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MCQOPTIONS
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.
| 801. |
The existence of the unique solution of the system \[x+y+z=\lambda ,\] \[5x-y+\mu z=10\], \[2x+3y-z=6\] depends on [Kurukshetra CEE 2002] |
| A. | \[\mu \]only |
| B. | \[\lambda \]only |
| C. | \[\lambda \]and \[\mu \] both |
| D. | Neither \[\lambda \]nor \[\mu \] |
| Answer» B. \[\lambda \]only | |
| 802. |
In a test of statistics marks were awarded out of 40. The average of 15 students was 38. Later it was decided to give marks out of 50. The new average marks will be |
| A. | 40 |
| B. | 47.5 |
| C. | 95 |
| D. | 41.5 |
| Answer» C. 95 | |
| 803. |
If \[\sum\nolimits_{i=1}^{9}{({{x}_{i}}-5)=9}\] and \[\sum\nolimits_{i=1}^{9}{{{({{x}_{i}}-5)}^{2}}=45,}\] then the standard deviation of the 9 items \[{{x}_{1}},{{x}_{2}},...{{x}_{9}}\] is |
| A. | 9 |
| B. | 4 |
| C. | 3 |
| D. | 2 |
| Answer» E. | |
| 804. |
Tangents drawn from origin to the circle \[{{x}^{2}}+{{y}^{2}}-2ax-2by+{{b}^{2}}=0\]are perpendicular to each other, if [MP PET 1995] |
| A. | \[a-b=1\] |
| B. | \[a+b=1\] |
| C. | \[{{a}^{2}}={{b}^{2}}\] |
| D. | \[{{a}^{2}}+{{b}^{2}}=1\] |
| Answer» D. \[{{a}^{2}}+{{b}^{2}}=1\] | |
| 805. |
The first of two samples has 100 items with mean 15 and SD 3. If the whole group has 250 items with mean 15.6 and \[SD=\sqrt{13.44}\] the SD of the second group is |
| A. | 5 |
| B. | 4 |
| C. | 6 |
| D. | 3.52 |
| Answer» C. 6 | |
| 806. |
Find the domain of the function \[f(x)=\sqrt{\left( \frac{2}{{{x}^{2}}-x+1}-\frac{1}{x+1}-\frac{2x-1}{{{x}^{3}}+1} \right)}\] |
| A. | \[(-\infty ,2]-\{-1\}\] |
| B. | \[(-\infty ,2)\] |
| C. | \[]-1,2]\] |
| D. | None of these |
| Answer» B. \[(-\infty ,2)\] | |
| 807. |
If \[{{x}^{2}}+2ax+10-3a>0\] for all \[x\in R\], then [IIT Screening 2004] |
| A. | \[-5<a<2\] |
| B. | \[a<-5\] |
| C. | \[a>5\] |
| D. | \[2<a<5\] |
| Answer» B. \[a<-5\] | |
| 808. |
A relation R is defined in the set Z of integers as follows \[(x,y)\in R\] iff \[{{x}^{2}}+{{y}^{2}}=9.\] Which of the following is false? |
| A. | \[R=\{(0,3),(0,-3),(3,0),(-3,0)\}\] |
| B. | Domain of \[R=\{-3,0,3\}\] |
| C. | Range of \[R=\{-3,0,3\}\] |
| D. | None of these |
| Answer» E. | |
| 809. |
The equation of the plane containing the line \[2x-5y+z=3;x+y+4z=5\], and parallel to the plane, \[x+3y+6z=1\], is: |
| A. | \[x+3y+6z=7\] |
| B. | \[2x+6y+12z=-13\] |
| C. | \[2x+6y+12z=13\] |
| D. | \[x+3y+6z=-7\] |
| Answer» B. \[2x+6y+12z=-13\] | |
| 810. |
A variable plane at a distance of 1 unit form the origin cuts the coordinate axes at A, B and C. if the centroid \[D(x,y,z)\] of triangle ABC satisfies the relation \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}=k\], then the value of k is |
| A. | 3 |
| B. | 1 |
| C. | 44256 |
| D. | 9 |
| Answer» E. | |
| 811. |
The set \[S=\{1,2,3,...,12\}\] is to be partitioned into three sets, A, B, C of equal size. Thus\[A\cup B\cup C=S,A\cap B=B\cap C=A\cap C=\phi \]. The number of ways to partition S is |
| A. | \[\frac{12!}{{{(4!)}^{3}}}\] |
| B. | \[\frac{12!}{{{(4!)}^{4}}}\] |
| C. | \[\frac{12!}{3!{{(4!)}^{3}}}\] |
| D. | \[\frac{12!}{3!{{(4!)}^{4}}}\] |
| Answer» B. \[\frac{12!}{{{(4!)}^{4}}}\] | |
| 812. |
5 - Digit numbers are to be formed using 2, 3, 5, 7, 9 without repeating the digits. If p be the number of such numbers that exceed 20000 and q be the number of those that lie between 30000 and 90000, then p:q is: |
| A. | \[6:5\] |
| B. | \[3:2\] |
| C. | \[4:3\] |
| D. | \[5:3\] |
| Answer» E. | |
| 813. |
The vectors \[\hat{i}-2x\hat{j}-3y\hat{k}\] and \[\hat{i}+3x\hat{j}+2y\hat{k}\] are orthogonal to each other. Then the locus of the point (x, y) is |
| A. | Hyperbola |
| B. | Ellipse |
| C. | Parabola |
| D. | Circle |
| Answer» E. | |
| 814. |
If a, b, c are the \[{{p}^{th}},\text{ }{{q}^{th}}.\text{ }{{\text{r}}^{th}}\] terms of an HP and \[\vec{u}=(q-r)\vec{i}+(r-p)\vec{j}+(p-q)\vec{k},\vec{v}=\frac{{\vec{i}}}{a}+\frac{{\vec{j}}}{b}+\frac{{\vec{k}}}{c}\] then |
| A. | \[\vec{u},\vec{v}\] are parallel vectors |
| B. | \[\vec{u},\vec{v}\] are orthogonal vectors |
| C. | \[\vec{u}.\vec{v}=1\] |
| D. | \[\vec{u}\times \vec{v}=\vec{i}+\vec{j}+\vec{k}\] |
| Answer» C. \[\vec{u}.\vec{v}=1\] | |
| 815. |
Given that the vectors \[\overline{\alpha }\] and \[\overset{\to }{\mathop{\beta }}\,\] are non-collinear. The values of x and y for which \[\overset{\to }{\mathop{u}}\,-\overset{\to }{\mathop{v}}\,=\overset{\to }{\mathop{w}}\,\] holds true if \[\overset{\to }{\mathop{u}}\,=2x\overset{\to }{\mathop{\alpha }}\,+y\overset{\to }{\mathop{\beta }}\,,\overset{\to }{\mathop{v}}\,=2\,y\overset{\to }{\mathop{\alpha }}\,+3x\overset{\to }{\mathop{\beta }}\,\] and \[\overset{\to }{\mathop{w}}\,=2\overset{\to }{\mathop{\alpha }}\,-5\overset{\to }{\mathop{\beta }}\,\]are |
| A. | \[x=2,y=1\] |
| B. | \[x=1,y=2\] |
| C. | \[x=-2,y=1\] |
| D. | \[x=-2,y=-1\] |
| Answer» B. \[x=1,y=2\] | |
| 816. |
The upper \[\frac{3}{4}\]th portion of a vertical pole subtends an angle \[{{\tan }^{-1}}\frac{3}{5}\] at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is |
| A. | 80 m |
| B. | 20 m |
| C. | 40 m |
| D. | 60 m |
| Answer» D. 60 m | |
| 817. |
If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] where \[a,\text{ }b,\text{ }c\text{ }\in \,\,R\text{ }\], then the maximum value of\[{{(4a-3b)}^{2}}+{{(5b-4c)}^{2}}+{{(3c-5a)}^{2}}\] is |
| A. | 25 |
| B. | 50 |
| C. | 144 |
| D. | None of these |
| Answer» C. 144 | |
| 818. |
For the matrix \[A=\left[ \begin{matrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \\ \end{matrix} \right]\], which of the following is correct [Kerala (Engg.)2001] |
| A. | \[{{A}^{3}}+3{{A}^{2}}-I=O\] |
| B. | \[{{A}^{3}}-3{{A}^{2}}-I=O\] |
| C. | \[{{A}^{3}}+2{{A}^{2}}-I=O\] |
| D. | \[{{A}^{3}}-{{A}^{2}}+I=O\] |
| Answer» C. \[{{A}^{3}}+2{{A}^{2}}-I=O\] | |
| 819. |
If \[\sqrt{3}\cos \,\theta +\sin \theta =\sqrt{2},\]then the most general value of \[\theta \] is [MP PET 1991, 2002; UPSEAT 1999] |
| A. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{4}\] |
| B. | \[{{(-1)}^{n}}\frac{\pi }{4}-\frac{\pi }{3}\] |
| C. | \[n\pi +\frac{\pi }{4}-\frac{\pi }{3}\] |
| D. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{4}-\frac{\pi }{3}\] |
| Answer» E. | |
| 820. |
If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=3\] and a, b are mutually perpendicular, then the area of the triangle whose vertices are \[\mathbf{0},\,\,\mathbf{a}+\mathbf{b},\,\,\mathbf{a}-\mathbf{b}\] is |
| A. | 5 |
| B. | 1 |
| C. | 6 |
| D. | 8 |
| Answer» D. 8 | |
| 821. |
From any point on the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] tangents are drawn to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}{{\sin }^{2}}\alpha \], the angle between them is [RPET 2002] |
| A. | \[\frac{\alpha }{2}\] |
| B. | \[\alpha \] |
| C. | \[2\alpha \] |
| D. | None of these |
| Answer» D. None of these | |
| 822. |
If \[\left| x \right| |
| A. | \[{{\left[ \frac{1-x}{1-2x} \right]}^{n}}\] |
| B. | \[{{(1-x)}^{n}}\] |
| C. | \[{{\left[ \frac{1-2x}{1-x} \right]}^{n}}\] |
| D. | \[{{\left( \frac{1}{1-x} \right)}^{n}}\] |
| Answer» B. \[{{(1-x)}^{n}}\] | |
| 823. |
The polar of the point (5, ?1/2) w.r.t circle \[{{(x-2)}^{2}}+{{y}^{2}}=4\]is [RPET 1996] |
| A. | \[5x-10y+2=0\] |
| B. | \[6x-y-20=0\] |
| C. | \[10x-y-10=0\] |
| D. | \[x-10y-2=0\] |
| Answer» C. \[10x-y-10=0\] | |
| 824. |
The position of a point in time ?t? is given by \[x=a+bt-c{{t}^{2}}\], \[y=at+b{{t}^{2}}\]. Its acceleration at time ?t? is [MP PET 2003] |
| A. | \[b-c\] |
| B. | \[b+c\] |
| C. | \[2b-2c\] |
| D. | \[2\sqrt{{{b}^{2}}+{{c}^{2}}}\] |
| Answer» E. | |
| 825. |
If \[\sin \theta =\frac{12}{13},(0 |
| A. | \[\frac{-56}{61}\] |
| B. | \[\frac{-56}{65}\] |
| C. | \[\frac{1}{65}\] |
| D. | -56 |
| Answer» C. \[\frac{1}{65}\] | |
| 826. |
The upper part of a tree broken over by the wind makes an angle of \[30{}^\circ \] with the ground and the distance from the root to the point where the top of the tree touches the ground is 10 m; what was the height of the tree |
| A. | \[8.66m\] |
| B. | \[15m\] |
| C. | \[17.32m\] |
| D. | \[25.98m\] |
| Answer» D. \[25.98m\] | |
| 827. |
The number of all possible matrices of order \[3\times 3\]with each entry 0 or 1 is |
| A. | 18 |
| B. | 512 |
| C. | 81 |
| D. | None of these |
| Answer» C. 81 | |
| 828. |
If \[A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\] and I is the unit matrix of order 3, then \[{{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}\] is equal to |
| A. | \[7{{A}^{8}}\] |
| B. | \[7{{A}^{7}}\] |
| C. | 8I |
| D. | 6I |
| Answer» B. \[7{{A}^{7}}\] | |
| 829. |
\[A=\left[ \begin{matrix} 1 & -1 \\ 2 & 3 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 2 & 3 \\ -1 & -2 \\ \end{matrix} \right]\] , then which of the following is/are correct? 1. \[AB({{A}^{-1}}{{B}^{-1}})\] is a unit matrix. 2. \[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\] Select the correct answer using the code given below: |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 only 2 |
| D. | Neither 1 nor 2 |
| Answer» E. | |
| 830. |
If \[\left[ \begin{matrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\]then find the value of x |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{5}\] |
| C. | No unique value of 'x' |
| D. | None of these |
| Answer» C. No unique value of 'x' | |
| 831. |
Let \[A=\left[ \begin{matrix} 0 & \alpha \\ 0 & 0 \\ \end{matrix} \right]\] and \[{{(A+I)}^{50}}-50A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right],\]find\[abc+abd+bcd+acd\] |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | None of these |
| Answer» B. -1 | |
| 832. |
If \[X=\left[ \begin{matrix} 1 & -2 \\ 0 & 3 \\ \end{matrix} \right]\], and I is a \[2\times 2\] identity matrix, then \[{{X}^{2}}-2X+3I\] equals to which one of the following? |
| A. | |
| B. | -2X |
| C. | 2X |
| D. | 4X |
| Answer» D. 4X | |
| 833. |
If \[A=\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 9 & a \\ b & c \\ \end{matrix} \right]\] and \[{{A}^{2}}=B\], then the value of a + b + c is |
| A. | 1 or -1 |
| B. | 5 or -1 |
| C. | 5 or 1 |
| D. | no real values |
| Answer» C. 5 or 1 | |
| 834. |
Consider the following in respect of the matrix \[A=\left( \begin{matrix} -1 & 1 \\ 1 & -1 \\ \end{matrix} \right):\] 1. \[{{A}^{2}}=-A\] 2. \[{{A}^{3}}=4A\] Which of the above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» C. Both 1 and 2 | |
| 835. |
If A and B are square matrices of size \[n\times n\] such that \[{{A}^{2}}-{{B}^{2}}=(A-B)(A+B)\], then which of the following will be always true? |
| A. | A = B |
| B. | AB = BA |
| C. | Either of A or B is a zero matrix |
| D. | Either of A or B is identity matrix |
| Answer» C. Either of A or B is a zero matrix | |
| 836. |
If Z is an idempotent matrix, then \[{{(I+Z)}^{n}}\] |
| A. | \[I+{{2}^{n}}Z\] |
| B. | \[I+({{2}^{n}}-1)Z\] |
| C. | \[I-({{2}^{n}}-1)Z\] |
| D. | None of these |
| Answer» C. \[I-({{2}^{n}}-1)Z\] | |
| 837. |
Which of the following is/are correct? |
| A. | B' AB is symmetric if A is symmetric |
| B. | B' AB is skew-symmetric if A is symmetric |
| C. | B' AB is symmetric if A is skew-symmetric |
| D. | None of these |
| Answer» B. B' AB is skew-symmetric if A is symmetric | |
| 838. |
If \[A=\left[ \begin{matrix} \alpha & \beta \\ \gamma & \delta \\ \end{matrix} \right]\] such that \[{{A}^{2}}\] is a two - rowed unit matrix, then \[\delta \] is equal to |
| A. | \[\alpha \] |
| B. | \[\beta \] |
| C. | \[\gamma \] |
| D. | None of these |
| Answer» B. \[\beta \] | |
| 839. |
If \[A=\left[ \begin{matrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right]\] then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}{{A}^{n}}\] is |
| A. | A null matrix |
| B. | An identity matrix |
| C. | \[\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix} \right]\] |
| D. | None of these |
| Answer» B. An identity matrix | |
| 840. |
If \[{{B}^{n}}-A=I\]and \[A=\left[ \begin{matrix} 26 & 26 & 18 \\ 25 & 37 & 17 \\ 52 & 39 & 50 \\ \end{matrix} \right],B=\left[ \begin{matrix} 1 & 4 & 2 \\ 3 & 5 & 1 \\ 7 & 1 & 6 \\ \end{matrix} \right]\] then n = |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» B. 3 | |
| 841. |
If \[A=\left[ \begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix} \right]\], I is the unit matrix of order 2 and a, b are arbitrary constants, then \[{{(aI+bA)}^{2}}\] is equal to |
| A. | \[{{a}^{2}}I+abA\] |
| B. | \[{{a}^{2}}I+2abA\] |
| C. | \[{{a}^{2}}I+{{b}^{2}}A\] |
| D. | None of these |
| Answer» C. \[{{a}^{2}}I+{{b}^{2}}A\] | |
| 842. |
If A and B are symmetric matrices of the same order and X = AB + BA and Y = AB - BA, then \[{{(XY)}^{T}}\] is equal to |
| A. | XY |
| B. | YX |
| C. | |
| D. | None of these |
| Answer» D. None of these | |
| 843. |
If a matrix A is such that\[3{{A}^{3}}+2{{A}^{2}}+5A+I=0,\] then what is \[{{A}^{-1}}\] equal to? |
| A. | \[-(3{{A}^{2}}+2A+5I)\] |
| B. | \[3{{A}^{2}}+2A+5I\] |
| C. | \[3{{A}^{2}}-2A-5I\] |
| D. | \[(3{{A}^{2}}+2A-5I)\] |
| Answer» B. \[3{{A}^{2}}+2A+5I\] | |
| 844. |
If the least number of zeroes in a lower triangular matrix is 10, then what is the order of the matrix? |
| A. | \[3\times 3\] |
| B. | \[4\times 4\] |
| C. | \[5\times 5\] |
| D. | \[10\times 10\] |
| Answer» C. \[5\times 5\] | |
| 845. |
If \[A=\left[ \begin{matrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} {{a}^{2}} & ab & ac \\ ab & {{b}^{2}} & bc \\ ac & bc & {{c}^{2}} \\ \end{matrix} \right]\], then AB is equal to |
| A. | B |
| B. | A |
| C. | O |
| D. | I |
| Answer» D. I | |
| 846. |
Let \[A+2B=\left[ \begin{matrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \\ \end{matrix} \right]\] and\[2A-B=\left[ \begin{matrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \\ \end{matrix} \right]\], then \[\operatorname{tr}(A) tr(B)\] is |
| A. | 1 |
| B. | 3 |
| C. | 2 |
| D. | 0 |
| Answer» D. 0 | |
| 847. |
If \[B=\left[ \begin{matrix} 3 & 4 \\ 2 & 3 \\ \end{matrix} \right]\] and \[C=\left[ \begin{matrix} 3 & -4 \\ -2 & 3 \\ \end{matrix} \right]\] and \[X=BC\],find \[{{X}^{n}}\] |
| A. | 0 |
| B. | I |
| C. | 2I |
| D. | None of these |
| Answer» C. 2I | |
| 848. |
The values of a, b, c if \[\left[ \begin{matrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \\ \end{matrix} \right]\] is orthogonal are |
| A. | \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{6}};c=\pm \frac{1}{\sqrt{3}}\] |
| B. | \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{3}};c=\pm \frac{1}{\sqrt{6}}\] |
| C. | \[a=\pm \frac{1}{\sqrt{6}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{3}}\] |
| D. | \[a=\pm \frac{1}{\sqrt{3}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{6}}\] |
| Answer» B. \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{3}};c=\pm \frac{1}{\sqrt{6}}\] | |
| 849. |
If A and B are two matrices such that AB = B and BA = A, then \[{{A}^{2}}+{{B}^{2}}\] is equal to |
| A. | 2AB |
| B. | 2BA |
| C. | A+B |
| D. | AB |
| Answer» D. AB | |
| 850. |
Let \[A=\left( \begin{matrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \\ \end{matrix} \right).\] and 10 \[B=\left( \begin{matrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \\ \end{matrix} \right).\] If B is the inverse of matrix A, then \[\alpha \] is |
| A. | 5 |
| B. | -1 |
| C. | 2 |
| D. | -2 |
| Answer» B. -1 | |