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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.
| 401. |
If \[\cos 3x+\sin \left( 2x-\frac{7\pi }{6} \right)=-2\], then \[x=\] (where \[k\in Z\]) |
| A. | \[\frac{\pi }{3}(6k+1)\] |
| B. | \[\frac{\pi }{3}(6k+1)\] |
| C. | \[\frac{\pi }{3}(2k+1)\] |
| D. | None of these |
| Answer» B. \[\frac{\pi }{3}(6k+1)\] | |
| 402. |
The roots of the equation\[({{a}^{2}}+{{b}^{2}}){{t}^{2}}-2(ac+bd)t+({{c}^{2}}+{{d}^{2}})=0\] are equal, then [MP PET 1996] |
| A. | \[ab=dc\] |
| B. | \[ac=bd\] |
| C. | \[ad+bc=0\] |
| D. | \[\frac{a}{b}=\frac{c}{d}\] |
| Answer» E. | |
| 403. |
The distance of the point (4, 3, 5) from the y-axis is [MP PET 2003] |
| A. | \[\sqrt{34}\] |
| B. | 5 |
| C. | \[\sqrt{41}\] |
| D. | \[\sqrt{15}\] |
| Answer» D. \[\sqrt{15}\] | |
| 404. |
.If \[\cos A=\frac{3}{4}\], then \[32\sin \frac{A}{2}\cos \frac{5}{2}A=\] [EAMCET 1982] |
| A. | \[\sqrt{7}\] |
| B. | \[-\sqrt{7}\] |
| C. | 7 |
| D. | -7 |
| Answer» C. 7 | |
| 405. |
If the ratio of the lengths of tangents drawn from the point \[(f,g)\]to the given circle \[{{x}^{2}}+{{y}^{2}}=6\]and \[{{x}^{2}}+{{y}^{2}}+3x+3y=0\]be 2 : 1, then |
| A. | \[{{f}^{2}}+{{g}^{2}}+2g+2f+2=0\] |
| B. | \[{{f}^{2}}+{{g}^{2}}+4g+4f+4=0\] |
| C. | \[{{f}^{2}}+{{g}^{2}}+4g+4f+2=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 406. |
If \[A,B\]are square matrices of order 3, A is non- singular and \[AB=O\], then B is a [EAMCET 2002] |
| A. | Null matrix |
| B. | Singular matrix |
| C. | Unit matrix |
| D. | Non- singular matrix |
| Answer» B. Singular matrix | |
| 407. |
Equations of diagonals of square formed by lines \[x=0,\] \[y=0,\]\[x=1\] and \[y=1\]are [MP PET 1984] |
| A. | \[y=x,\ y+x=1\] |
| B. | \[y=x,\ x+y=2\] |
| C. | \[2y=x,\ y+x=\frac{1}{3}\] |
| D. | \[y=2x,\ y+2x=1\] |
| Answer» B. \[y=x,\ x+y=2\] | |
| 408. |
The horizontal distance between two towers is 60 meters and the angular depression of the top of the first tower as seen from the top of the second. is\[30{}^\circ \]. If the height of the second tower be 150 meters, then the highest of the first tower is |
| A. | \[150-60\sqrt{3}m\] |
| B. | 90 m |
| C. | \[150-20\sqrt{3}m\] |
| D. | None of these |
| Answer» D. None of these | |
| 409. |
The line passing through \[(-1,\pi /2)\] and perpendicular to \[\sqrt{3}\sin \theta +2\cos \theta =\frac{4}{r}\] is [EAMCET 2003] |
| A. | \[2=\sqrt{3}\,r\cos \theta -2\,r\sin \theta \] |
| B. | \[5=-2\sqrt{3}\,r\sin \theta +4\,r\cos \theta \] |
| C. | \[2=\sqrt{3}\,r\cos \theta +2\,r\cos \theta \] |
| D. | \[5=2\sqrt{3}\,r\sin \theta +4\,r\cos \theta \] |
| Answer» B. \[5=-2\sqrt{3}\,r\sin \theta +4\,r\cos \theta \] | |
| 410. |
If the co-ordinates of the points \[P,\,Q,R,\,S\] be (1, 2, 3), (4, 5, 7), (? 4, 3, ? 6) and (2, 0, 2) respectively, then |
| A. | \[PQ||RS\] |
| B. | \[PQ\,\bot \,RS\] |
| C. | \[PQ=RS\] |
| D. | None of these |
| Answer» E. | |
| 411. |
The value of \[\theta \] satisfying the given equation \[\cos \theta +\sqrt{3}\sin \theta \] = 2, is [MNR 1981; EAMCET 1989] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{5\pi }{3}\] |
| C. | \[\frac{2\pi }{3}\] |
| D. | \[\frac{4\pi }{3}\] |
| Answer» B. \[\frac{5\pi }{3}\] | |
| 412. |
If \[{{S}_{n}}\] denotes the sum of first n terms of an A.P. whose first term is a and \[\frac{{{S}_{nx}}}{{{S}_{x}}}\] is independent of x, then \[{{S}_{p}}=\] |
| A. | \[{{P}^{3}}\] |
| B. | \[{{P}^{2}}a\] |
| C. | \[P{{a}^{2}}\] |
| D. | \[{{a}^{3}}\] |
| Answer» C. \[P{{a}^{2}}\] | |
| 413. |
Let P be a prime number such that \[p\ge 11.\] Let \[n=p!+1.\] The number of primes in the list \[n+1,\] \[n+2,n+3,...n+P-1,\] is |
| A. | \[p-1\] |
| B. | 2 |
| C. | 1 |
| D. | None of these |
| Answer» E. | |
| 414. |
If x co-ordinates of a point P of line joining the points \[Q(2,\,2,\,1)\] and \[R\,(5,\,2,-2)\]is 4, then the z-coordinates of P is [RPET 2000] |
| A. | ? 2 |
| B. | ?1 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 415. |
The angle between the tangents from \[(\alpha ,\beta )\]to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], is |
| A. | \[{{\tan }^{-1}}\left( \frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}} \right)\] |
| B. | \[{{\tan }^{-1}}\left( \frac{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}}{a} \right)\] |
| C. | \[2{{\tan }^{-1}}\left( \frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}} \right)\] |
| D. | None of these |
| Answer» D. None of these | |
| 416. |
The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are |
| A. | \[\frac{2}{7},\frac{3}{7},\frac{6}{7}\] |
| B. | 2, 3, 6 |
| C. | \[\frac{2}{11},\frac{3}{11},\frac{6}{11}\] |
| D. | None of these |
| Answer» B. 2, 3, 6 | |
| 417. |
Two dice are thrown n times in succession. The probability of obtaining a double - six at least once is |
| A. | \[{{\left( \frac{1}{36} \right)}^{n}}\] |
| B. | \[1-{{\left( \frac{35}{36} \right)}^{n}}\] |
| C. | \[{{\left( \frac{1}{12} \right)}^{n}}\] |
| D. | None of these |
| Answer» C. \[{{\left( \frac{1}{12} \right)}^{n}}\] | |
| 418. |
If q be the angle between the vectors a and b and \[|\mathbf{a}\times \mathbf{b}|\,=\mathbf{a}\,.\,\mathbf{b},\] then \[\theta =\] [RPET 1990; MP PET 1990; UPSEAT 2003] |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | 0 |
| Answer» D. 0 | |
| 419. |
The area of the region bounded by the curve \[9{{x}^{2}}+4{{y}^{2}}-36=0\] is [Karnataka CET 2005] |
| A. | \[9\pi \] |
| B. | \[4\pi \] |
| C. | \[36\pi \] |
| D. | \[6\pi \] |
| Answer» E. | |
| 420. |
If the position vectors of two point P and Q are respectively \[9\mathbf{i}-\mathbf{j}+5\mathbf{k}\] and \[\mathbf{i}+3\mathbf{j}+5\mathbf{k}\], and the line segment PQ intersects the YOZ plane at a point R, the PR : RQ is equal to [J & K 2005] |
| A. | 9 : 1 |
| B. | 1 : 9 |
| C. | ?1 : 9 |
| D. | ? 9 : 1 |
| Answer» E. | |
| 421. |
In tossing 10 coins, the probability of getting exactly 5 heads is [MP PET 1996] |
| A. | \[\frac{9}{128}\] |
| B. | \[\frac{63}{256}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{193}{256}\] |
| Answer» C. \[\frac{1}{2}\] | |
| 422. |
The approximate value of \[{{(7.995)}^{1/3}}\]correct to four decimal places is [MNR 1991; UPSEAT 2000] |
| A. | 1.9995 |
| B. | 1.9996 |
| C. | 1.999 |
| D. | 1.9991 |
| Answer» B. 1.9996 | |
| 423. |
If A and B are any two sets, then \[A\cup (A\cap B)\]is equal to [Karnataka CET 1996] |
| A. | A |
| B. | B |
| C. | \[{{A}^{c}}\] |
| D. | \[{{B}^{c}}\] |
| Answer» B. B | |
| 424. |
The greatest value of the function \[F(x)=\int_{1}^{x}{\,\,|t|\,dt}\] on the interval \[\left[ -\frac{1}{2},\,\,\frac{1}{2} \right]\] is given by [IIT Screening] |
| A. | \[\frac{3}{8}\] |
| B. | \[-\frac{1}{2}\] |
| C. | \[-\frac{3}{8}\] |
| D. | \[\frac{2}{5}\] |
| Answer» D. \[\frac{2}{5}\] | |
| 425. |
The area of smaller part between the circle \[{{x}^{2}}+{{y}^{2}}=4\]and the line \[x=1\] is [RPET 1999] |
| A. | \[\frac{4\pi }{3}-\sqrt{3}\] |
| B. | \[\frac{8\pi }{3}-\sqrt{3}\] |
| C. | \[\frac{4\pi }{3}+\sqrt{3}\] |
| D. | \[\frac{5\pi }{3}+\sqrt{3}\] |
| Answer» C. \[\frac{4\pi }{3}+\sqrt{3}\] | |
| 426. |
If \[A(-a,0)\] and \[B(a,0)\]are two fixed points, then the locus of the point on which the line AB subtends the right angle, is |
| A. | \[{{x}^{2}}+{{y}^{2}}=2{{a}^{2}}\] |
| B. | \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+{{a}^{2}}=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] |
| Answer» E. | |
| 427. |
If \[x=2+{{2}^{2/3}}+{{2}^{1/3}},\]then \[{{x}^{3}}-6{{x}^{2}}+6x=\] [MNR 1985] |
| A. | 3 |
| B. | 2 |
| C. | 1 |
| D. | None of these |
| Answer» C. 1 | |
| 428. |
If A and B are sets, then \[A\text{ }\cap \text{ }\left( B\text{ }-\text{ }A \right)\] is |
| A. | \[\varphi \] |
| B. | A |
| C. | B |
| D. | None of these |
| Answer» B. A | |
| 429. |
The adjoint of \[\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \\ \end{matrix} \right]\]is [RPET 1993] |
| A. | \[\left[ \begin{matrix} 3 & -9 & -5 \\ -4 & 1 & 3 \\ -5 & 4 & 1 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 3 & -4 & -5 \\ -9 & 1 & 4 \\ -5 & 3 & 1 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} -3 & \,\,4 & 5 \\ 9 & -1 & -4 \\ 5 & -3 & -1 \\ \end{matrix} \right]\] |
| D. | None of these |
| Answer» C. \[\left[ \begin{matrix} -3 & \,\,4 & 5 \\ 9 & -1 & -4 \\ 5 & -3 & -1 \\ \end{matrix} \right]\] | |
| 430. |
If the direction cosines of a line are \[\left( \frac{1}{c},\frac{1}{c},\frac{1}{c} \right)\], then [JET 1989; CET 1993] |
| A. | \[c>0\] |
| B. | \[c=\pm \sqrt{3}\] |
| C. | \[0<c<1\] |
| D. | \[c>2\] |
| Answer» C. \[0<c<1\] | |
| 431. |
For \[0\le x\le \pi ,\] the area bounded by \[y=x\] and \[y=x+\sin x,\] is [Roorkee Qualifying 1998] |
| A. | 2 |
| B. | 4 |
| C. | \[2\pi \] |
| D. | \[4\pi \] |
| Answer» B. 4 | |
| 432. |
If two coins are tossed 5 times, then the probability of getting 5 heads and 5 tails is [AMU 2002] |
| A. | \[\frac{63}{256}\] |
| B. | \[\frac{1}{1024}\] |
| C. | \[\frac{2}{205}\] |
| D. | \[\frac{9}{64}\] |
| Answer» B. \[\frac{1}{1024}\] | |
| 433. |
Let \[S=\{0,\,1,\,5,\,4,\,7\}\]. Then the total number of subsets of S is |
| A. | 64 |
| B. | 32 |
| C. | 40 |
| D. | 20 |
| Answer» C. 40 | |
| 434. |
If a, b, g are the roots of the equation \[2{{x}^{3}}-3{{x}^{2}}+6x+1=0\], then \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}\] is equal to [Karnataka CET 2005] |
| A. | -\[\frac{15}{4}\] |
| B. | \[\frac{15}{4}\] |
| C. | \[\frac{9}{4}\] |
| D. | 4 |
| Answer» B. \[\frac{15}{4}\] | |
| 435. |
If A and B are two matrices such that \[AB=B\]and \[BA=A,\] then \[{{A}^{2}}+{{B}^{2}}=\] [EAMCET 1994] |
| A. | \[2AB\] |
| B. | \[2BA\] |
| C. | \[A+B\] |
| D. | \[AB\] |
| Answer» D. \[AB\] | |
| 436. |
If the middle point of the line segment joining the points (5, a) and (b,7) be (3,5), then (a, b) = |
| A. | (3, 1) |
| B. | (1, 3) |
| C. | (-2,-2) |
| D. | (-3, -1) |
| Answer» B. (1, 3) | |
| 437. |
The equation of line, which bisect the line joining two points (2, -19) and (6, 1) and perpendicular to the line joining two points (-1, 3) and (5, - 1), is [RPET 1987] |
| A. | \[3x-2y=30\] |
| B. | \[2x-y-3=0\] |
| C. | \[2x+3y=20\] |
| D. | None of these |
| Answer» B. \[2x-y-3=0\] | |
| 438. |
A force \[\vec{F}=3\hat{i}+4\hat{j}-3\hat{k}\] is applied at the point P, whose position vector is \[\overrightarrow{r}=\widehat{2i}-2\hat{j}-3\hat{k}\]. What is the magnitude of the moment of the force about the origin? |
| A. | 23 units |
| B. | 19 units |
| C. | 18 units |
| D. | 21 units |
| Answer» B. 19 units | |
| 439. |
What is the interior acute angle of the parallelogram whose sides are represented by the vectors \[\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\hat{k}\] and \[\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}+\hat{k}\]? |
| A. | \[60{}^\circ \] |
| B. | \[45{}^\circ \] |
| C. | \[30{}^\circ \] |
| D. | \[15{}^\circ \] |
| Answer» B. \[45{}^\circ \] | |
| 440. |
If \[\theta \]is an acute angle and \[\sin \frac{\theta }{2}=\sqrt{\frac{x-1}{2x}}\], then \[\tan \theta \] is equal to [Orissa JEE 2005] |
| A. | \[{{x}^{2}}-1\] |
| B. | \[\sqrt{{{x}^{2}}-1}\] |
| C. | \[\sqrt{{{x}^{2}}+1}\] |
| D. | \[{{x}^{2}}+1\] |
| Answer» C. \[\sqrt{{{x}^{2}}+1}\] | |
| 441. |
If \[A\text{ }\cap \text{ }B\text{ }=\text{ }B,\] then [JMIEE 2000] |
| A. | \[A\subset B\] |
| B. | \[B\subset A\] |
| C. | \[A=\varphi \] |
| D. | \[B=\varphi \] |
| Answer» C. \[A=\varphi \] | |
| 442. |
The length of common chord of the circles \[{{x}^{2}}+{{y}^{2}}=12\]and \[{{x}^{2}}+{{y}^{2}}-4x+3y-2=0\], is [RPET 1990, 99] |
| A. | \[4\sqrt{2}\] |
| B. | \[5\sqrt{2}\] |
| C. | \[2\sqrt{2}\] |
| D. | \[6\sqrt{2}\] |
| Answer» B. \[5\sqrt{2}\] | |
| 443. |
\[\int_{0}^{a}{{{x}^{4}}\sqrt{{{a}^{2}}-{{x}^{2}}}}\,dx=\] |
| A. | \[\frac{\pi }{32}\] |
| B. | \[\frac{\pi }{32}{{a}^{6}}\] |
| C. | \[\frac{\pi }{16}{{a}^{6}}\] |
| D. | \[\frac{\pi }{8}{{a}^{6}}\] |
| Answer» C. \[\frac{\pi }{16}{{a}^{6}}\] | |
| 444. |
If \[\sin A,\sin B,\cos A\] are in G.P., then roots of \[{{x}^{2}}+2x\cot B+1=0\] are always [Orissa JEE 2005] |
| A. | Real |
| B. | Imaginary |
| C. | Greater than 1 |
| D. | Equal |
| Answer» B. Imaginary | |
| 445. |
If \[\sin 5x+\sin 3x+\sin x=0\], then the value of x other than 0 lying between \[0\le x\le \frac{\pi }{2}\]is [MNR 1985] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{12}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» D. \[\frac{\pi }{4}\] | |
| 446. |
\[\tan 3A-\tan 2A-\tan A=\] [MNR 1982; Pb. CET 1991] |
| A. | \[\tan 3A\tan 2A\tan A\] |
| B. | \[-\tan 3A\tan 2A\tan A\] |
| C. | \[\tan A\tan 2A-\tan 2A\tan 3A-\tan 3A\tan A\] |
| D. | None of these |
| Answer» B. \[-\tan 3A\tan 2A\tan A\] | |
| 447. |
The sum of an infinite GP is x and the common ratio r is such that \[\left| r \right|<1\]. If the first term of the GP is 2, then which one of the following is correct? |
| A. | \[-1<x<1\] |
| B. | \[-\infty <x<1\] |
| C. | \[1<x<\infty \] |
| D. | None of these |
| Answer» D. None of these | |
| 448. |
If\[\frac{2\sin \alpha }{\{1+\cos \alpha +\sin \alpha \}}=y,\]then \[\frac{\{1-\cos \alpha +\sin \alpha \}}{1+\sin \alpha }=\] [BIT Ranchi 1996; Orissa JEE 2004] |
| A. | \[\frac{1}{y}\] |
| B. | \[y\] |
| C. | \[1-y\] |
| D. | \[1+y\] |
| Answer» C. \[1-y\] | |
| 449. |
The equation \[8{{x}^{2}}+8xy+2{{y}^{2}}+26x+13y+15=0\] represents a pair of straight lines. The distance between them is |
| A. | \[7/\sqrt{5}\] |
| B. | \[7/2\sqrt{5}\] |
| C. | \[\sqrt{7}/5\] |
| D. | None of these |
| Answer» C. \[\sqrt{7}/5\] | |
| 450. |
The line \[x+3y-2=0\]bisects the angle between a pair of straight lines of which one has equation\[x-7y+5=0\]. The equation of the other line is |
| A. | \[3x+3y-1=0\] |
| B. | \[x-3y+2=0\] |
| C. | \[5x+5y-3=0\] |
| D. | None of these |
| Answer» D. None of these | |