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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.
| 601. |
If \[\varphi (x)=\int_{1/x}^{\sqrt{x}}{\sin ({{t}^{2}})\,dt,}\] then \[{\varphi }'(1)=\] |
| A. | \[\sin 1\] |
| B. | \[2\sin 1\] |
| C. | \[\frac{3}{2}\sin 1\] |
| D. | None of these |
| Answer» D. None of these | |
| 602. |
If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=5\] and \[|\mathbf{a}\times \mathbf{b}|\,=8,\] then a . b is equal to [AI CBSE 1984; RPET 1991] |
| A. | 0 |
| B. | 2 |
| C. | 4 |
| D. | 6 |
| Answer» E. | |
| 603. |
The probability that in the random arrangement of the letters of the word 'UNIVERSITY' the two I's does not come together is |
| A. | \[\frac{4}{5}\] |
| B. | 44317 |
| C. | 44470 |
| D. | 44478 |
| Answer» B. 44317 | |
| 604. |
A bag contains an assortment of blue and red balls. If two balls are drawn at random, the probability of drawing two red balls is five times the probability of drawing two blue balls furthermore, the probability of drawing one ball of each color is six times the probability of drawing two blue balls. The number of red and blue balls I the bag is |
| A. | 6, 3 |
| B. | 3, 6 |
| C. | 2, 7 |
| D. | None of these |
| Answer» B. 3, 6 | |
| 605. |
Two dice are thrown. What is the probability that the sum of the faces equals or exceeds 10? |
| A. | 44531 |
| B. | ¼ |
| C. | 44256 |
| D. | 44348 |
| Answer» E. | |
| 606. |
If the circle \[{{x}^{2}}+{{y}^{2}}+6x-2y+k=0\] bisects the circumference of the circle \[{{x}^{2}}+{{y}^{2}}+2x-6y-15=0,\] then k = [EAMCET 2003] |
| A. | 21 |
| B. | - 21 |
| C. | 23 |
| D. | - 23 |
| Answer» E. | |
| 607. |
If an integer q be chosen at random in the interval \[-10\le q\le 10,\] then the probability that the roots of the equation \[{{x}^{2}}+qx+\frac{3q}{4}+1=0\] are real is |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{15}{21}\] |
| C. | \[\frac{16}{21}\] |
| D. | \[\frac{17}{21}\] |
| Answer» E. | |
| 608. |
Let A and B be two events. Then \[1+P(A\cap B)-P(B)-P(A)\] is equal to |
| A. | \[P(\bar{A}\cup \bar{B})\] |
| B. | \[P(A\cap \bar{B})\] |
| C. | \[P(\bar{A}\cap B)\] |
| D. | \[P(\bar{A}\cap \bar{B})\] |
| Answer» E. | |
| 609. |
A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds and red for 30 seconds. At a randomly chosen time, the probability that the light will not be green is |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{1}{4}\] |
| C. | \[\frac{4}{3}\] |
| D. | \[\frac{7}{12}\] |
| Answer» E. | |
| 610. |
The position of a moving point in the XY-plane at time t is given by \[\left( (u\cos \alpha )t,(u\sin \alpha )t-\frac{1}{2}g{{t}^{2}} \right),\] where \[u,\,\alpha ,\,g\]are constants. The locus of the moving point is |
| A. | A circle |
| B. | A parabola |
| C. | An ellipse |
| D. | None of these |
| Answer» C. An ellipse | |
| 611. |
Chord AB is a diameter of the sphere \[\left| \vec{r}-2\vec{i}-\vec{j}+6\vec{k} \right|=\sqrt{18.}\] if the coordinates of A are (3, 2,-2), then the coordinates of B are |
| A. | (1, 0, 10) |
| B. | (1, 0, -10) |
| C. | (-1, 0, 10) |
| D. | None of these |
| Answer» C. (-1, 0, 10) | |
| 612. |
The equation of a sphere is \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10z=0\]. If one end point of a diameter of the sphere is (-3, -4, 5), what is the other end point? |
| A. | \[(-3,-4,-5)\] |
| B. | \[(3,4,5)\] |
| C. | \[(3,4,-5)\] |
| D. | \[(-3,4,-5)\] |
| Answer» C. \[(3,4,-5)\] | |
| 613. |
The equation of the plane through (1, 1, 1) and passing through the line of intersection of the planes \[x+2y-z+1=0\] and \[3x-y-4z+3=0\]is |
| A. | \[8x+5y-11z+8=0\] |
| B. | \[8x+5y-11z+8=0\] |
| C. | \[8x-5y-11z+8=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 614. |
If \[\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b},\] then what is \[\frac{\tan x}{\tan y}\] equal to? |
| A. | \[\frac{b}{a}\] |
| B. | \[\frac{a}{b}\] |
| C. | \[ab\] |
| D. | \[1\] |
| Answer» C. \[ab\] | |
| 615. |
The length of the shadow of a pole inclined at \[10{}^\circ \] to the vertical towards the sun is 2.05 meters, when the elevation of the sun is \[38{}^\circ .\] The length of the pole is |
| A. | \[\frac{2.05\sin 38{}^\circ }{\sin 42{}^\circ }\] |
| B. | \[\frac{2.05\sin 42{}^\circ }{\sin 38{}^\circ }\] |
| C. | \[\frac{2.05\cos 38{}^\circ }{\cos 42{}^\circ }\] |
| D. | None of these |
| Answer» B. \[\frac{2.05\sin 42{}^\circ }{\sin 38{}^\circ }\] | |
| 616. |
If \[\left[ \begin{matrix} x & 0 \\ 1 & y \\ \end{matrix} \right]+\left[ \begin{matrix} -2 & 1 \\ 3 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} 3 & 5 \\ 6 & 3 \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & 4 \\ 2 & 1 \\ \end{matrix} \right]\], then [RPET 1994] |
| A. | \[x=-3,y=-2\] |
| B. | \[x=3,y=-2\] |
| C. | \[x=3,y=2\] |
| D. | \[x=-3,y=2\] |
| Answer» C. \[x=3,y=2\] | |
| 617. |
The circumradius of the triangle formed by the three lines \[y+3x-5=0;y=x\] and \[3y-x+10=0\] is |
| A. | \[\frac{25}{4\sqrt{2}}\] |
| B. | \[\frac{25}{3\sqrt{2}}\] |
| C. | \[\frac{25}{2\sqrt{2}}\] |
| D. | \[\frac{25}{\sqrt{2}}\] |
| Answer» B. \[\frac{25}{3\sqrt{2}}\] | |
| 618. |
The intercept cut off by a line from y-axis twice than that form x-axis, and the line passes through the point (1, 2). The equation of the line is |
| A. | \[2x+y=4\] |
| B. | \[2x+y+4=0\] |
| C. | \[2x-y=4\] |
| D. | \[2x-y+4=0\] |
| Answer» B. \[2x+y+4=0\] | |
| 619. |
Two points \[P(a,0)\] and \[Q(-a,0)\] are given, R is a variable point on one side of the line PQ such that \[\angle RPQ-\angle RQP\] is \[2\alpha \]. Then, the locus of R is |
| A. | \[{{x}^{2}}-{{y}^{2}}+2xy\,\,\cot \,\,2\alpha -{{a}^{2}}=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+2xy\,\,\cot \,\,2\alpha -{{a}^{2}}=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+2xy\,\,\cot \,\,2\alpha +{{a}^{2}}=0\] |
| D. | None of the above |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}+2xy\,\,\cot \,\,2\alpha -{{a}^{2}}=0\] | |
| 620. |
If \[A=\left[ \begin{matrix} ab & {{b}^{2}} \\ -{{a}^{2}} & -ab \\ \end{matrix} \right]\]and \[{{A}^{n}}=O\], then the minimum value of n is |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» B. 3 | |
| 621. |
Co-ordinate of a point equidistant from the points (0,0,0), (a, 0, 0), (0, b, 0), (0, 0, c) is [RPET 2003] |
| A. | \[\left( \frac{a}{4},\frac{b}{4},\frac{c}{4} \right)\] |
| B. | \[\left( \frac{a}{2},\frac{b}{4},\frac{c}{4} \right)\] |
| C. | \[\left( \frac{a}{2},\frac{b}{2},\frac{c}{2} \right)\] |
| D. | (a, b, c) |
| Answer» D. (a, b, c) | |
| 622. |
If out of 20 consecutive whole numbers two are chosen at random, then the probability that their sum is odd, is |
| A. | \[\frac{5}{19}\] |
| B. | \[\frac{10}{19}\] |
| C. | \[\frac{9}{19}\] |
| D. | None of these |
| Answer» C. \[\frac{9}{19}\] | |
| 623. |
The shortest distance between the skew lines\[{{l}_{1}}:\vec{r}={{\vec{a}}_{1}}+\lambda {{\vec{b}}_{1}}{{l}_{2}}:\vec{r}={{\vec{a}}_{2}}+\mu {{\vec{b}}_{2}}\] is |
| A. | \[\frac{|({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}{|{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}\] |
| B. | \[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{a}}}_{2}}\times {{{\vec{b}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\] |
| C. | \[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{b}}}_{2}}).{{{\vec{a}}}_{1}}\times {{{\vec{b}}}_{1}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\] |
| D. | \[\frac{\left| ({{{\vec{a}}}_{1}}-{{{\vec{b}}}_{2}}).{{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{a}}}_{2}} \right|}\] |
| Answer» B. \[\frac{\left| ({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}}).{{{\vec{a}}}_{2}}\times {{{\vec{b}}}_{2}} \right|}{\left| {{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}} \right|}\] | |
| 624. |
The distance of point A (-2, 3, 1) from the line PQ through P (- 3, 5, 2), which makes equal angles with the axes is |
| A. | \[2/\sqrt{3}\] |
| B. | \[\sqrt{14/3}\] |
| C. | \[16/\sqrt{3}\] |
| D. | \[5/\sqrt{3}\] |
| Answer» C. \[16/\sqrt{3}\] | |
| 625. |
Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+\mathbf{j}.\] If c is a vector such that \[\mathbf{a}\,.\,\mathbf{c}=\,|\mathbf{c}|,\,\,|\mathbf{c}-\mathbf{a}|\,=2\sqrt{2}\]and the angle between \[(\mathbf{a}\times \mathbf{b})\] and c is \[{{30}^{o}}\], then \[|\,(\mathbf{a}\times \mathbf{b})\times \mathbf{c}|\,=\] [IIT 1999] |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{3}{2}\] |
| C. | 2 |
| D. | 3 |
| Answer» C. 2 | |
| 626. |
If \[|\mathbf{a}|\,=4,\,|\mathbf{b}|\,=2\] and the angle between a and b is \[\frac{\pi }{6}\], then \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [AIEEE 2002] |
| A. | 48 |
| B. | 16 |
| C. | 8 |
| D. | None of these |
| Answer» C. 8 | |
| 627. |
If two spheres of radii \[{{r}_{1}}\] and \[{{r}_{2}}\] cut orthogonally, then the radius of the common circle is |
| A. | \[{{r}_{1}}{{r}_{2}}\] |
| B. | \[\sqrt{(r_{1}^{2}+r_{2}^{2}})\] |
| C. | \[{{r}_{1}}{{r}_{2}}\sqrt{(r_{1}^{2}+r_{2}^{2})}\] |
| D. | \[\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{(r_{1}^{2}+r_{2}^{2})}}\] |
| Answer» E. | |
| 628. |
The area of the smaller segment cut off from the circle \[{{x}^{2}}+{{y}^{2}}=9\] by \[x=1\] is [RPET 2002] |
| A. | \[\frac{1}{2}(9{{\sec }^{-1}}3-\sqrt{8})\] |
| B. | \[9{{\sec }^{-1}}(3)-\sqrt{8}\] |
| C. | \[\sqrt{8}-9{{\sec }^{-1}}(3)\] |
| D. | None of these |
| Answer» C. \[\sqrt{8}-9{{\sec }^{-1}}(3)\] | |
| 629. |
If \[P(3,\,4,\,5),\] \[Q(4,\,6,\,3),\] \[R(-1,\,2,\,4),\] \[S(1,\,0,\,5)\] then the projection of RS on PQ is [Orissa JEE 2002; RPET 2002] |
| A. | ? 2/3 |
| B. | ? 4/3 |
| C. | ½ |
| D. | 2 |
| Answer» C. ½ | |
| 630. |
The angle between the straight lines \[x-y\sqrt{3}=5\] and \[\sqrt{3x}+y=7\]is [MP PET 2003] |
| A. | \[{{90}^{o}}\] |
| B. | \[{{60}^{o}}\] |
| C. | \[{{75}^{o}}\] |
| D. | \[{{30}^{o}}\] |
| Answer» B. \[{{60}^{o}}\] | |
| 631. |
Probability that India will win against Pakistan in a cricket match is 2/3, in series of 5 matches what is the probability that India will win the series? |
| A. | \[161/81\] |
| B. | \[192/243\] |
| C. | \[172/243\] |
| D. | None of these |
| Answer» C. \[172/243\] | |
| 632. |
In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is \[\frac{1}{3}.\]The probability that he copies is \[\frac{1}{6}\] and the probability that his answer is correct given that he copied it is \[\frac{1}{8}.\] The probability that he knew the answer to the question given that he correctly answered it, is |
| A. | \[\frac{24}{29}\] |
| B. | \[\frac{1}{4}\] |
| C. | \[\frac{3}{4}\] |
| D. | \[\frac{1}{2}\] |
| Answer» B. \[\frac{1}{4}\] | |
| 633. |
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then the conditional probabilities that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl are |
| A. | \[\frac{1}{2}and\frac{1}{4}\] |
| B. | \[\frac{1}{3}and\frac{1}{2}\] |
| C. | \[\frac{1}{3}and\frac{1}{4}\] |
| D. | \[\frac{1}{2}and\frac{1}{3}\] |
| Answer» B. \[\frac{1}{3}and\frac{1}{2}\] | |
| 634. |
The standard deviation of \[9,\,\,16,\,\,23,\,\,30,\,\,37,\,\,44,\,\,51\] is |
| A. | 7 |
| B. | 9 |
| C. | 12 |
| D. | 14 |
| Answer» E. | |
| 635. |
If the mean of the numbers \[27+x,31+x,89+x,107+x,156+x\] is 82, then the mean of \[130+x\], \[126+x,\,\,68+x,\,\,50+x,\,\,1+x\] |
| A. | 75 |
| B. | 157 |
| C. | 82 |
| D. | 80 |
| Answer» B. 157 | |
| 636. |
\[\int_{\,0}^{\,\infty }{\frac{xdx}{(1+x)(1+{{x}^{2}})}=}\] [Karnataka CET 2003] |
| A. | 0 |
| B. | \[\pi /2\] |
| C. | \[\pi /4\] |
| D. | 1 |
| Answer» D. 1 | |
| 637. |
If \[\overrightarrow{A}=i-2j-3k,\,\overrightarrow{B}=2i+j-k,\,\overrightarrow{C}=i+3j-2k\], then \[(\overrightarrow{A}\times \overrightarrow{B})\times \overrightarrow{C}\] is [MP PET 2001] |
| A. | \[5(-\,i+3j+4k)\] |
| B. | \[4(-\,i+3j+4k)\] |
| C. | \[5(-\,i-3j-4k)\] |
| D. | \[4(i+3j+4k)\] |
| Answer» B. \[4(-\,i+3j+4k)\] | |
| 638. |
\[\sqrt{\frac{1-\sin A}{1+\sin A}}=\] |
| A. | \[\sec A+\tan A\] |
| B. | \[\tan \left( \frac{\pi }{4}-A \right)\] |
| C. | \[\tan \left( \frac{\pi }{4}+\frac{A}{2} \right)\] |
| D. | \[\tan \left( \frac{\pi }{4}-\frac{A}{2} \right)\] |
| Answer» E. | |
| 639. |
The volume of the solid generated by revolving about the y-axis the figure bounded by the parabola \[y={{x}^{2}}\] and \[x={{y}^{2}}\] is [UPSEAT 2002] |
| A. | \[\frac{21}{5}\pi \] |
| B. | \[\frac{24}{5}\pi \] |
| C. | \[\frac{2}{15}\pi \] |
| D. | \[\frac{5}{24}\pi \] |
| Answer» D. \[\frac{5}{24}\pi \] | |
| 640. |
The locus of the centre of a circle which passes through the point (a, 0) and touches the line \[x+1=0\], is |
| A. | Circle |
| B. | Ellipse |
| C. | Parabola |
| D. | Hyperbola |
| Answer» D. Hyperbola | |
| 641. |
The value of the integral \[\sum\limits_{k=1}^{n}{\int_{0}^{1}{f(k-1+x)\,dx}}\] is |
| A. | \[\int_{0}^{1}{f(x)\,dx}\] |
| B. | \[\int_{0}^{2}{f(x)\,dx}\] |
| C. | \[\int_{0}^{n}{f(x)\,dx}\] |
| D. | \[n\int_{0}^{1}{f(x)\,dx}\] |
| Answer» D. \[n\int_{0}^{1}{f(x)\,dx}\] | |
| 642. |
The equations of the lines which cuts off an intercept -1 from y-axis are equally inclined to the axes are |
| A. | \[x-y+1=0,\ \ x+y+1=0\] |
| B. | \[x-y-1=0,\ \ x+y-1=0\] |
| C. | \[x-y-1=0,\ \ x+y+1=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 643. |
If \[\left| \,\begin{matrix} \cos (A+B) & -\sin (A+B) & \cos 2B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B \\ \end{matrix}\, \right|=0\], then B = [EAMCET 2003] |
| A. | \[(2n+1)\frac{\pi }{2}\] |
| B. | \[n\pi \] |
| C. | \[(2n+1)\frac{\pi }{2}\] |
| D. | \[2n\pi \] |
| Answer» B. \[n\pi \] | |
| 644. |
Consider the following statements. I. If \[{{A}_{n}}\]is the set of first n prime numbers, then \[\underset{n=2}{\overset{10}{\mathop{U}}}\,{{A}_{n}}\]is equal to {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} II. If A and B are two sets such that \[n(A\cup B)=50,\]\[n(A)=28,\,\,n(B)=32,\] then \[n(A\cap B)=10.\] Which of these is correct? |
| A. | Only I is true |
| B. | Only II is true |
| C. | Both are true |
| D. | Both are false |
| Answer» D. Both are false | |
| 645. |
The points P is equidistant from A(1,3), B (-3,5) and C(5,-1). Then PA = [EAMCET 2003] |
| A. | 5 |
| B. | \[5\sqrt{5}\] |
| C. | 25 |
| D. | \[5\sqrt{10}\] |
| Answer» E. | |
| 646. |
If the direction ratios of a line are \[1,-3,\,2\], then the direction cosines of the line are [MP PET 1997, Pb. CET 2002] |
| A. | \[\frac{1}{\sqrt{14}},\frac{-3}{\sqrt{14}},\frac{2}{\sqrt{14}}\] |
| B. | \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] |
| C. | \[\frac{-1}{\sqrt{14}},\frac{3}{\sqrt{14}},\frac{-2}{\sqrt{14}}\] |
| D. | \[\frac{-1}{\sqrt{14}},\frac{-2}{\sqrt{14}},\frac{-3}{\sqrt{14}}\] |
| Answer» B. \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] | |
| 647. |
The value of \[k\]for which \[2{{x}^{2}}-kx+x+8=0\]has equal and real roots are [BIT Ranchi 1990] |
| A. | -9 and -7 |
| B. | 9 and 7 |
| C. | -9 and 7 |
| D. | 9 and -7 |
| Answer» E. | |
| 648. |
The equation \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=0\]represents |
| A. | (0, 0, 0) |
| B. | A circle |
| C. | A plane |
| D. | None of these |
| Answer» B. A circle | |
| 649. |
\[2\cos x-\cos 3x-\cos 5x=\] [Roorkee 1974] |
| A. | \[16{{\cos }^{3}}x{{\sin }^{2}}x\] |
| B. | \[16{{\sin }^{3}}x{{\cos }^{2}}x\] |
| C. | \[4{{\cos }^{3}}x{{\sin }^{2}}x\] |
| D. | \[4{{\sin }^{3}}x{{\cos }^{2}}x\] |
| Answer» B. \[16{{\sin }^{3}}x{{\cos }^{2}}x\] | |
| 650. |
The area of the triangle formed by the tangents from the points (h, k) to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]and the line joining their points of contact is [MNR 1980] |
| A. | \[a\text{ }\frac{{{({{h}^{2}}+{{k}^{2}}-{{a}^{2}})}^{3/2}}}{{{h}^{2}}+{{k}^{2}}}\] |
| B. | \[a\text{ }\frac{{{({{h}^{2}}+{{k}^{2}}-{{a}^{2}})}^{1/2}}}{{{h}^{2}}+{{k}^{2}}}\] |
| C. | \[\frac{{{({{h}^{2}}+{{k}^{2}}-{{a}^{2}})}^{3/2}}}{{{h}^{2}}+{{k}^{2}}}\] |
| D. | \[\frac{{{({{h}^{2}}+{{k}^{2}}-{{a}^{2}})}^{1/2}}}{{{h}^{2}}+{{k}^{2}}}\] |
| Answer» B. \[a\text{ }\frac{{{({{h}^{2}}+{{k}^{2}}-{{a}^{2}})}^{1/2}}}{{{h}^{2}}+{{k}^{2}}}\] | |