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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. \[f(a-x)=-f(x)\] | |