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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.
| 351. |
Let A and B be two events such that \[P(A\cap B')=0.20,P(A'\cap B)=0.15,\]\[P(A'\cap B')=0.1,\] Then \[P(A/B)\] is equal to |
| A. | 41944 |
| B. | 44502 |
| C. | 44379 |
| D. | 44378 |
| Answer» B. 44502 | |
| 352. |
If (2, 3, 5) is one end of a diameter of the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-6x-12y-2z+20=0\]then co-ordinates of the other end of the diameter are [Kerala (Engg.) 2005] |
| A. | (4, 3, 5) |
| B. | (4, 9, -3) |
| C. | (4, 9, 3) |
| D. | (4, 3, -3) |
| E. | (4, 9, 5) |
| Answer» C. (4, 9, 3) | |
| 353. |
If a, b, c are any vectors, then the true statement is [RPET 1988] |
| A. | \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] |
| B. | \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{a}\] |
| C. | \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\mathbf{a}\,.\,\mathbf{b}\times \mathbf{a}\,.\,\mathbf{c}\] |
| D. | \[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=\mathbf{a}\,.\,\mathbf{b}-\mathbf{a}\,.\,\mathbf{c}\] |
| Answer» E. | |
| 354. |
If the circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]and \[{{x}^{2}}+{{y}^{2}}-2gx+{{g}^{2}}-{{b}^{2}}=0\] touch each other externally, then |
| A. | \[g=ab\] |
| B. | \[{{g}^{2}}={{a}^{2}}+{{b}^{2}}\] |
| C. | \[{{g}^{2}}=ab\] |
| D. | \[g=a+b\] |
| Answer» E. | |
| 355. |
Let \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] be three vectors such that \[\mathbf{a}\ne 0,\] and \[\mathbf{a}\times \mathbf{b}=2\mathbf{a}\times \mathbf{c},\,\,|\mathbf{a}|\,=\,|\mathbf{c}|\,=\,1,\,|\mathbf{b}|\,=4\] and \[|\mathbf{b}\times \mathbf{c}|\,=15.\] If \[\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a},\] then l equals to [Orissa JEE 2004] |
| A. | 1 |
| B. | \[\pm \,4\] |
| C. | 3 |
| D. | ? 2 |
| Answer» C. 3 | |
| 356. |
Consider the sequence \[8A+2B,\text{ }6A+B,\text{ }4A,\,\,2A-B,........\] Which term of this sequence will have a coefficient of A which is twice the coefficient of B? |
| A. | 10th |
| B. | 14th |
| C. | 16th |
| D. | None of these |
| Answer» E. | |
| 357. |
If \[P=n({{n}^{2}}-{{1}^{2}})({{n}^{2}}-{{2}^{2}})({{n}^{2}}-{{3}^{2}})...({{n}^{2}}-{{r}^{2}}),\] \[n>r,n\in N\] then P is necessarily divisible by |
| A. | \[(2r+2)!\] |
| B. | \[(2r+4)!\] |
| C. | \[(2r+1)!\] |
| D. | None of these |
| Answer» D. None of these | |
| 358. |
A sample of 4 items is drawn at a random without replacement from a lot of 10 items. Containing 3 defective. If X denotes the number of defective items in the sample then \[P(0 |
| A. | \[\frac{3}{10}\] |
| B. | \[\frac{4}{5}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{1}{6}\] |
| Answer» C. \[\frac{1}{2}\] | |
| 359. |
\[\frac{\sin 3\theta +\sin 5\theta +\sin 7\theta +\sin 9\theta }{\cos 3\theta +\cos 5\theta +\cos 7\theta +\cos 9\theta }=\] [Roorkee 1973] |
| A. | \[\tan 3\theta \] |
| B. | \[\cot 3\theta \] |
| C. | \[\tan 6\theta \] |
| D. | \[\cot 6\theta \] |
| Answer» D. \[\cot 6\theta \] | |
| 360. |
If \[A=\left[ \begin{matrix} \lambda & 1 \\ -1 & -\lambda \\ \end{matrix} \right]\], then for what value of \[\lambda ,\,{{A}^{2}}=O\] [MP PET 1992] |
| A. | 0 |
| B. | \[\pm \text{ }1\] |
| C. | -1 |
| D. | 1 |
| Answer» C. -1 | |
| 361. |
A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and IV. The probabilities of the student passing in tests I, II, III are p, q and \[\frac{1}{2}\] respectively. The probability that the student is successful is \[\frac{1}{2}\] then the relation between p and q is given by |
| A. | \[pq+p=1\] |
| B. | \[{{p}^{2}}+q=1\] |
| C. | \[pq-1=p\] |
| D. | None of these |
| Answer» B. \[{{p}^{2}}+q=1\] | |
| 362. |
A die is rolled three times. Let \[{{E}_{1}}\] denote the event of getting a number larger than the previous number each time and \[{{E}_{2}}\] denote the event that the numbers (in order) form an increasing AP then |
| A. | \[P({{E}_{2}})\ge P({{E}_{1}})\] |
| B. | \[P({{E}_{2}}\cap {{E}_{1}})=\frac{3}{10}\] |
| C. | \[P({{E}_{2}}/{{E}_{1}})=\frac{1}{36}\] |
| D. | \[P({{E}_{1}})=\frac{10}{3}P({{E}_{2}})\] |
| Answer» E. | |
| 363. |
A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is |
| A. | 8/729 |
| B. | 8/243 |
| C. | 1/729 |
| D. | 8/9 |
| Answer» C. 1/729 | |
| 364. |
The equation to the locus of a point which moves so that its distance from x-axis is always one half its distance from the origin, is |
| A. | \[{{x}^{2}}+3{{y}^{2}}=0\] |
| B. | \[{{x}^{2}}-3{{y}^{2}}=0\] |
| C. | \[3{{x}^{2}}+{{y}^{2}}=0\] |
| D. | \[3{{x}^{2}}-{{y}^{2}}=0\] |
| Answer» C. \[3{{x}^{2}}+{{y}^{2}}=0\] | |
| 365. |
All the words that can be formed using alphabets A, H, L, U and R are written as in a dictionary (no alphabet is repeated). Rank of the word RAHUL is |
| A. | 71 |
| B. | 72 |
| C. | 73 |
| D. | 74 |
| Answer» E. | |
| 366. |
The vectors \[\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}\] are the sides of a triangle ABC. The length of the median through A is: |
| A. | \[\sqrt{13}\]units |
| B. | \[2\sqrt{5}\] units |
| C. | 5 units |
| D. | 10 units |
| Answer» D. 10 units | |
| 367. |
If \[\vec{a},\text{ }\vec{b}\] and \[\vec{c}\] are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct? |
| A. | \[\vec{a}+\vec{b}+\vec{c}=\vec{0}\] |
| B. | \[\vec{a}+\vec{b}+\vec{c}=\,\,unit\,\,vector\] |
| C. | \[\vec{a}+\vec{b}=\vec{c}\] |
| D. | \[\vec{a}=\vec{b}+\vec{c}\] |
| Answer» B. \[\vec{a}+\vec{b}+\vec{c}=\,\,unit\,\,vector\] | |
| 368. |
A force \[F=2i+j-k\] acts at a point A, whose position vector is \[2i-j\]. The moment of F about the origin is |
| A. | \[i+2j-4k\] |
| B. | \[i-2j-4k\] |
| C. | \[i+2j+4k\] |
| D. | \[i-2j+4k\] |
| Answer» D. \[i-2j+4k\] | |
| 369. |
The angle between the two tangents from the origin to the circle \[{{(x-7)}^{2}}+{{(y+1)}^{2}}=25\] is [MNR 1990; RPET 1997; DCE 2000] |
| A. | 0 |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 370. |
The solution of differential equation \[dy-\sin x\sin ydx=0\] is [MP PET 1996] |
| A. | \[{{e}^{\cos x}}\tan \frac{y}{2}=c\] |
| B. | \[{{e}^{\cos x}}\tan y=c\] |
| C. | \[\cos x\tan y=c\] |
| D. | \[\cos x\sin y=c\] |
| Answer» B. \[{{e}^{\cos x}}\tan y=c\] | |
| 371. |
The number of solutions for the equation \[{{x}^{2}}-5|x|+\,6=0\] is [Karnataka CET 2004] |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 1 |
| Answer» B. 3 | |
| 372. |
If \[1+\cot \theta =\text{cosec}\theta \], then the general value of \[\theta \] is [Roorkee 1981] |
| A. | \[n\pi +\frac{\pi }{2}\] |
| B. | \[2n\pi -\frac{\pi }{2}\] |
| C. | \[2n\pi +\frac{\pi }{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 373. |
The area between the curve \[y=4+3x-{{x}^{2}}\] and x-axis is [RPET 2001] |
| A. | 125/6 |
| B. | 125/3 |
| C. | 125/2 |
| D. | None of these |
| Answer» B. 125/3 | |
| 374. |
The locus of the centres of the circles which touch externally the circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]and \[{{x}^{2}}+{{y}^{2}}=4ax\], will be |
| A. | \[12{{x}^{2}}-4{{y}^{2}}-24ax+9{{a}^{2}}=0\] |
| B. | \[12{{x}^{2}}+4{{y}^{2}}-24ax+9{{a}^{2}}=0\] |
| C. | \[12{{x}^{2}}-4{{y}^{2}}+24ax+9{{a}^{2}}=0\] |
| D. | \[12{{x}^{2}}+4{{y}^{2}}+24ax+9{{a}^{2}}=0\] |
| Answer» B. \[12{{x}^{2}}+4{{y}^{2}}-24ax+9{{a}^{2}}=0\] | |
| 375. |
Three expressions are given below:\[{{Q}_{1}}=\sin (A+B)+\sin (B+C)+\sin (C+A)\]\[{{Q}_{2}}=\cos (A-B)+\cos (B-C)+\cos (C-A)\]\[{{Q}_{3}}=\sin A(\cos B+\cos C)+\sin B(\cos C+\cos A)+\]\[\sin C(\cos A+\cos B)\]Which one of the following is correct? |
| A. | \[{{Q}_{1}}={{Q}_{2}}\] |
| B. | \[{{Q}_{2}}={{Q}_{3}}\] |
| C. | \[{{Q}_{1}}={{Q}_{3}}\] |
| D. | All the expressions are different |
| Answer» D. All the expressions are different | |
| 376. |
The shortest distance between the lines \[{{\mathbf{r}}_{1}}=4\mathbf{i}-3\mathbf{j}-\mathbf{k}+\lambda (\mathbf{i}-4\mathbf{j}+7\mathbf{k})\] and \[{{\mathbf{r}}_{2}}=\mathbf{i}-\mathbf{j}-10\mathbf{k}+\lambda (2\mathbf{i}-3\mathbf{j}+8\mathbf{k})\]is [J & K 2005] |
| A. | 3 |
| B. | 1 |
| C. | 2 |
| D. | 0 |
| Answer» E. | |
| 377. |
The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is |
| A. | 28 |
| B. | 30 |
| C. | 35 |
| D. | 38 |
| Answer» E. | |
| 378. |
Let \[P=\{(x,y):\left| {{x}^{2}}+{{y}^{2}} \right|=1,x,y\in R\}.\] Then P is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | Anti-symmetric |
| Answer» C. Transitive | |
| 379. |
The area between the parabola \[y={{x}^{2}}\] and the line \[y=x\] is [UPSEAT 2004] |
| A. | \[\frac{1}{6}\]sq. unit |
| B. | \[\frac{1}{3}\]sq. unit |
| C. | \[\frac{1}{2}\]sq. unit |
| D. | None of these |
| Answer» B. \[\frac{1}{3}\]sq. unit | |
| 380. |
If \[\cos \theta +\sec \theta =\frac{5}{2}\], then the general value of \[\theta \]is |
| A. | \[n\pi \pm \frac{\pi }{3}\] |
| B. | \[2n\pi \pm \frac{\pi }{6}\] |
| C. | \[n\pi \pm \frac{\pi }{6}\] |
| D. | \[2n\pi \pm \frac{\pi }{3}\] |
| Answer» E. | |
| 381. |
The remainder when \[{{5}^{4n}}\] is divided by 13, is |
| A. | 1 |
| B. | 8 |
| C. | 9 |
| D. | 10 |
| Answer» B. 8 | |
| 382. |
If the mean deviation of the numbers \[1,\,\,1+d,\] \[1+2d,...1+100d\] from their mean is 255, then d is equal to: |
| A. | 20 |
| B. | 10.1 |
| C. | 20.2 |
| D. | 10 |
| Answer» C. 20.2 | |
| 383. |
Let f and g be functions from R To R defined as \[f(x)=\left\{ \begin{matrix} 7{{x}^{2}}+x-8,x\le 1 \\ 4x+5,1
|
| A. | \[(fog)(-3)=8\] |
| B. | \[(fog)(9)=683\] |
| C. | \[(gof)(0)=-8\] |
| D. | \[(gof)(6)=427\] |
| Answer» C. \[(gof)(0)=-8\] | |
| 384. |
\[f(x)=\left| x-1 \right|,f:{{R}^{+}}\to R\] and \[g(x)={{e}^{x}},\] \[g:[(-1,\infty )\to R].\] If the function fog (x) is defined, then its domain and range respectively are |
| A. | \[(0,\infty )\,\,and\,\,[0,\infty )\] |
| B. | \[[-1,\infty )\,\,and\,\,[0,\infty )\] |
| C. | \[[-1,\infty )and\left[ 1-\frac{1}{e},\infty \right)\] |
| D. | \[[-1,\infty )and\left[ \frac{1}{e}-1,\infty \right)\] |
| Answer» C. \[[-1,\infty )and\left[ 1-\frac{1}{e},\infty \right)\] | |
| 385. |
If the system of equation\[3x-2y+z=0\], \[\lambda x-14y+15z=0\], \[x+2y+3z=0\]have a non-trivial solution, then \[\lambda =\] [EAMCET 1993] |
| A. | 5 |
| B. | -5 |
| C. | -29 |
| D. | 29 |
| Answer» E. | |
| 386. |
If \[\tan x=\frac{b}{a},\]then \[\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}=\] [MP PET 1990, 2002] |
| A. | \[\frac{2\sin x}{\sqrt{\sin 2x}}\] |
| B. | \[\frac{2\cos x}{\sqrt{\cos 2x}}\] |
| C. | \[\frac{2\cos x}{\sqrt{\sin 2x}}\] |
| D. | \[\frac{2\sin x}{\sqrt{\cos 2x}}\] |
| Answer» C. \[\frac{2\cos x}{\sqrt{\sin 2x}}\] | |
| 387. |
The matrix \[A=\left[ \begin{matrix} i & 1-2i \\ -1-2i & 0 \\ \end{matrix} \right]\]is which of the following [Kurukshetra CEE 2002] |
| A. | Symmetric |
| B. | Skew-symmetric |
| C. | Hermitian |
| D. | Skew-hermitian |
| Answer» E. | |
| 388. |
The number of real values of x for which the equality \[\left| \,3{{x}^{2}}+12x+6\, \right|=5x+16\] holds good is [AMU 1999] |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 1 |
| Answer» D. 1 | |
| 389. |
Let\[\frac{d}{dx}F(x)=\left( \frac{{{e}^{\sin x}}}{x} \right)\,;\,x>0\]. If \[\int_{\,1}^{\,4}{\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=F(k)-F(1)}\], then one of the possible value of k, is [AIEEE 2003] |
| A. | 15 |
| B. | 16 |
| C. | 63 |
| D. | 64 |
| Answer» E. | |
| 390. |
The number of different words which can be formed from the letters of the word LUCKNOW when the vowels always occupy even places in |
| A. | 120 |
| B. | 720 |
| C. | 400 |
| D. | None of these |
| Answer» C. 400 | |
| 391. |
The general solution of the differential equation \[\frac{dy}{dx}=\cot x\cot y\] is [AISSE 1983; MP PET 1994] |
| A. | \[\cos x=c\,\cos \text{ec}y\] |
| B. | \[\sin x=c\sec y\] |
| C. | \[\sin x=c\cos y\] |
| D. | \[\cos x=c\sin y\] |
| Answer» C. \[\sin x=c\cos y\] | |
| 392. |
\[\frac{1}{4}\left[ \sqrt{3}\cos 23{}^\circ -\sin 23{}^\circ \right]=\] |
| A. | \[\cos 43{}^\circ \] |
| B. | \[\cos 7{}^\circ \] |
| C. | \[\cos 53{}^\circ \] |
| D. | None of these |
| Answer» E. | |
| 393. |
How many numbers with no more than three digits can be formed using only the digits 1 through 7 with no digit used more than once in a given number? |
| A. | 259 |
| B. | 249 |
| C. | 257 |
| D. | 252 |
| Answer» B. 249 | |
| 394. |
If \[A,B,C\]are three \[n\times n\]matrices, then \[(ABC{)}'=\] [MP PET 1988] |
| A. | \[{A}'\,{B}'\,{C}'\] |
| B. | \[{C}'\,{B}'\,{A}'\] |
| C. | \[{B}'\,{C}'\,{A}'\] |
| D. | \[{B}'\,{A}'\,{C}'\] |
| Answer» C. \[{B}'\,{C}'\,{A}'\] | |
| 395. |
If \[a>2b>0\]then the positive value of m for which \[y=mx-b\sqrt{1+{{m}^{2}}}\]is a common tangent to \[{{x}^{2}}+{{y}^{2}}={{b}^{2}}\]and \[{{(x-a)}^{2}}+{{y}^{2}}={{b}^{2}}\], is [IIT Screening 2002] |
| A. | \[\frac{2b}{\sqrt{{{a}^{2}}-4{{b}^{2}}}}\] |
| B. | \[\frac{\sqrt{{{a}^{2}}-4{{b}^{2}}}}{2b}\] |
| C. | \[\frac{2b}{a-2b}\] |
| D. | \[\frac{b}{a-2b}\] |
| Answer» B. \[\frac{\sqrt{{{a}^{2}}-4{{b}^{2}}}}{2b}\] | |
| 396. |
The equation of a circle that intersects the circle \[{{x}^{2}}+{{y}^{2}}+14x+6y+2=0\]orthogonally and whose centre is (0, 2) is [MP PET 1998] |
| A. | \[{{x}^{2}}+{{y}^{2}}-4y-6=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+4y-14=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+4y+14=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}-4y-14=0\] |
| Answer» E. | |
| 397. |
The combined equation of the pair of lines through the point (1, 0) and parallel to the lines represented by \[2{{x}^{2}}-xy-{{y}^{2}}=0\] is |
| A. | \[2{{x}^{2}}-xy-{{y}^{2}}-4x-y=0\] |
| B. | \[2{{x}^{2}}-xy-{{y}^{2}}-4x+y+2=0\] |
| C. | \[2{{x}^{2}}+xy+{{y}^{2}}-2x+y=0\] |
| D. | None of these |
| Answer» C. \[2{{x}^{2}}+xy+{{y}^{2}}-2x+y=0\] | |
| 398. |
If \[{{\log }_{10}}2,\,{{\log }_{10}}({{2}^{x}}-1)\] and \[{{\log }_{10}}({{2}^{x}}+3)\] are three consecutive terms of an A.P, then the value of x is |
| A. | 1 |
| B. | \[lo{{g}_{5}}2\] |
| C. | \[lo{{g}_{2}}5\] |
| D. | \[lo{{g}_{10}}5\] |
| Answer» D. \[lo{{g}_{10}}5\] | |
| 399. |
Solution of \[\frac{dy}{dx}=\frac{x\log {{x}^{2}}+x}{\sin y+y\,\,\cos y}\] is [EAMCET 2003] |
| A. | \[y\sin y={{x}^{2}}\log x+c\] |
| B. | \[y\sin y={{x}^{2}}+c\] |
| C. | \[y\sin y={{x}^{2}}+\log x+c\] |
| D. | \[y\sin y=x\log x+c\] |
| Answer» B. \[y\sin y={{x}^{2}}+c\] | |
| 400. |
The solution of \[\cos (x+y)\,dy=\,\,dx\] is [DCE 1999] |
| A. | \[y=\tan \,\left( \frac{x+y}{2} \right)+c\] |
| B. | \[y+{{\cos }^{-1}}\left( \frac{y}{x} \right)=c\] |
| C. | \[y=x\,\,\sec \left( \frac{y}{x} \right)+c\] |
| D. | None of these |
| Answer» B. \[y+{{\cos }^{-1}}\left( \frac{y}{x} \right)=c\] | |