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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.
| 6451. |
The solution of the differential equation \[{{x}^{4}}\frac{dy}{dx}+{{x}^{3}}y+\text{cosec}\,(xy)=0\] is equal to [Pb. CET 2004] |
| A. | \[2\cos \,(xy)+{{x}^{-2}}=c\] |
| B. | \[2\cos \,(xy)+{{y}^{-2}}=c\] |
| C. | \[2\sin \,(xy)+{{x}^{-2}}=c\] |
| D. | \[2\sin \,(xy)+{{y}^{-2}}=c\] |
| Answer» B. \[2\cos \,(xy)+{{y}^{-2}}=c\] | |
| 6452. |
If \[\int_{{}}^{{}}{\frac{4{{e}^{x}}+6{{e}^{-x}}}{9{{e}^{x}}-4{{e}^{-x}}}dx=Ax+B\log (9{{e}^{2x}}-4)}+C\], then A, B and C are [IIT 1990] |
| A. | \[A=\frac{3}{2},\ B=\frac{36}{35},\ C=\frac{3}{2}\log 3+\]constant |
| B. | \[A=\frac{3}{2},\ B=\frac{35}{36},\ C=\frac{3}{2}\log 3+\]constant |
| C. | \[A=-\frac{3}{2},\ B=-\frac{35}{36},\ C=-\frac{3}{2}\log 3+\]constant |
| D. | None of these |
| Answer» E. | |
| 6453. |
If \[y=4x-5\] is tangent to the curve \[{{y}^{2}}=p{{x}^{3}}+q\] at (2, 3), then [IIT 1994; UPSEAT 2001] |
| A. | \[p=2,q=-7\] |
| B. | \[p=-2,q=7\] |
| C. | \[p=-2,q=-7\] |
| D. | \[p=2,q=7\] |
| Answer» B. \[p=-2,q=7\] | |
| 6454. |
The values of a and b such that \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x(1+a\cos x)-b\sin x}{{{x}^{3}}}=1\], are [Roorkee 1996] |
| A. | \[\frac{5}{2},\ \frac{3}{2}\] |
| B. | \[\frac{5}{2},\ -\frac{3}{2}\] |
| C. | \[-\frac{5}{2},\ -\frac{3}{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6455. |
If the vectors \[a\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}+b\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}+c\mathbf{k}\] \[(a\ne b\ne c\ne 1)\] are coplanar, then the value of \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\] [BIT Ranchi 1988; RPET 1987; IIT 1987; DCE 2001; MP PET 2004; Orissa JEE 2005] |
| A. | -1 |
| B. | \[-\frac{1}{2}\] |
| C. | \[\frac{1}{2}\] |
| D. | 1 |
| Answer» E. | |
| 6456. |
\[\left( \begin{matrix} n \\ 0 \\ \end{matrix} \right)+2\,\left( \begin{matrix} n \\ 1 \\ \end{matrix} \right)+{{2}^{2}}\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)+.....+{{2}^{n}}\left( \begin{matrix} n \\ n \\ \end{matrix} \right)\] is equal to [AMU 2000] |
| A. | \[{{2}^{n}}\] |
| B. | 0 |
| C. | \[{{3}^{n}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6457. |
In triangle \[ABC\]if \[\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\], then the triangle is [Karnataka 1991; Pb. CET 1989] |
| A. | Right angled |
| B. | Obtuse angled |
| C. | Equilateral |
| D. | Isosceles |
| Answer» D. Isosceles | |
| 6458. |
The lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] will be equidistant from the origin, if |
| A. | \[{{f}^{2}}+{{g}^{2}}=c(b-a)\] |
| B. | \[{{f}^{4}}+{{g}^{4}}=c(b{{f}^{2}}+a{{g}^{2}})\] |
| C. | \[{{f}^{4}}-{{g}^{4}}=c(b{{f}^{2}}-a{{g}^{2}})\] |
| D. | \[{{f}^{2}}+{{g}^{2}}=a{{f}^{2}}+b{{g}^{2}}\] |
| Answer» D. \[{{f}^{2}}+{{g}^{2}}=a{{f}^{2}}+b{{g}^{2}}\] | |
| 6459. |
The solution of \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\cos x-\sin x\]is |
| A. | \[y=-\cos x+\sin x+{{c}_{1}}x+{{c}_{2}}\] |
| B. | \[y=-\cos x-\sin x+{{c}_{1}}x+{{c}_{2}}\] |
| C. | \[y=\cos x-\sin x+{{c}_{1}}{{x}^{2}}+{{c}_{2}}x\] |
| D. | \[y=\cos x+\sin x+{{c}_{1}}{{x}^{2}}+{{c}_{2}}x\] |
| Answer» B. \[y=-\cos x-\sin x+{{c}_{1}}x+{{c}_{2}}\] | |
| 6460. |
If \[x\in \left( \frac{\pi }{4},\frac{3\pi }{4} \right)\], then \[\int_{{}}^{{}}{\frac{\sin x-\cos x}{\sqrt{1-\sin 2x}}{{e}^{\sin x}}\cos x\ dx=}\] |
| A. | \[{{e}^{\sin x}}+c\] |
| B. | \[{{e}^{\sin x-\cos x}}+c\] |
| C. | \[{{e}^{\sin x+\cos x}}+c\] |
| D. | \[{{e}^{\cos x-\sin x}}+c\] |
| Answer» B. \[{{e}^{\sin x-\cos x}}+c\] | |
| 6461. |
The distance travelled s (in metre) by a particle in t seconds is given by, \[s={{t}^{3}}+2{{t}^{2}}+t.\]The speed of the particle after 1 second will be [UPSEAT 2003] |
| A. | 8 cm/sec |
| B. | 6 cm/sec |
| C. | 2 cm/sec |
| D. | None of these |
| Answer» B. 6 cm/sec | |
| 6462. |
If [.] denotes the greatest integer less than or equal to x, then the value of \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\]is |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | None of these |
| Answer» D. None of these | |
| 6463. |
The moment about the point \[M(-2,\,4,\,-6)\] of the force represented in magnitude and position by \[\overrightarrow{AB}\] where the points A and B have the co-ordinates \[(1,\,2,\,-3)\] and \[(3,\,-4,\,2)\] respectively, is [MP PET 2000] |
| A. | 8i - 9j - 14k |
| B. | 2i - 6j + 5k |
| C. | - 3i + 2j - 3k |
| D. | - 5i + 8j - 8k |
| Answer» B. 2i - 6j + 5k | |
| 6464. |
Two of the lines represented by the equation \[a{{y}^{4}}+bx{{y}^{3}}+c{{x}^{2}}{{y}^{2}}+d{{x}^{3}}y+e{{x}^{4}}=0\] will be perpendicular, then [Kurukshetra CEE 1998] |
| A. | \[(b+d)(ad+be)+{{(e-a)}^{2}}(a+c+e)=0\] |
| B. | \[(b+d)(ad+be)+{{(e+a)}^{2}}(a+c+e)=0\] |
| C. | \[(b-d)(ad-be)+{{(e-a)}^{2}}(a+c+e)=0\] |
| D. | \[(b-d)(ad-be)+{{(e+a)}^{2}}(a+c+e)=0\] |
| Answer» B. \[(b+d)(ad+be)+{{(e+a)}^{2}}(a+c+e)=0\] | |
| 6465. |
The rate of increase of bacteria in a certain culture is proportional to the number present. If it double in 5 hours then in 25 hours, its number would be [Pb. CET 2004] |
| A. | 8 times the original |
| B. | 16 times the original |
| C. | 32 times the original |
| D. | 64 times the original |
| Answer» D. 64 times the original | |
| 6466. |
\[\int_{{}}^{{}}{\sqrt{\frac{a-x}{x}}\ dx=}\] |
| A. | \[a\left[ {{\sin }^{-1}}\sqrt{\frac{x}{a}}+\sqrt{\frac{x}{a}}\sqrt{\frac{a-x}{a}} \right]+c\] |
| B. | \[{{\sin }^{-1}}\frac{x}{a}+\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}+c\] |
| C. | \[a\left[ {{\sin }^{-1}}\frac{x}{a}-\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}} \right]+c\] |
| D. | \[{{\sin }^{-1}}\frac{x}{a}-\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}+c\] |
| Answer» B. \[{{\sin }^{-1}}\frac{x}{a}+\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}+c\] | |
| 6467. |
\[\int_{{}}^{{}}{x{{\sin }^{-1}}x\ dx}=\] [MP PET 1991] |
| A. | \[\left( \frac{{{x}^{2}}}{2}-\frac{1}{4} \right){{\sin }^{-1}}x+\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] |
| B. | \[\left( \frac{{{x}^{2}}}{2}+\frac{1}{4} \right){{\sin }^{-1}}x+\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] |
| C. | \[\left( \frac{{{x}^{2}}}{2}-\frac{1}{4} \right){{\sin }^{-1}}x-\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] |
| D. | \[\left( \frac{{{x}^{2}}}{2}+\frac{1}{4} \right){{\sin }^{-1}}x-\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] |
| Answer» B. \[\left( \frac{{{x}^{2}}}{2}+\frac{1}{4} \right){{\sin }^{-1}}x+\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] | |
| 6468. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\sin [\pi \sqrt{{{n}^{2}}+1}]=\] |
| A. | \[\infty \] |
| B. | 0 |
| C. | Does not exist |
| D. | None of these |
| Answer» C. Does not exist | |
| 6469. |
Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then [AIEEE 2003] |
| A. | \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}+\frac{1}{c{{'}^{2}}}=0\] |
| B. | \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] |
| C. | \[\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] |
| D. | \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}-\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] |
| Answer» E. | |
| 6470. |
The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000] |
| A. | \[\frac{1}{4}\,|a\times b+b\times d+d\times a|\] |
| B. | \[\frac{1}{4}\,\left| b\times c+c\times d+a\times d+b\times a \right|\] |
| C. | \[\frac{1}{4}\,\left| a\times b+b\times c+c\times d+d\times a \right|\] |
| D. | \[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\]. |
| Answer» D. \[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\]. | |
| 6471. |
If the two angle on the base of a triangle are \[{{\left( 22\frac{1}{2} \right)}^{o}}\] and \[{{\left( 112\frac{1}{2} \right)}^{o}}\], then the ratio of the height of the triangle to the length of the base is [MP PET 1993; Pb CET 2002] |
| A. | 0.0430555555555556 |
| B. | 0.0840277777777778 |
| C. | 0.0854166666666667 |
| D. | 0.0423611111111111 |
| Answer» B. 0.0840277777777778 | |
| 6472. |
A variable line passes through a fixed point P. The algebraic sum of the perpendicular drawn from (2,0), (0, 2) and (1, 1) on the line is zero, then the coordinates of the P are [IIT 1991; AMU 2005] |
| A. | (1, -1) |
| B. | (1, 1) |
| C. | (2, 1) |
| D. | (2, 2) |
| Answer» C. (2, 1) | |
| 6473. |
The equation of the pair of straight lines parallel to x-axis and touching the circle \[{{x}^{2}}+{{y}^{2}}-6x-4y-12=0\] [Kerala (Engg.) 2002] |
| A. | \[{{y}^{2}}-4y-21=0\] |
| B. | \[{{y}^{2}}+4y-21=0\] |
| C. | \[{{y}^{2}}-4y+21=0\] |
| D. | \[{{y}^{2}}+4y+21=0\] |
| Answer» B. \[{{y}^{2}}+4y-21=0\] | |
| 6474. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{e}^{x}}}=0\] for [IIT 1992] |
| A. | No value of n |
| B. | n is any whole number |
| C. | \[n=0\] only |
| D. | \[n=2\] only |
| Answer» C. \[n=0\] only | |
| 6475. |
The d.r-s of normal to the plane through \[(1,\,0,\,0),\,\,(0,\,1,\,0)\] which makes an angle \[\frac{\pi }{4}\] with plane \[x+y=3\], are [AIEEE 2002] |
| A. | \[1,\sqrt{2},1\] |
| B. | 1,1, \[\sqrt{2}\] |
| C. | 1, 1, 2 |
| D. | \[\sqrt{2},\,1,\,1\] |
| Answer» C. 1, 1, 2 | |
| 6476. |
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{a}\,.\,\mathbf{b}=1\] and \[\mathbf{a}\times \mathbf{b}=\mathbf{j}-\mathbf{k},\] then \[\mathbf{b}=\] [IIT Screening 2004] |
| A. | \[\mathbf{i}\] |
| B. | \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] |
| C. | \[2\mathbf{j}-\mathbf{k}\] |
| D. | \[2\mathbf{i}\] |
| Answer» B. \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] | |
| 6477. |
Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A U B [MNR 1987; Karnataka CET 1996] |
| A. | 3 |
| B. | 6 |
| C. | 9 |
| D. | 18 |
| Answer» C. 9 | |
| 6478. |
If \[\cos \theta +\cos 7\theta +\cos 3\theta +\cos 5\theta =0\], then \[\theta \] [Dhanbad Engg. 1972] |
| A. | \[\frac{n\pi }{4}\] |
| B. | \[\frac{n\pi }{2}\] |
| C. | \[\frac{n\pi }{8}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6479. |
If \[A(a{{t}^{2}},\,2at),\ B(a/{{t}^{2}},\,-2a/t)\] and \[C(a,\,0)\], then 2a is equal to [RPET 2000] |
| A. | A.M. of CA and CB |
| B. | G.M. of CA and CB |
| C. | H.M. of CA and CB |
| D. | None of these |
| Answer» D. None of these | |
| 6480. |
If the co-ordinates of the middle point of the portion of a line intercepted between coordinate axes (3,2), then the equation of the line will be [RPET 1985; MP PET 1984] |
| A. | \[2x+3y=12\] |
| B. | \[3x+2y=12\] |
| C. | \[4x-3y=6\] |
| D. | \[5x-2y=10\] |
| Answer» B. \[3x+2y=12\] | |
| 6481. |
Shaded region is represented by [MP PET 1997] |
| A. | \[4x-2y\le 3\] |
| B. | \[4x-2y\le -3\] |
| C. | \[4x-2y\ge 3\] |
| D. | \[4x-2y\ge -3\] |
| Answer» C. \[4x-2y\ge 3\] | |
| 6482. |
For the circuits shown below, the Boolean polynomial is [Karnataka CET 1999] |
| A. | \[(\tilde{\ }p\vee q)\vee (p\ \vee \tilde{\ }q)\] |
| B. | \[(\tilde{\ }p\wedge p)\wedge (q\wedge q)\] |
| C. | \[(\tilde{\ }p\ \wedge \tilde{\ }q)\wedge (q\wedge p)\] |
| D. | \[(\tilde{\ }p\wedge q)\vee (p\ \wedge \tilde{\ }q)\] |
| Answer» E. | |
| 6483. |
Mean of 100 observations is 45. It was later found that two observations 19 and 31 were incorrectly recorded as 91 and 13. The correct mean is |
| A. | 44.0 |
| B. | 44.46 |
| C. | 45.00 |
| D. | 45.54 |
| Answer» C. 45.00 | |
| 6484. |
The differential equation representing the family of curves \[{{y}^{2}}=2c(x+\sqrt{c}),\]where c is a positive parameter, is of [IIT 1999; AIEEE 2005; MP PET 2002] |
| A. | Order 1 |
| B. | Order 2 |
| C. | Degree 3 |
| D. | Degree 4 |
| Answer» B. Order 2 | |
| 6485. |
\[\int_{{}}^{{}}{\frac{3\cos x+3\sin x}{4\sin x+5\cos x}\ dx=}\] [EAMCET 1991] |
| A. | \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x+5\cos x)\] |
| B. | \[\frac{27}{41}x+\frac{3}{41}\log (4\sin x+5\cos x)\] |
| C. | \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x-5\cos x)\] |
| D. | None of these |
| Answer» B. \[\frac{27}{41}x+\frac{3}{41}\log (4\sin x+5\cos x)\] | |
| 6486. |
If \[y={{\cot }^{-1}}{{(\cos 2x)}^{1/2}}\], then the value of \[\frac{dy}{dx}\]at \[x=\frac{\pi }{6}\]will be [IIT 1992] |
| A. | \[{{\left( \frac{2}{3} \right)}^{1/2}}\] |
| B. | \[{{\left( \frac{1}{3} \right)}^{1/2}}\] |
| C. | \[{{(3)}^{1/2}}\] |
| D. | \[{{(6)}^{1/2}}\] |
| Answer» B. \[{{\left( \frac{1}{3} \right)}^{1/2}}\] | |
| 6487. |
Suppose \[f:[2,\ 2]\to R\] is defined by \[f(x)=\left\{ \begin{align} & -1\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\ -2\le x\le 0 \\ & x-1\ \ \ \ \ \text{for}\ 0\le x\le 2 \\ \end{align} \right.\], then \[\{x\in (-2,\ 2):x\le 0\] and \[f(|x|)=x\}=\] [EAMCET 2003] |
| A. | \[\{-1\}\] |
| B. | {0} |
| C. | \[\{-1/2\}\] |
| D. | \[\varphi \] |
| Answer» D. \[\varphi \] | |
| 6488. |
In a \[\Delta ABC,\] if \[(\sin A+\sin B+\sin C)\] \[(\sin A+\sin B-\sin C)\] = \[3\sin A\sin B,\] then the angle C is equal to [AMU 1999] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | \[\frac{\pi }{6}\] |
| Answer» C. \[\frac{\pi }{4}\] | |
| 6489. |
The equation of the line joining the point (3, 5)to the point of intersection of the lines \[4x+y-1=0\] and \[7x-3y-35=0\] is equidistant from the points (0, 0) and (8, 34) [Roorkee 1984] |
| A. | True |
| B. | False |
| C. | Nothing can be said |
| D. | None of these |
| Answer» B. False | |
| 6490. |
If the pair of lines \[a{{x}^{2}}+2(a+b)xy+b{{y}^{2}}=0\] lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then [AIEEE 2005] |
| A. | \[3{{a}^{2}}+10ab+3{{b}^{2}}=0\] |
| B. | \[3{{a}^{2}}+2ab+3{{b}^{2}}=0\] |
| C. | \[3{{a}^{2}}-10ab+3{{b}^{2}}=0\] |
| D. | \[3{{a}^{2}}-2ab+3{{b}^{2}}=0\] |
| Answer» C. \[3{{a}^{2}}-10ab+3{{b}^{2}}=0\] | |
| 6491. |
The equation of family of curves for which the length of the normal is equal to the radius vector is |
| A. | \[{{y}^{2}}\pm {{x}^{2}}=k\] |
| B. | \[y\pm x=k\] |
| C. | \[{{y}^{2}}=kx\] |
| D. | None of these |
| Answer» B. \[y\pm x=k\] | |
| 6492. |
\[\int_{{}}^{{}}{{{\tan }^{3}}}2x\sec 2x\ dx=\] [IIT 1977] |
| A. | \[\frac{1}{6}{{\sec }^{3}}2x-\frac{1}{2}\sec 2x+c\] |
| B. | \[\frac{1}{6}{{\sec }^{3}}2x+\frac{1}{2}\sec 2x+c\] |
| C. | \[\frac{1}{9}{{\sec }^{2}}2x-\frac{1}{3}\sec 2x+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{6}{{\sec }^{3}}2x+\frac{1}{2}\sec 2x+c\] | |
| 6493. |
The volume of a spherical balloon is increasing at the rate of 40 cubic centrimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 8 centimetre, is [Roorkee 1983] |
| A. | \[\frac{5}{2}\] sq cm/min |
| B. | 5 sq cm/min |
| C. | 10 sq cm/min |
| D. | 20 sq cm/min |
| Answer» D. 20 sq cm/min | |
| 6494. |
True statement for \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{2+3x}-\sqrt{2-3x}}\] is [BIT Ranchi 1982] |
| A. | Does not exist |
| B. | Lies between 0 and \[\frac{1}{2}\] |
| C. | Lies between \[\frac{1}{2}\] and 1 |
| D. | Greater then 1 |
| Answer» C. Lies between \[\frac{1}{2}\] and 1 | |
| 6495. |
The equation of the plane through the intersection of the planes \[x+2y+3z-4=0\], \[4x+3y+2z+1=0\] and passing through the origin will be [MP PET 1998] |
| A. | \[x+y+z=0\] |
| B. | \[17x+14y+11z=0\] |
| C. | \[7x+4y+z=0\] |
| D. | \[17x+14y+z=0\] |
| Answer» C. \[7x+4y+z=0\] | |
| 6496. |
Let the vectors a, b, c and d be such that\[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=0\]. Let \[{{P}_{1}}\] and \[{{P}_{2}}\] be planes determined by pair of vectors a, b and c, d respectively. Then the angle between \[{{P}_{1}}\] and \[{{P}_{2}}\] is [IIT Screening 2000; MP PET 2004] |
| A. | \[{{0}^{o}}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» B. \[\frac{\pi }{4}\] | |
| 6497. |
In a triangle \[ABC\], \[\tan \frac{A}{2}=\frac{5}{6}\] and \[\tan \frac{C}{2}=\frac{2}{5},\] then [EAMCET 1994] |
| A. | \[a,\ b,\ c\]are in A.P. |
| B. | \[\cos A,\ \cos B,\ \cos C\]are in A.P. |
| C. | \[\sin A,\ \sin B,\ \sin C\]are in A.P. |
| D. | (a) and (c) both |
| Answer» E. | |
| 6498. |
The vertices of a triangle are (2, 1), (5, 2) and (4, 4). The lengths of the perpendicular from these vertices on the opposite sides are [IIT 1962] |
| A. | \[\frac{7}{\sqrt{5}},\frac{7}{\sqrt{13}},\frac{7}{\sqrt{6}}\] |
| B. | \[\frac{7}{\sqrt{6}},\frac{7}{\sqrt{8}},\frac{7}{\sqrt{10}}\] |
| C. | \[\frac{7}{\sqrt{5}},\frac{7}{\sqrt{8}},\frac{7}{\sqrt{15}}\] |
| D. | \[\frac{7}{\sqrt{5}},\frac{7}{\sqrt{13}},\frac{7}{\sqrt{10}}\] |
| Answer» E. | |
| 6499. |
The slope of the tangent at \[(x,y)\]to a curve passing through \[\left( 1,\frac{\pi }{4} \right)\]is given by \[\frac{y}{x}-{{\cos }^{2}}\left( \frac{y}{x} \right)\], then the equation of the curve is [Kurukshetra CEE 2002] |
| A. | \[y={{\tan }^{-1}}\left[ \log \left( \frac{e}{x} \right) \right]\] |
| B. | \[y=x{{\tan }^{-1}}\left[ \log \left( \frac{x}{e} \right) \right]\] |
| C. | \[y=x{{\tan }^{-1}}\left[ \log \left( \frac{e}{x} \right) \right]\] |
| D. | None of these |
| Answer» D. None of these | |
| 6500. |
The value of \[\int{\frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{x}dx}\] will be [UPSEAT 1999] |
| A. | \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,-a{{\tan }^{-1}}\left[ \frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{a} \right]\] |
| B. | \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,+a{{\tan }^{-1}}\left[ \frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{a} \right]\] |
| C. | \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,+{{a}^{2}}{{\tan }^{-1}}[\sqrt{{{x}^{2}}-{{a}^{2}}}]\] |
| D. | \[{{\tan }^{-1}}x/a+c\] |
| Answer» B. \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,+a{{\tan }^{-1}}\left[ \frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{a} \right]\] | |