\infty ,\,\,0)\]
Answer» E.
8364.

\[\int_{-\pi /2}^{\pi /2}{{{\sin }^{2}}x{{\cos }^{2}}x(\sin x+\cos x)\,dx=}\] [EAMCET 1992]

A. \[\frac{2}{15}\]
B. \[\frac{4}{15}\]
C. \[\frac{6}{15}\]
D. \[\frac{8}{15}\]
Answer» C. \[\frac{6}{15}\]
8365.

The value of the integral \[\int_{-1}^{1}{\frac{d}{dx}\left( {{\tan }^{-1}}\frac{1}{x} \right)}\,dx\] is

A. \[\frac{\pi }{2}\]
B. \[\frac{\pi }{4}\]
C. \[-\frac{\pi }{2}\]
D. None of these
Answer» D. None of these
8366.

\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{1}^{3}}+{{n}^{3}}}+\frac{4}{{{2}^{3}}+{{n}^{3}}}+....+\frac{1}{2n}\]is equal to [RPET 1997]

A. \[\frac{1}{3}{{\log }_{e}}3\]
B. \[\frac{1}{3}{{\log }_{e}}2\]
C. \[\frac{1}{3}{{\log }_{e}}\frac{1}{3}\]
D. None of these
Answer» C. \[\frac{1}{3}{{\log }_{e}}\frac{1}{3}\]
8367.

The points of intersection of \[{{F}_{1}}(x)=\int_{2}^{x}{(2t-5)\,dt}\] and \[{{F}_{2}}(x)=\int_{0}^{x}{2t\,dt,}\] are [IIT Screening]

A. \[\left( \frac{6}{5},\,\frac{36}{25} \right)\]
B. \[\left( \frac{2}{3},\,\frac{4}{9} \right)\]
C. \[\left( \frac{1}{3},\,\frac{1}{9} \right)\]
D. \[\left( \frac{1}{5},\,\frac{1}{25} \right)\]
Answer» B. \[\left( \frac{2}{3},\,\frac{4}{9} \right)\]
8368.

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

A. \[\pi \log \frac{1}{2}\]
B. \[\pi \log 2\]
C. \[2\pi \log \frac{1}{2}\]
D. \[2\pi \log 2\]
Answer» C. \[2\pi \log \frac{1}{2}\]
8369.

The triangle formed by the tangent to the curve \[f(x)={{x}^{2}}+bx-b\] at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is

A. -1
B. 3
C. -3
D. 1
Answer» D. 1
8370.

What is the area of the region enclosed by \[~y=2\left| x \right|\] and\[y=4\]?

A. 2 square unit
B. 4 square unit
C. 8 square unit
D. 16 square unit
Answer» D. 16 square unit
8371.

What is the area under the curve \[y=\left| x \right|+\left| x-1 \right|\]between \[x=0\] and\[x=1\]?

A. \[\frac{1}{2}\]
B. 1
C. \[\frac{3}{2}\]
D. 2
Answer» C. \[\frac{3}{2}\]
8372.

What is the area enclosed by the equation\[{{x}^{2}}+{{y}^{2}}=2\]?

A. \[4\pi \] square units
B. \[2\pi \] square units
C. \[4{{\pi }^{2}}\] square units
D. 4 square units
Answer» C. \[4{{\pi }^{2}}\] square units
8373.

The line y = mx bisects the area enclosed by lines \[x=0,\text{ }y=0\] and \[x=3/2\] and the curve\[y=1+4x-{{x}^{2}}\]. Then the value of m is

A. \[\frac{13}{6}\]
B. \[\frac{13}{2}\]
C. \[\frac{13}{5}\]
D. \[\frac{13}{7}\]
Answer» B. \[\frac{13}{2}\]
8374.

What is the area bounded by the curve \[y=4x-{{x}^{2}}-3\] and the x-axis?

A. 2/3 sq. unit
B. 4/3 sq. unit
C. 5/3 sq. unit
D. 4/5 sq. unit
Answer» C. 5/3 sq. unit
8375.

The area bounded by the curves y = ln x, y = ln \[\left| x \right|,y=\left| ln\text{ }x \right|\] and \[,y=\left| ln\text{ }\left| x \right| \right|\] is

A. 4 sq. units
B. 6 sq. units
C. 10 sq. units
D. None of these
Answer» B. 6 sq. units
8376.

If \[{{c}_{1}}=y=\frac{1}{1+{{x}^{2}}}\] and \[{{c}_{2}}=y=\frac{{{x}^{2}}}{2}\] be two curves lying in XY-plane, then

A. Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and \[y=0\] is \[\frac{\pi }{2}\]
B. Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[\frac{\pi }{2}-1\]
C. Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[1-\frac{\pi }{2}\]
D. Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and x-axis is \[\frac{\pi }{2}\]
Answer» C. Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[1-\frac{\pi }{2}\]
8377.

Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is \[\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos a\], then \[f\left( \frac{\pi }{2} \right)=\]

A. 1
B. \[\frac{1}{2}\]
C. \[\frac{1}{3}\]
D. None of these
Answer» C. \[\frac{1}{3}\]
8378.

The area of the smaller segment cut off from the circle \[{{x}^{2}}+{{y}^{2}}=9\,\,by\,\,x=1\] is

A. \[\frac{1}{2}(9\,se{{c}^{-1}}3-\sqrt{8})\] sq. unit
B. \[(9\,se{{c}^{-1}}3-\sqrt{8})\] sq. unit
C. \[(\sqrt{8}-9\,\,se{{c}^{-1}}3)\]sq. unit
D. None of the above
Answer» C. \[(\sqrt{8}-9\,\,se{{c}^{-1}}3)\]sq. unit
8379.

The area bounded by \[y={{x}^{2}}+3\] and \[y=2x+3\] is (in sq. units)

A. \[\frac{12}{7}\]
B. \[\frac{4}{3}\]
C. \[\frac{3}{4}\]
D. \[\frac{8}{3}\]
Answer» C. \[\frac{3}{4}\]
8380.

Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is \[\frac{{{a}^{2}}}{2}+\frac{a}{2}\sin a+\frac{\pi }{2}\cos \] a, then \[f\left( \frac{\pi }{2} \right)=\]

A. 1
B. \[\frac{1}{2}\]
C. \[\frac{1}{3}\]
D. None of these
Answer» C. \[\frac{1}{3}\]
8381.

The area of the region enclosed by the curves \[y=x\,\,\log \,\,x\] and \[y=2x-2{{x}^{2}}\] is

A. \[\frac{5}{12}\]
B. \[\frac{7}{12}\]
C. 1
D. \[\frac{4}{7}\]
Answer» C. 1
8382.

The area bounded by the curves \[y=f(x),\] the x-axis, and the ordinates \[x=1\]and \[x=b\] is \[(b-1)\sin (3b+4)\]. Then \[f(x)\] is

A. \[(x-1)\cos (3x+4)\]
B. \[sin(3x+4)\]
C. \[\sin (3x+4)+3(x-1)\cos (3x+4)\]
D. None of these
Answer» D. None of these
8383.

The area enclosed by the curve \[x=a\text{ }co{{s}^{3}}t,\]\[y=b\text{ }si{{n}^{3}}t\] and the positive directions of x-axis and y-axis is

A. \[\frac{\pi ab}{4}\]
B. \[\frac{\pi ab}{32}\]
C. \[\frac{3\pi ab}{32}\]
D. \[\frac{5\pi ab}{32}\]
Answer» D. \[\frac{5\pi ab}{32}\]
8384.

The area bounded by the x-axis, the curve \[y=f(x)\] and the lines \[x=1,\text{ }x=b,\] is equal to\[\sqrt{{{b}^{2}}+1}-\sqrt{2}\] for all \[b>1\], then f(x) is

A. \[\sqrt{x-1}\]
B. \[\sqrt{x+1}\]
C. \[\sqrt{{{x}^{2}}+1}\]
D. \[\frac{x}{\sqrt{1+{{x}^{2}}}}\]
Answer» E.
8385.

The area bounded by the curve \[y=x{{(3-x)}^{2}}\], the x-axis and the ordinates of the maximum and minimum points of the curve, is given by

A. 1 sq. unit
B. 2 sq. unit
C. 4 sq. unit
D. None of these
Answer» D. None of these
8386.

What is the area bounded by the curves \[y={{e}^{x}},y={{e}^{-x}}\] and the straight line\[x=1\]?

A. \[\left( e+\frac{1}{e} \right)\] sq. unit
B. \[\left( e-\frac{1}{e} \right)\] sq. unit
C. \[\left( e+\frac{1}{e}-2 \right)\] sq. unit
D. \[\left( e-\frac{1}{e}-2 \right)\] sq. unit
Answer» D. \[\left( e-\frac{1}{e}-2 \right)\] sq. unit
8387.

What is the area bounded by y = tan x, y = 0 and\[x=\frac{\pi }{4}\]?

A. \[\ell \,n\,\,2\] square units
B. \[\frac{\ell \,n\,\,2}{2}\]square units
C. \[2(\ell n\,2)\] square units
D. None of these
Answer» C. \[2(\ell n\,2)\] square units
8388.

The value of c + 2 for which the area of the figure bounded by the curve\[y=8{{x}^{2}}-{{x}^{5}}\], the straight lines \[x=1\] and \[x=c\] and x-axis is equal to \[\frac{16}{3},\] is

A. 1
B. 3
C. -1
D. 4
Answer» B. 3
8389.

The area bounded by the curve \[y=f(x),y=x\]and the lines \[x=1,x=t\] is \[(t+\sqrt{1+{{t}^{2}}})-\sqrt{2}-1\]sq. unit, for all t > 1. If f(x) satisfying f(x)>x for all x>1, then f(x) is equal to

A. \[x+1+\frac{x}{\sqrt{1+{{x}^{2}}}}\]
B. \[x+\frac{x}{\sqrt{1+{{x}^{2}}}}\]
C. \[1+\frac{x}{\sqrt{1+{{x}^{2}}}}\]
D. \[\frac{x}{\sqrt{1+{{x}^{2}}}}\]
Answer» B. \[x+\frac{x}{\sqrt{1+{{x}^{2}}}}\]
8390.

\[f(x)=f(2-x),\] then \[\int_{\,0.5}^{\,1.5}{\,xf(x)\,dx}\] equals [AMU 1999]

A. \[\int_{\,0}^{\,1}{\,f(x)\,dx}\]
B. \[\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]
C. \[2\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]
D. 0
Answer» C. \[2\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]
8391.

\[\int_{-\,\pi /2}^{\,\pi /2}{\,\frac{\sin x}{1+{{\cos }^{2}}x}{{e}^{-{{\cos }^{2}}x}}dx}\] is equal to [AMU 1999]

A. \[2{{e}^{-1}}\]
B. 1
C. 0
D. None of these
Answer» D. None of these
8392.

The value of \[\int_{0}^{2\pi }{|{{\sin }^{3}}\theta |\,d\theta }\] is [Roorkee Qualifying 1998]

A. 0
B. \[3/8\]
C. \[8/3\]
D. \[\pi \]
Answer» D. \[\pi \]
8393.

\[\int_{0}^{\pi /2}{\,\,\log \tan x\,dx=}\] [MP PET 1999; RPET 2001, 02; Karnataka CET 1999, 2000, 01, 02]

A. \[\frac{\pi }{2}{{\log }_{e}}2\]
B. \[-\frac{\pi }{2}{{\log }_{e}}2\]
C. \[\pi {{\log }_{e}}2\]
D. 0
Answer» E.
8394.

\[\int_{-3}^{3}{\frac{{{x}^{2}}\sin 2x}{{{x}^{2}}+1}\,dx=}\]

A. 0
B. 1
C. \[2{{\log }_{e}}3\]
D. None of these
Answer» B. 1
8395.

\[\int_{0}^{\pi /2}{\frac{\sin x}{\sin x+\cos x}\,dx}\] equals [RPET 1996; Kerala (Engg.) 2002]

A. \[\frac{\pi }{2}\]
B. \[\frac{\pi }{3}\]
C. \[\frac{\pi }{4}\]
D. \[\frac{\pi }{6}\]
Answer» D. \[\frac{\pi }{6}\]
8396.

\[\int_{\,0}^{\,2\pi }{|\sin x|\,dx=}\]

A. 0
B. 1
C. 2
D. 4
Answer» E.
8397.

The function \[F(x)=\int_{0}^{x}{\log \left( \frac{1-x}{1+x} \right)}\,dx\] is

A. An even function
B. An odd function
C. A periodic function
D. None of these
Answer» B. An odd function
8398.

If \[(n-m)\] is odd and \[|m|\,\ne \,|n|,\] then \[\int_{0}^{\pi }{\cos mx\sin nx}\,dx\] is

A. \[\frac{2n}{{{n}^{2}}-{{m}^{2}}}\]
B. 0
C. \[\frac{2n}{{{m}^{2}}-{{n}^{2}}}\]
D. \[\frac{2m}{{{n}^{2}}-{{m}^{2}}}\]
Answer» B. 0
8399.

\[\int_{0}^{\pi /2}{\frac{1}{1+\sqrt{\tan x}}}\,dx=\] [RPET 1995; Kurukshetra CEE 1998]

A. \[\frac{\pi }{2}\]
B. \[\frac{\pi }{4}\]
C. \[\frac{\pi }{6}\]
D. 1
Answer» C. \[\frac{\pi }{6}\]
8400.

If \[\int_{0}^{2a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}}\] then [SCRA 1986]

A. \[f(2a-x)=-f(x)\]
B. \[f(2a-x)=f(x)\]
C. \[f(a-x)=-f(x)\]
D. \[f(a-x)=f(x)\]
Answer» C. \[f(a-x)=-f(x)\]