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.

701.

A unit vector perpendicular to the plane determined by the points \[P\,(1,\,\,-1,\,\,2),\,\,Q\,(2,\,\,0,\,-1)\] and \[R\,(0,\,\,2,\,\,1)\] is           [IIT 1994]

A. \[\frac{2\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{6}}\]
B. \[\frac{2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\]
C. \[\frac{-2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\]
D. \[\frac{2\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{6}}\]
Answer» C. \[\frac{-2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\]
702.

If \[(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c})\] where \[\vec{a},\vec{b}\] and \[\vec{c}\] are any three vectors such that \[\vec{a}.\vec{b}\ne 0,\vec{b}.\vec{c}\ne 0\] then \[\vec{a}\] and \[\vec{c}\] are

A. Inclined at an angle of \[\frac{\pi }{3}\] between them
B. Inclined at an angle of \[\frac{\pi }{6}\] between them
C. Perpendicular
D. Parallel
Answer» E.
703.

Consider the parallelepiped with side \[\vec{a}=3\hat{i}+2\hat{j}+\hat{k},\text{ }\vec{b}=\hat{i}+\hat{j}+2\hat{k}\] and  \[\vec{c}=\hat{i}+3\hat{j}+3\hat{k}\] then the angle between \[\vec{a}\]and the plane containing the face determined by \[\vec{b}\] and \[\vec{c}\] is

A. \[\sin {{\,}^{-1}}\frac{1}{3}\]
B. \[\cos {{\,}^{-1}}\frac{1}{14}\]
C. \[sin{{\,}^{-1}}\frac{9}{14}\]
D. \[sin{{\,}^{-1}}\frac{2}{3}\]
Answer» D. \[sin{{\,}^{-1}}\frac{2}{3}\]
704.

If \[\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=4\hat{i}+3\hat{j}+4\hat{k}\] and\[\vec{c}=\hat{i}+\alpha \hat{j}+\beta \hat{k}\] are coplanar and \[\left| {\vec{c}} \right|=\sqrt{3}\], then

A. \[\alpha =\sqrt{2},\beta =1\]
B. \[\alpha =1,\beta =\pm 1\]
C. \[\alpha =\pm 1,\beta =1\]
D. \[\alpha =\pm 1,\beta =-1\]
Answer» D. \[\alpha =\pm 1,\beta =-1\]
705.

Consider the following statements. I. If (a, 1), (b, 2) and (c, 1) are in \[A\times B\] and \[n(A)=3,\,\,n(B)=2\] then \[A=\{a,b,c\}\] and \[B=\{1,2\}\] II. If \[A=\{1,2\}\] and \[B=\{3,4\},\] then \[A\times (B\cap \phi )\]is equal to \[A\times B.\] Choose the correct option.

A. Only I is true
B. Only II is true
C. Both are true
D. Neither I nor II is true
Answer» B. Only II is true
706.

Given the line \[L:\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-3}{-1}\]  and the plane\[\pi :x-2y=z\]. of the following assertions, the only one that is always true is

A. L is \[\bot \] to \[\pi \]
B. L lies in \[\pi \]
C. L is paralel to \[\pi \]
D. None of these
Answer» C. L is paralel to \[\pi \]
707.

A line makes the same angle \[\alpha \] with each of the x and y axes. If the angle\[\theta \], which it makes with the z-axis, is such that\[si{{n}^{2}}\theta =2\,{{\sin }^{2}}\alpha \], then what is the value of\[\alpha \]?

A. \[\pi /4\]
B. \[\pi /6\]
C. \[\pi /3\]
D. \[\pi /2\]
Answer» B. \[\pi /6\]
708.

The distance of the point (1, -2, 3) from the plane \[x-y+z=5\] measured parallel to the line \[\frac{x}{2}=\frac{y}{3}=\frac{z-1}{-6}\] is

A. 1
B. 2
C. 4
D. \[2\sqrt{3}\]
Answer» B. 2
709.

The ratio of the areas bounded by the curves \[y=\cos x\] and \[y=\cos 2x\] between \[x=0,\] \[x=\pi /3\] and \[x-\]axis, is                                                                       [MP PET 1997]

A. \[\sqrt{2}:1\]
B. \[1:1\]
C. \[1:2\]
D. \[2:1\]
Answer» E.
710.

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then \[{{P}^{2}}{{R}^{3}}:{{S}^{3}}\] is equal to

A. 0.0423611111111111
B. Common ratio: 1
C. \[{{\left( \text{first term} \right)}^{\text{2}}}\text{: }{{\left( \text{common ratio} \right)}^{\text{2}}}\]
D. \[{{\left( common\text{ }ratio \right)}^{n}}:1\]
Answer» B. Common ratio: 1
711.

The value of \[\theta \]lying between 0 and \[\pi /2\]and satisfying the equation  \[\left| \,\begin{matrix}    1+{{\sin }^{2}}\theta  & {{\cos }^{2}}\theta  & 4\sin 4\theta   \\    {{\sin }^{2}}\theta  & 1+{{\cos }^{2}}\theta  & 4\sin 4\theta   \\    {{\sin }^{2}}\theta  & {{\cos }^{2}}\theta  & 1+4\sin 4\theta   \\ \end{matrix}\, \right|=0\] [IIT 1988; MNR 1992; Kurukshetra CEE 1998; DCE 1996]

A. \[\frac{7\pi }{24}\] or \[\frac{11\pi }{24}\]
B. \[\frac{5\pi }{24}\]
C. \[\frac{\pi }{24}\]
D. None of these
Answer» B. \[\frac{5\pi }{24}\]
712.

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

A. \[\frac{2\,\,m\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\]
B. \[\frac{(2m)\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\]
C. \[\frac{2m\,\,!}{{{2}^{m}}.\,{{(m\,\,!)}^{2}}}.\frac{\pi }{2}\]
D. None of these
Answer» C. \[\frac{2m\,\,!}{{{2}^{m}}.\,{{(m\,\,!)}^{2}}}.\frac{\pi }{2}\]
713.

If \[A=\{2,\,4,\,5\},\,\,B=\{7,\,\,8,\,9\},\] then \[n(A\times B)\] is equal to

A. 6
B. 9
C. 3
D. 0
Answer» C. 3
714.

If \[A+B=\frac{\pi }{4},\]then \[(1+\tan A)(1+\tan B)=\]

A. 1
B. 2
C. \[\infty \]
D. -2
Answer» C. \[\infty \]
715.

If \[A\] and \[B\] are square matrices of same order, then [Roorkee 1995]

A. \[A+B=B+A\]
B. \[A+B=A-B\]
C. \[A-B=B-A\]
D. \[AB=BA\]
Answer» B. \[A+B=A-B\]
716.

If \[\cos A=m\cos B,\]then [MNR 1990]

A. \[\cot \frac{A+B}{2}=\frac{m+1}{m-1}\tan \frac{B-A}{2}\]
B. \[\tan \frac{A+B}{2}=\frac{m+1}{m-1}\cot \frac{B-A}{2}\]
C. \[\cot \frac{A+B}{2}=\frac{m+1}{m-1}\tan \frac{A-B}{2}\]
D. None of these
Answer» B. \[\tan \frac{A+B}{2}=\frac{m+1}{m-1}\cot \frac{B-A}{2}\]
717.

The equation of line passing through the point of intersection of the lines \[4x-3y-1=0\]and \[5x-2y-3=0\] and parallel to the line \[2y-3x+2=0,\] is                                                            [RPET 1985, 86, 88]

A. \[x-3y=1\]
B. \[3x-2y=1\]
C. \[2x-3y=1\]
D. \[2x-y=1\]
Answer» C. \[2x-3y=1\]
718.

If \[A=\{4n+2|n\] is a natural number} and \[B=\{3n|n\] is a natural number,}, then what is \[(A\cap B)\] equal to?

A. \[\{12{{n}^{2}}+6n|n\] is a natural number}
B. \[\{24n-12|n\] is a natural number}
C. \[\{60n+30|n\] is a natural number}
D. \[\{12n-6|n\] is a natural number}
Answer» E.
719.

If the projections of a line on the co-ordinate axes be           2, ?1, 2, then the length of the lines is

A. 3
B. 4
C. 2
D. \[\frac{1}{2}\]
Answer» B. 4
720.

If \[A=\left[ \begin{matrix}    1 & -2  \\    5 & 3  \\ \end{matrix} \right]\], then \[A+{{A}^{T}}\]equals [RPET 1994]

A. \[\left[ \begin{matrix}    2 & 3  \\    3 & 6  \\ \end{matrix} \right]\]
B. \[\left[ \begin{matrix}    2 & -4  \\    10 & 6  \\ \end{matrix} \right]\]
C. \[\left[ \begin{matrix}    2 & 4  \\    -10 & 6  \\ \end{matrix} \right]\]
D. None of these
Answer» B. \[\left[ \begin{matrix}    2 & -4  \\    10 & 6  \\ \end{matrix} \right]\]
721.

The co-ordinates of the point in which the line joining the points \[(3,\,\ 5,\ -7)\] and \[(-2,\,\ 1,\,\ 8)\] is intersected by the plane yz are given by                                                                      [MP PET 1993]

A. \[\left( 0,\,\frac{13}{5},\,\,2 \right)\]
B. \[\left( 0,\,-\frac{13}{5},\,-2 \right)\]
C. \[\left( 0,-\frac{13}{5},\frac{2}{5} \right)\]
D. \[\left( 0,\frac{13}{5},\frac{2}{5} \right)\]
Answer» B. \[\left( 0,\,-\frac{13}{5},\,-2 \right)\]
722.

The direction cosines of the normal to the plane \[x+2y-3z+4=0\] are       [MP PET  1996; Pb. CET  2000]

A. \[-\frac{1}{\sqrt{14}},-\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\]
B. \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\]
C. \[-\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\]
D. \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},-\frac{3}{\sqrt{14}}\]
Answer» B. \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\]
723.

In an organization number of women are \[\mu \]times than that of men. If n things are distributed among them and the probability that the number of things Received by men are odd is\[\frac{1}{2}-{{\left( \frac{1}{2} \right)}^{n+1}}\], then \[\mu \] equal to

A. 1
B. 2
C. 3
D. \[\frac{1}{4}\]
Answer» D. \[\frac{1}{4}\]
724.

An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is

A. \[\frac{1}{10}\]
B. \[\frac{3}{10}\]
C. \[\frac{3}{5}\]
D. \[\frac{1}{2}\]
Answer» E.
725.

If \[A(3,\,5),B(-5,\,-4),C(7,\,10)\] are the vertices of a parallelogram, taken in the order, then the co-ordinates of the fourth vertex are [Kerala (Engg.) 2005]

A. (10, 19)
B. (15, 10)
C. (19, 10)
D. (19, 15)
E. (e) (15, 19)
Answer» F.
726.

\[\sqrt{2+\sqrt{2+2\cos 4\theta }}=\]

A. \[\cos \theta \]
B. \[\sin \theta \]
C. \[2\cos \theta \]
D. \[2\sin \theta \]
Answer» D. \[2\sin \theta \]
727.

The real roots of the equation \[{{x}^{2}}+5|x|+\,\,4=0\] are  [UPSEAT 1993, 99; Orissa JEE 2004]

A. - 1, 4
B. 1, 4
C. - 4, 4
D. None of these
Answer» E.
728.

A box contains 20 identical balls of which 10 are blue and 10 are green. The balls are drawn at random from the box. One at a time with replacement. The probability that a blue ball is drawn 4th time on the 7th draw is

A. \[\frac{27}{32}\]
B. \[\frac{5}{64}\]
C. \[\frac{5}{32}\]
D. \[\frac{1}{2}\]
Answer» D. \[\frac{1}{2}\]
729.

A bag contains p white and q black ball. Two players A and B alternately draw a ball from the bag, replacing the balls each time after the draw till one of them draws a white ball and wins the game. If A begins the game and the probability of A winning the game is three times chat of B, then the ratio p:q is:

A. 0.127777777777778
B. 0.16875
C. 0.0840277777777778
D. 1 : 2
Answer» D. 1 : 2
730.

The bisector of the acute angle formed between the lines \[4x-3y+7=0\] and \[3x-4y+14=0\] has the equation:

A. \[x+y+3=0\]
B. \[x-y-3=0\]
C. \[x-y+3=0\]
D. \[3x+y-7=0\]
Answer» D. \[3x+y-7=0\]
731.

If the point \[P(x,y)\] is equidistant from points\[A(a+b,b-a)\] and \[B(a-b,a+b)\], then

A. \[ax=by\]
B. \[bx=ay\] and P ca be (a, b)
C. \[{{x}^{2}}-{{y}^{2}}=2(ax+by)\]
D. None of the above
Answer» C. \[{{x}^{2}}-{{y}^{2}}=2(ax+by)\]
732.

Let \[A\left( \alpha ,\frac{1}{\alpha } \right),B\left( \alpha ,\frac{1}{\beta } \right),C\left( \gamma ,\frac{1}{\gamma } \right)\] be the vertices of a \[\Delta ABC,\] where \[\alpha ,\beta \] are the roots of the equation \[{{x}^{2}}-6{{p}_{1}}x+2=0,\,\,\beta ,\,\,\gamma \]\[{{x}^{2}}-6{{p}_{1}}x+2=0,\beta ,\gamma \] are the roots of the equation \[{{x}^{2}}-6{{p}_{2}}x+3=0\] and \[\gamma ,\alpha \] are the roots of the equation \[{{x}^{2}}-6{{p}_{3}}x+6=0,{{p}_{1}},{{p}_{2}},{{p}_{3}}\] being positive. Then, the coordinates of the centroid of \[\Delta ABC\] is

A. \[\left( 1,\frac{11}{18} \right)\]
B. \[\left( 0,\frac{11}{8} \right)\]
C. \[\left( 2,\frac{11}{18} \right)\]
D. None of these
Answer» D. None of these
733.

If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}\] (gof) \[(x)=2{{x}^{2}}-5x+2,\] then \[f(x)\] is equal to

A. \[2x-3\]
B. \[2x+3\]
C. \[2{{x}^{2}}+3x+1\]
D. \[2{{x}^{2}}-3x-1\]
Answer» B. \[2x+3\]
734.

The number of objective functions from a set A to itself when A contains 106 elements, is

A. 106
B. \[{{(106)}^{2}}\]
C. \[(106)!\]
D. \[{{2}^{106}}\]
Answer» D. \[{{2}^{106}}\]
735.

The graph of the function \[\cos x\,\cos \,(x+2)-co{{s}^{2}}(x+1)\] is

A. A straight line passing through \[(0,-si{{n}^{2}}1)\] with slope 2
B. A straight line passing through (0, 0)
C. A parabola with vertex \[(1,-si{{n}^{2}}1)\]
D. A straight line passing through the point \[\left( \frac{\pi }{2},-{{\sin }^{2}}1 \right)\]and parallel to the x-axis.
Answer» E.
736.

Period of the function \[\left| {{\sin }^{3}}\frac{x}{2} \right|+\left| {{\cos }^{5}}\frac{x}{5} \right|\]is:

A. \[2\pi \]
B. \[10\pi \]
C. \[8\pi \]
D. \[5\pi \]
Answer» C. \[8\pi \]
737.

Let\[0

A. \[\tan \left( x-\frac{\pi }{4} \right)\]
B. \[\tan \left( \frac{\pi }{4}-x \right)\]
C. \[\tan \left( x+\frac{\pi }{4} \right)\]
D. \[{{\tan }^{2}}\left( x+\frac{\pi }{4} \right)\]
Answer» C. \[\tan \left( x+\frac{\pi }{4} \right)\]
738.

The point of contact of the tangent to the circle \[{{x}^{2}}+{{y}^{2}}=5\]at the point (1, ?2) which touches the circle \[{{x}^{2}}+{{y}^{2}}-8x+6y+20=0\], is                                                                [Roorkee 1989]

A. (2, -1)
B. (3, -1)
C. (4, -1)
D. (5, -1)
Answer» C. (4, -1)
739.

If a set A has n  elements, then the total number of subsets of A  is [Roorkee 1991; Karnataka CET 1992,  2000]

A. n
B. \[{{n}^{2}}\]
C. \[{{2}^{n}}\]
D. \[2n\]
Answer» D. \[2n\]
740.

If \[f:R\to R\] is defined by \[f(x)=3x+\left| x \right|,\] then\[f(2x)-f(-x)-6x=\]

A. \[f(x)\]
B. \[2f(x)\]
C. \[-f(x)\]
D. \[f(-x)\]
Answer» B. \[2f(x)\]
741.

In the following frequency distribution. Class limits of some of the class intervals and mid-vale of a class are missing. However, the mean of the distribution is known to be 46.5. Class intervals Mid-values Frequency \[{{x}_{1}}-{{x}_{2}}\] 15 10 \[{{x}_{2}}-{{x}_{3}}\] 30 40 \[{{x}_{3}}-{{x}_{4}}\] M 30 \[{{x}_{4}}-{{x}_{5}}\] 75 10 \[{{x}_{5}}-100\] 90 10 The values of \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},{{x}_{5}}\] respectively will be

A. \[(0,20,40,60,80)\]
B. \[(40,50,60,70,80)\]
C. \[(10,20,40,70,80)\]
D. \[(0,19.5,39.5,69.5,80)\]
Answer» D. \[(0,19.5,39.5,69.5,80)\]
742.

Let \[\bar{x}\] be the mean of n observations \[{{x}_{1}},{{x}_{2}},...{{x}_{n}},\]if \[(a-b)\] is added to each observation, then what is the mean of new set of observations?

A. 0
B. \[\bar{x}\]
C. \[\bar{x}-(a-b)\]
D. \[\bar{x}+(a-b)\]
Answer» E.
743.

\[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is coplanar with

A. b and c
B. c and a
C. a and b
D. a, b and c
Answer» B. c and a
744.

\[f(x)=\frac{x(x-p)}{q-p}+\frac{x(x-q)}{p-q},\] \[p\ne q\]. What is the value of\[f\left( q \right)+f\left( q \right)\]?

A. \[f(p-q)\]
B. \[f(p+q)\]
C. \[f(p(p+q))\]
D. \[f(q(p-q))\]
Answer» C. \[f(p(p+q))\]
745.

If \[\sin A=\frac{1}{\sqrt{10}}\]and \[\sin B=\frac{1}{\sqrt{5}},\]where A and B are positive acute angles, then \[A+B=\] [MP PET 1986]

A. \[\pi \]
B. \[\pi /2\]
C. \[\pi /3\]
D. \[\pi /4\]
Answer» E.
746.

The range of a random variable x is \[\{1,2,3,...\}.\] If \[P(x=r)=\frac{1}{{{2}^{r}}}\], then the mean of the distribution is

A. 8
B. 16
C. 1
D. 2
Answer» E.
747.

A 10cm long rod AB moves with its ends on two mutually perpendicular straight lines OX and OY. If the end A be moving at the rate of \[2cm/\sec \], then when the distance of A from O is \[8cm\], the rate at which the end B is moving, is                              [SCRA 1996]

A. \[\frac{8}{3}cm/\sec \]
B. \[\frac{4}{3}cm/\sec \]
C. \[\frac{2}{9}cm/\sec \]
D. None of these
Answer» B. \[\frac{4}{3}cm/\sec \]
748.

A vector of magnitude 3, bisecting the angle between the vectors \[\overset{\to }{\mathop{a}}\,=2\hat{i}+\hat{j}-\hat{k}\] and \[\overset{\to }{\mathop{b}}\,=\hat{i}-2\hat{j}+\hat{k}\] and making an obtuse angle with \[\overset{\to }{\mathop{b}}\,\] is

A. \[\frac{3\hat{i}-\hat{j}}{\sqrt{6}}\]
B. \[\frac{\hat{i}+3\hat{j}-2\hat{k}}{\sqrt{14}}\]
C. \[\frac{3(\hat{i}+3\hat{j}-2\hat{k})}{\sqrt{14}}\]
D. \[\frac{3\hat{i}-\hat{j}}{\sqrt{10}}\]
Answer» D. \[\frac{3\hat{i}-\hat{j}}{\sqrt{10}}\]
749.

If vector \[a=2i-3j+6k\] and vector \[b=-2i+2j-k\],\[\text{then}\,\,\,\frac{\text{Projection}\,\,\text{of}\,\,\text{vector}\,\,\text{a}\,\,\text{on}\,\,\text{vector}\,\,\text{b}}{\text{Projection}\,\,\text{of}\,\,\text{vector}\,\,\text{b}\,\,\text{on}\,\,\text{vector}\,\,\text{a}}\text{=}\]

A. \[\frac{3}{7}\]
B. \[\frac{7}{3}\]
C. 3
D. 7
Answer» C. 3
750.

One set containing five members has mean 8, variance 18 and the second set containing three members has mean 8 and variance 24. The variance of combined set of numbers is

A. 24
B. 20.25
C. 22.25
D. None of these
Answer» C. 22.25