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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.
| 651. |
\[\int_{0}^{b-c}{\,\,{f}''(x+a)\,dx=}\] [SCRA 1990] |
| A. | \[{f}'(a)-{f}'(b)\] |
| B. | \[{f}'(b-c+a)-{f}'(a)\] |
| C. | \[{f}'(b+c-a)+{f}'(a)\] |
| D. | None of these |
| Answer» C. \[{f}'(b+c-a)+{f}'(a)\] | |
| 652. |
The equation of the circle which passes through the intersection of \[{{x}^{2}}+{{y}^{2}}+13x-3y=0\]and \[2{{x}^{2}}+2{{y}^{2}}+4x-7y-25=0\] and whose centre lies on \[13x+30y=0\] is [DCE 2001] |
| A. | \[{{x}^{2}}+{{y}^{2}}+30x-13y-25=0\] |
| B. | \[4{{x}^{2}}+4{{y}^{2}}+30x-13y-25=0\] |
| C. | \[2{{x}^{2}}+2{{y}^{2}}+30x-13y-25=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+30x-13y+25=0\] |
| Answer» C. \[2{{x}^{2}}+2{{y}^{2}}+30x-13y-25=0\] | |
| 653. |
The correct evaluation of \[\int_{0}^{\pi }{\left| \,{{\sin }^{4}}x\, \right|\,dx}\] is [MP PET 1993] |
| A. | \[\frac{8\pi }{3}\] |
| B. | \[\frac{2\pi }{3}\] |
| C. | \[\frac{4\pi }{3}\] |
| D. | \[\frac{3\pi }{8}\] |
| Answer» E. | |
| 654. |
The unit vector perpendicular to the vectors \[6\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}-6\mathbf{j}-2\mathbf{k},\] is [IIT 1989; RPET 1996] |
| A. | \[\frac{2\mathbf{i}-3\mathbf{j}+6\mathbf{k}}{7}\] |
| B. | \[\frac{2\mathbf{i}-3\mathbf{j}-6\mathbf{k}}{7}\] |
| C. | \[\frac{2\mathbf{i}+3\mathbf{j}-6\mathbf{k}}{7}\] |
| D. | \[\frac{2\mathbf{i}+3\mathbf{j}+6\mathbf{k}}{7}\] |
| Answer» D. \[\frac{2\mathbf{i}+3\mathbf{j}+6\mathbf{k}}{7}\] | |
| 655. |
The area bounded by the curves \[y=\sqrt{x},\] \[2y+3=x\] and \[x-\]axis in the 1st quadrant is [IIT Screening 2003] |
| A. | 9 |
| B. | \[\frac{27}{4}\] |
| C. | 36 |
| D. | 18 |
| Answer» B. \[\frac{27}{4}\] | |
| 656. |
If O is the origin and \[OP=3\]with direction ratios \[-1,\,2,-2\], then co-ordinates of P are [RPET 2000; DCE 2005] |
| A. | (1, 2, 2) |
| B. | \[(-1,\,2,\,-2)\] |
| C. | (?3, 6, ?9) |
| D. | \[(-1/3\,,\,2/3,\,-2/3)\] |
| Answer» C. (?3, 6, ?9) | |
| 657. |
If A and B are two events. The probability that at most one of A, B occurs, is |
| A. | \[1-P(A\cap B)\] |
| B. | \[P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B})\] |
| C. | \[P(\bar{A})+P(\bar{B})+P(A\cup B)-1\] |
| D. | All of these |
| Answer» E. | |
| 658. |
If A and B are any two sets, then \[A\text{ }\cap \text{ }\left( A\text{ }\cup \text{ }B \right)\] is equal to |
| A. | A |
| B. | B |
| C. | \[{{A}^{c}}\] |
| D. | \[{{B}^{c}}\] |
| Answer» B. B | |
| 659. |
If the radius of the cirumcircle of isosceles triangle ABC is equal to AB=AC, then the angle A is: |
| A. | \[30{}^\circ \] |
| B. | \[60{}^\circ \] |
| C. | \[90{}^\circ \] |
| D. | \[120{}^\circ \] |
| Answer» E. | |
| 660. |
If \[\tan \theta -\sqrt{2}\sec \theta =\sqrt{3}\], then the general value of \[\theta \] is |
| A. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{4}-\frac{\pi }{3}\] |
| B. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{3}-\frac{\pi }{4}\] |
| C. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{3}+\frac{\pi }{4}\] |
| D. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{4}+\frac{\pi }{3}\] |
| Answer» C. \[n\pi +{{(-1)}^{n}}\frac{\pi }{3}+\frac{\pi }{4}\] | |
| 661. |
What is the acute angle between the planes \[x+y+2z=3\] and \[-2x+y-z=11?\] |
| A. | \[\pi /5\] |
| B. | \[\pi /4\] |
| C. | \[\pi /6\] |
| D. | \[\pi /3\] |
| Answer» E. | |
| 662. |
If \[\theta \] is the acute angle between the diagonals of a cube, then which one of the following is correct? |
| A. | \[\theta <30{}^\circ \] |
| B. | \[\theta =60{}^\circ \] |
| C. | \[30{}^\circ <\theta <60{}^\circ \] |
| D. | \[\theta >60{}^\circ \] |
| Answer» E. | |
| 663. |
If \[A=\left[ \begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{matrix} \right]\], then \[{{A}^{2}}\]is [MNR 1980] |
| A. | Null matrix |
| B. | Unit matrix |
| C. | A |
| D. | 2A |
| Answer» C. A | |
| 664. |
If a, b, c are position vector of vertices of a triangle \[ABC\], then unit vector perpendicular to its plane is [RPET 1999] |
| A. | \[a\times b+b\times c+c\times a\] |
| B. | \[\frac{a\times b+b\times c+c\times a}{|a\times b+b\times c+c\times a|}\] |
| C. | \[\frac{a\times b}{|a\times b|}\] |
| D. | None of these |
| Answer» C. \[\frac{a\times b}{|a\times b|}\] | |
| 665. |
What must be the matrix X if \[2X+\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} 3 & 8 \\ 7 & 2 \\ \end{matrix} \right]\] [Karnataka CET 2004] |
| A. | \[\left[ \begin{matrix} 1 & 3 \\ 2 & -1 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 1 & -3 \\ 2 & -1 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} 2 & 6 \\ 4 & -2 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 2 & -6 \\ 4 & -2 \\ \end{matrix} \right]\] |
| Answer» B. \[\left[ \begin{matrix} 1 & -3 \\ 2 & -1 \\ \end{matrix} \right]\] | |
| 666. |
If \[\sin \beta \]is the geometric mean between \[\sin \alpha \]and \[\cos \alpha ,\]then \[\cos 2\beta \]is equal to |
| A. | \[2{{\sin }^{2}}\left( \frac{\pi }{4}-\alpha \right)\] |
| B. | \[2{{\cos }^{2}}\left( \frac{\pi }{4}-\alpha \right)\] |
| C. | \[2{{\cos }^{2}}\left( \frac{\pi }{4}+\alpha \right)\] |
| D. | \[2{{\sin }^{2}}\left( \frac{\pi }{4}+\alpha \right)\] |
| Answer» D. \[2{{\sin }^{2}}\left( \frac{\pi }{4}+\alpha \right)\] | |
| 667. |
What is the angle between the lines \[x+y=1\]and \[x-y=1?\] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 668. |
Through the point \[P(\alpha ,\beta )\], where \[\alpha \beta >0.\] the straight line \[\frac{x}{a}+\frac{y}{b}=1\] is drawn so as the form with axes a triangle of area S. if \[ab>0,\] then least value of S is |
| A. | \[\alpha \beta \] |
| B. | \[2\alpha \beta \] |
| C. | \[3\alpha \beta \] |
| D. | None of these |
| Answer» C. \[3\alpha \beta \] | |
| 669. |
P is point on the line \[y+2x=1,\] and Q and R are two points on the line \[3y+6x=6\]such that triangle PQR is an equilateral triangle. The length of the side of the triangle is |
| A. | \[2/\sqrt{15}\] |
| B. | \[3/\sqrt{5}\] |
| C. | \[4/\sqrt{5}\] |
| D. | None of these |
| Answer» B. \[3/\sqrt{5}\] | |
| 670. |
If \[f(x)=2x+\left| x \right|,g(x)=\frac{1}{3}(2x-\left| x \right|)\] and \[h(x)=f(g(x)),\] then domain of \[{{\sin }^{-1}}\]\[\underbrace{(h(h(h(h...h(x)...))))}_{n\,times}\] is |
| A. | \[[-1,1]\] |
| B. | \[\left[ -1,-\frac{1}{2} \right]\cup \left[ \frac{1}{2},1 \right]\] |
| C. | \[\left[ -1,-\frac{1}{2} \right]\] |
| D. | \[\left[ \frac{1}{2},1 \right]\] |
| Answer» B. \[\left[ -1,-\frac{1}{2} \right]\cup \left[ \frac{1}{2},1 \right]\] | |
| 671. |
Let \[A=Z\cup \{\sqrt{2}\}.\] Define a relation R in A by aRb if and only if \[a+b\in Z.\] The relation R is |
| A. | Reflexive |
| B. | Symmetric and transitive |
| C. | Only transitive |
| D. | None of these |
| Answer» C. Only transitive | |
| 672. |
If \[f(x)=\left\{ \begin{matrix} {{x}^{3}}+1,x |
| A. | \[x,\forall x\in R\] |
| B. | \[x-1,\forall x\in R\] |
| C. | \[x+1,\forall x\in R\] |
| D. | None of these |
| Answer» B. \[x-1,\forall x\in R\] | |
| 673. |
The domain of the function \[f(x)=lo{{g}_{e}}\{sgn(9-{{x}^{2}})\}+\sqrt{{{[x]}^{3}}-4[x]}\] (where [.] represents the greatest integer function) is |
| A. | \[\left[ -2,1 \right)\cup \left[ 2,3 \right)\] |
| B. | \[\left[ -4,1 \right)\cup \left[ 2,3 \right)\] |
| C. | \[\left[ 4,1 \right)\cup \left[ 2,3 \right)\] |
| D. | \[\left[ 2,1 \right)\cup \left[ 2,3 \right)\] |
| Answer» B. \[\left[ -4,1 \right)\cup \left[ 2,3 \right)\] | |
| 674. |
Let \[f(x)=-1+\left| x-1 \right|,-1\le x\le 3\] and \[\le g(x)=2-\left| x+1 \right|,-2\le x\le 2,\] then \[(fog)(x)\], is equal to |
| A. | \[\left\{ \begin{matrix} x+1-2\le x\le 0 \\ x-10<x\le 2 \\ \end{matrix} \right.\] |
| B. | \[\left\{ \begin{matrix} x-1-2\le x\le 0 \\ x+10<x\le 2 \\ \end{matrix} \right.\] |
| C. | \[\left\{ \begin{matrix} -1-x-2\le x\le 0 \\ x-10<x\le 2 \\ \end{matrix} \right.\] |
| D. | None of these |
| Answer» E. | |
| 675. |
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},........,{{a}_{n}},......\] are in G.P. and \[{{a}_{i}}>0\]for each i, then the value of the determinant \[\Delta =\left| \,\begin{matrix} \log {{a}_{n}} & \log {{a}_{n+2}} & \log {{a}_{n+4}} \\ \log {{a}_{n+6}} & \log {{a}_{n+8}} & \log {{a}_{n+10}} \\ \log {{a}_{n+12}} & \log {{a}_{n+14}} & \log {{a}_{n+16}} \\ \end{matrix} \right|\] is equal to |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 676. |
For real numbers x and y, we define x R y iff \[x-y+\sqrt{5}\] is an irrational number. The relation R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» B. Symmetric | |
| 677. |
If \[f(x)=si{{n}^{2}}x+{{\sin }^{2}}\left( x+\frac{\pi }{3} \right)+\cos x\cos \left( x+\frac{\pi }{3} \right)\]and \[g\left( \frac{5}{4} \right)=1,\] then gof(x)= |
| A. | 1 |
| B. | 0 |
| C. | \[\sin x\] |
| D. | None |
| Answer» B. 0 | |
| 678. |
If a, b, c are the sides of a triangle, then the minimum value of \[\frac{a}{b+c-a}+\frac{b}{c+a-b}+\]\[\frac{c}{a+b-c}\] is equal to |
| A. | 3 |
| B. | 6 |
| C. | 9 |
| D. | 12 |
| Answer» B. 6 | |
| 679. |
In a triangle ABC, \[c=2,A=45{}^\circ ,a=2\sqrt{2}\], than what is C equal to? |
| A. | \[30{}^\circ \] |
| B. | \[15{}^\circ \] |
| C. | \[45{}^\circ \] |
| D. | None of these |
| Answer» B. \[15{}^\circ \] | |
| 680. |
From an aeroplane above a straight road the angle of depression of two positions at a distance 20 m apart on the road are observed to be \[30{}^\circ \] and \[45{}^\circ \]. The height of the aeroplane above the ground is: |
| A. | \[10\sqrt{3}\,m\] |
| B. | \[10(\sqrt{3}-1)m\] |
| C. | \[10(\sqrt{3}+1)m\] |
| D. | 20 m |
| Answer» D. 20 m | |
| 681. |
The gradient of the tangent line at the point \[(a\cos \alpha ,a\sin \alpha )\]to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], is |
| A. | \[\tan \alpha \] |
| B. | \[\tan (\pi -\alpha )\] |
| C. | \[\cot \alpha \] |
| D. | \[-\cot \alpha \] |
| Answer» E. | |
| 682. |
The number of real solutions of the equation\[|x{{|}^{2}}\]- \[3|x|+2=0\] are [IIT 1982, 89; MP PET 1997; DCE 2002; AMU 2000; UPSEAT 1999; AIEEE 2003] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 683. |
If \[A=\left( \begin{matrix} 1 & -2 & 1 \\ 2 & 1 & 3 \\ \end{matrix} \right)\]and \[B=\left( \begin{matrix} 2 & 1 \\ 3 & 2 \\ 1 & 1 \\ \end{matrix} \right)\], then \[{{(AB)}^{T}}\]is equal to [RPET 2001] |
| A. | \[\left( \begin{matrix} -3 & -2 \\ 10 & 7 \\ \end{matrix} \right)\] |
| B. | \[\left( \begin{matrix} -3 & 10 \\ -2 & 7 \\ \end{matrix} \right)\] |
| C. | \[\left( \begin{matrix} -3 & 7 \\ 10 & 2 \\ \end{matrix} \right)\] |
| D. | None of these |
| Answer» C. \[\left( \begin{matrix} -3 & 7 \\ 10 & 2 \\ \end{matrix} \right)\] | |
| 684. |
A pole stands vertically inside a triangular park ABC. If the angle of elevation of the top of the pole form each corner of the park is same, then the foot of the pole is at the |
| A. | Centroid |
| B. | circumcentre |
| C. | Incentre |
| D. | orthocentre |
| Answer» B. circumcentre | |
| 685. |
If the standard deviation of the observations \[-5,-4,-3,-2,-1,0,1,2,3,4,5\] is \[\sqrt{10}\]. The standard deviation of observations \[15,16,17,18,19,20,21,22,23,24,25\] will be |
| A. | \[\sqrt{10}+20\] |
| B. | \[\sqrt{10}+10\] |
| C. | \[\sqrt{10}\] |
| D. | None of these |
| Answer» D. None of these | |
| 686. |
For the data 3, 5, 1, 6, 5, 9, 2, 8, 6 the mean, median and mode are x, y, and z respectively. Which one of the following is correct? |
| A. | \[x=y\ne z\] |
| B. | \[x\ne y=z\] |
| C. | \[x\ne y\ne z\] |
| D. | \[x=y=z\] |
| Answer» E. | |
| 687. |
Consider the following relations :(1) \[A-B=A-(A\cap B)\](2) \[A=(A\cap B)\cup (A-B)\](3) \[A-(B\cup C)=(A-B)\cup (A-C)\]which of these is/are correct [NDA 2003] |
| A. | 1 and 3 |
| B. | 2 only |
| C. | 2 and 3 |
| D. | 1 and 2 |
| Answer» E. | |
| 688. |
If mean of the n observations \[{{x}_{1}},{{x}_{2}},{{x}_{3}},...{{x}_{n}}\] be \[\bar{x},\] then the mean of n observations \[2{{x}_{1}}+3,\,\,2{{x}_{2}}+3,\,\,2{{x}_{3}}+3,...,2{{x}_{n}}+3\] is |
| A. | \[3\bar{x}+2\] |
| B. | \[2\bar{x}+3\] |
| C. | \[\bar{x}+3\] |
| D. | \[2\bar{x}\] |
| Answer» C. \[\bar{x}+3\] | |
| 689. |
The angle between the pair of lines represented by \[2{{x}^{2}}-7xy+3{{y}^{2}}=0\], is [Kurukshetra CEE 2002] |
| A. | \[{{60}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{\tan }^{-1}}\left( \frac{7}{6} \right)\] |
| D. | \[{{30}^{o}}\] |
| Answer» B. \[{{45}^{o}}\] | |
| 690. |
If \[(\sin \theta =3\sin (\theta +2\alpha ),\] then the value of \[\tan (\theta +\alpha )+2\tan \alpha \]is |
| A. | 3 |
| B. | 2 |
| C. | \[-1\] |
| D. | 0 |
| Answer» E. | |
| 691. |
\[(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})=\] [MP PET 1997] |
| A. | [b c a] a |
| B. | [c a b] b |
| C. | [a b c] c |
| D. | [a c b] b |
| Answer» D. [a c b] b | |
| 692. |
If \[A=\left[ \begin{matrix} 2 & 3 \\ 4 & 6 \\ \end{matrix} \right]\], then \[{{A}^{-1}}\]= [Karnataka CET 2001] |
| A. | \[\left[ \begin{matrix} 1 & 2 \\ -3/2 & 3 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 2 & -3 \\ 4 & 6 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} -2 & 4 \\ -3 & 6 \\ \end{matrix} \right]\] |
| D. | Does not exist |
| Answer» E. | |
| 693. |
The equation of straight line passing through point of intersection of the straight lines \[3x-y+2=0\] and \[5x-2y+7=0\] and having infinite slope is [UPSEAT 2001] |
| A. | \[x=2\] |
| B. | \[x+y=3\] |
| C. | \[x=3\] |
| D. | \[x=4\] |
| Answer» D. \[x=4\] | |
| 694. |
The angle between the lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]is given by [RPET 1995] |
| A. | \[\tan \theta =\frac{2({{h}^{2}}-ab)}{(a+b)}\] |
| B. | \[\tan \theta =\frac{2\sqrt{{{h}^{2}}-ab}}{(a+b)}\] |
| C. | \[\tan \theta =\frac{2({{h}^{2}}-ab)}{\sqrt{a+b}}\] |
| D. | \[\tan \theta =\frac{2\sqrt{{{h}^{2}}+ab}}{(a+b)}\] |
| Answer» C. \[\tan \theta =\frac{2({{h}^{2}}-ab)}{\sqrt{a+b}}\] | |
| 695. |
\[\frac{1-i}{1+i}\]is equal to [RPET 1984] |
| A. | \[\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\] |
| B. | \[\cos \frac{\pi }{2}-i\sin \frac{\pi }{2}\] |
| C. | \[\sin \frac{\pi }{2}+i\cos \frac{\pi }{2}\] |
| D. | None of these |
| Answer» C. \[\sin \frac{\pi }{2}+i\cos \frac{\pi }{2}\] | |
| 696. |
If A = {2, 3, 5}, B = {2, 5, 6}, then \[\left( A\text{ }-\text{ }B \right)\text{ }\times \text{ }\left( A\text{ }\cap \text{ }B \right)\] is |
| A. | {(3, 2), (3, 3), (3, 5)} |
| B. | {(3, 2), (3, 5), (3, 6)} |
| C. | {(3, 2), (3, 5)} |
| D. | None of these |
| Answer» D. None of these | |
| 697. |
Matrix \[A=\left[ \begin{matrix} 1 & 0 & -k \\ 2 & 1 & 3 \\ k & 0 & 1 \\ \end{matrix} \right]\]is invertible for [UPSEAT 2002] |
| A. | \[k=1\] |
| B. | \[k=-1\] |
| C. | \[k=0\] |
| D. | All real k |
| Answer» E. | |
| 698. |
The area in the first quadrant between \[{{x}^{2}}+{{y}^{2}}={{\pi }^{2}}\] and \[y=\sin x\] is [MP PET 1997] |
| A. | \[\frac{({{\pi }^{3}}-8)}{4}\] |
| B. | \[\frac{{{\pi }^{3}}}{4}\] |
| C. | \[\frac{({{\pi }^{3}}-16)}{4}\] |
| D. | \[\frac{({{\pi }^{3}}-8)}{2}\] |
| Answer» B. \[\frac{{{\pi }^{3}}}{4}\] | |
| 699. |
A point \[(x,y,z)\] moves parallel to x-axis. Which of the three variable\[x,y,z\]remain fixed |
| A. | x |
| B. | y and z |
| C. | x and y |
| D. | z and x |
| Answer» C. x and y | |
| 700. |
If \[\sin (A+B)\]=1 and \[\cos (A-B)=\frac{\sqrt{3}}{2},\]then the smallest positive values of A and B are |
| A. | \[{{60}^{o}},\text{ }{{30}^{o}}\] |
| B. | \[{{75}^{o}},\text{ }{{15}^{o}}\] |
| C. | \[{{45}^{o}},\text{ }{{60}^{o}}\] |
| D. | \[{{45}^{o}},\text{ }{{45}^{o}}\] |
| Answer» B. \[{{75}^{o}},\text{ }{{15}^{o}}\] | |