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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.
| 4351. |
The eccentricity of the hyperbola \[5{{x}^{2}}-4{{y}^{2}}+20x+8y=4\] is [UPSEAT 2004] |
| A. | \[\sqrt{2}\] |
| B. | \[\frac{3}{2}\] |
| C. | 2 |
| D. | 3 |
| Answer» C. 2 | |
| 4352. |
\[{{x}^{2}}-4{{y}^{2}}-2x+16y-40=0\] represents [DCE 1999] |
| A. | A pair of straight lines |
| B. | An ellipse |
| C. | A hyperbola |
| D. | A parabola |
| Answer» D. A parabola | |
| 4353. |
The equation of the hyperbola whose directrix is \[2x+y=1\], focus (1, 1) and eccentricity \[=\sqrt{3}\], is |
| A. | \[7{{x}^{2}}+12xy-2{{y}^{2}}-2x+4y-7=0\] |
| B. | \[11{{x}^{2}}+12xy+2{{y}^{2}}-10x-4y+1=0\] |
| C. | \[11{{x}^{2}}+12xy+2{{y}^{2}}-14x-14y+1=0\] |
| D. | None of these |
| Answer» B. \[11{{x}^{2}}+12xy+2{{y}^{2}}-10x-4y+1=0\] | |
| 4354. |
The latus rectum of the hyperbola \[9{{x}^{2}}-16{{y}^{2}}-18x-32y-151=0\] is [MP PET 1996] |
| A. | \[\frac{9}{4}\] |
| B. | 9 |
| C. | \[\frac{3}{2}\] |
| D. | \[\frac{9}{2}\] |
| Answer» E. | |
| 4355. |
The auxiliary equation of circle of hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\], is |
| A. | \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] |
| B. | \[{{x}^{2}}+{{y}^{2}}={{b}^{2}}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\] |
| D. | \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}-{{b}^{2}}\] |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}={{b}^{2}}\] | |
| 4356. |
The equation of the hyperbola whose foci are (6, 4) and (?4, 4) and eccentricity 2 is given by [MP PET 1993] |
| A. | \[12{{x}^{2}}-4{{y}^{2}}-24x+32y-127=0\] |
| B. | \[12{{x}^{2}}+4{{y}^{2}}+24x-32y-127=0\] |
| C. | \[12{{x}^{2}}-4{{y}^{2}}-24x-32y+127=0\] |
| D. | \[12{{x}^{2}}-4{{y}^{2}}+24x+32y+127=0\] |
| Answer» B. \[12{{x}^{2}}+4{{y}^{2}}+24x-32y-127=0\] | |
| 4357. |
Centre of hyperbola \[9{{x}^{2}}-16{{y}^{2}}+18x+32y-151=0\] is |
| A. | (1, ?1) |
| B. | (?1, 1) |
| C. | (?1, ?1) |
| D. | (1, 1) |
| Answer» C. (?1, ?1) | |
| 4358. |
The equation of the directrices of the conic \[{{x}^{2}}+2x-{{y}^{2}}+5=0\] are |
| A. | \[x=\pm 1\] |
| B. | \[y=\pm 2\] |
| C. | \[y=\pm \sqrt{2}\] |
| D. | \[x=\pm \sqrt{3}\] |
| Answer» D. \[x=\pm \sqrt{3}\] | |
| 4359. |
The equation \[{{x}^{2}}+4xy+{{y}^{2}}+2x+4y+2=0\] represents |
| A. | An ellipse |
| B. | A pair of straight lines |
| C. | A hyperbola |
| D. | None of these |
| Answer» D. None of these | |
| 4360. |
The vertices of a hyperbola are at (0, 0) and (10, 0) and one of its foci is at (18, 0). The equation of the hyperbola is |
| A. | \[\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{144}=1\] |
| B. | \[\frac{{{(x-5)}^{2}}}{25}-\frac{{{y}^{2}}}{144}=1\] |
| C. | \[\frac{{{x}^{2}}}{25}-\frac{{{(y-5)}^{2}}}{144}=1\] |
| D. | \[\frac{{{(x-5)}^{2}}}{25}-\frac{{{(y-5)}^{2}}}{144}=1\] |
| Answer» C. \[\frac{{{x}^{2}}}{25}-\frac{{{(y-5)}^{2}}}{144}=1\] | |
| 4361. |
The equation of the hyperbola whose directrix is \[x+2y=1\], focus (2, 1) and eccentricity 2 will be [MP PET 1988, 89] |
| A. | \[{{x}^{2}}-16xy-11{{y}^{2}}-12x+6y+21=0\] |
| B. | \[3{{x}^{2}}+16xy+15{{y}^{2}}-4x-14y-1=0\] |
| C. | \[{{x}^{2}}+16xy+11{{y}^{2}}-12x-6y+21=0\] |
| D. | None of these |
| Answer» B. \[3{{x}^{2}}+16xy+15{{y}^{2}}-4x-14y-1=0\] | |
| 4362. |
The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6. The equation of the hyperbola referred to its axes as axes of co-ordinates is |
| A. | \[3{{x}^{2}}-{{y}^{2}}=3\] |
| B. | \[{{x}^{2}}-3{{y}^{2}}=3\] |
| C. | \[3{{x}^{2}}-{{y}^{2}}=9\] |
| D. | \[{{x}^{2}}-3{{y}^{2}}=9\] |
| Answer» D. \[{{x}^{2}}-3{{y}^{2}}=9\] | |
| 4363. |
The eccentricity of the hyperbola \[2{{x}^{2}}-{{y}^{2}}=6\] is [MP PET 1992] |
| A. | \[\sqrt{2}\] |
| B. | 2 |
| C. | 3 |
| D. | \[\sqrt{3}\] |
| Answer» E. | |
| 4364. |
The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is [Karnataka CET 2003] |
| A. | A straight line |
| B. | A circle |
| C. | An ellipse |
| D. | A hyperbola |
| Answer» E. | |
| 4365. |
If P is a point on the hyperbola \[16{{x}^{2}}-9{{y}^{2}}=144\] whose foci are \[{{S}_{1}}\] and \[{{S}_{2}}\], then \[P{{S}_{1}}\tilde{\ }P{{S}_{2}}=\] |
| A. | 4 |
| B. | 6 |
| C. | 8 |
| D. | 12 |
| Answer» C. 8 | |
| 4366. |
Locus of the point of intersection of straight lines \[\frac{x}{a}-\frac{y}{b}=m\] and \[\frac{x}{a}+\frac{y}{b}=\frac{1}{m}\] is [MP PET 1991, 2003] |
| A. | An ellipse |
| B. | A circle |
| C. | A hyperbola |
| D. | A parabola |
| Answer» D. A parabola | |
| 4367. |
The directrix of the hyperbola is \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1\] [UPSEAT 2003] |
| A. | \[x=9/\sqrt{13}\] |
| B. | \[y=9/\sqrt{13}\] |
| C. | \[x=6/\sqrt{13}\] |
| D. | \[y=6/\sqrt{13}\] |
| Answer» B. \[y=9/\sqrt{13}\] | |
| 4368. |
The length of transverse axis of the parabola \[3{{x}^{2}}-4{{y}^{2}}=32\] is [Karnataka CET 2001] |
| A. | \[\frac{8\sqrt{2}}{\sqrt{3}}\] |
| B. | \[\frac{16\sqrt{2}}{\sqrt{3}}\] |
| C. | \[\frac{3}{32}\] |
| D. | \[\frac{64}{3}\] |
| Answer» B. \[\frac{16\sqrt{2}}{\sqrt{3}}\] | |
| 4369. |
The foci of the hyperbola \[9{{x}^{2}}-16{{y}^{2}}=144\] are [MP PET 2001] |
| A. | \[(\pm 4,\ 0)\] |
| B. | \[(0,\ \pm 4)\] |
| C. | \[(\pm 5,\ 0)\] |
| D. | \[(0,\ \pm 5)\] |
| Answer» D. \[(0,\ \pm 5)\] | |
| 4370. |
The foci of the hyperbola \[2{{x}^{2}}-3{{y}^{2}}=5\], is [MP PET 2000] |
| A. | \[\left( \pm \frac{5}{\sqrt{6}},\ 0 \right)\] |
| B. | \[\left( \pm \frac{5}{6},\ 0 \right)\] |
| C. | \[\left( \pm \frac{\sqrt{5}}{6},\ 0 \right)\] |
| D. | None of these |
| Answer» B. \[\left( \pm \frac{5}{6},\ 0 \right)\] | |
| 4371. |
The locus of the centre of a circle, which touches externally the given two circles, is [Karnataka CET 1999] |
| A. | Circle |
| B. | Parabola |
| C. | Hyperbola |
| D. | Ellipse |
| Answer» D. Ellipse | |
| 4372. |
The eccentricity of the hyperbola \[4{{x}^{2}}-9{{y}^{2}}=16\], is |
| A. | \[\frac{8}{3}\] |
| B. | \[\frac{5}{4}\] |
| C. | \[\frac{\sqrt{13}}{3}\] |
| D. | \[\frac{4}{3}\] |
| Answer» D. \[\frac{4}{3}\] | |
| 4373. |
The eccentricity of the conic \[{{x}^{2}}-4{{y}^{2}}=1\], is [MP PET 1999] |
| A. | \[\frac{2}{\sqrt{3}}\] |
| B. | \[\frac{\sqrt{3}}{2}\] |
| C. | \[\frac{2}{\sqrt{5}}\] |
| D. | \[\frac{\sqrt{5}}{2}\] |
| Answer» E. | |
| 4374. |
If the eccentricities of the hyperbolas \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and \[\frac{{{y}^{2}}}{{{b}^{2}}}-\frac{{{x}^{2}}}{{{a}^{2}}}=1\] be e and \[{{e}_{1}}\], then \[\frac{1}{{{e}^{2}}}+\frac{1}{e_{1}^{2}}=\] [MNR 1984; MP PET 1995; DCE 2000] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» B. 2 | |
| 4375. |
The locus of the point of intersection of the lines \[\sqrt{3}x-y-4\sqrt{3}k=0\] and \[\sqrt{3}kx+ky-4\sqrt{3}=0\] for different value of k is |
| A. | Circle |
| B. | Parabola |
| C. | Hyperbola |
| D. | Ellipse |
| Answer» D. Ellipse | |
| 4376. |
A hyperbola passes through the points (3, 2) and (?17, 12) and has its centre at origin and transverse axis is along x-axis. The length of its transverse axis is |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | None of these |
| Answer» B. 4 | |
| 4377. |
The locus of the point of intersection of the lines \[ax\sec \theta +by\tan \theta =a\] and \[ax\tan \theta +by\sec \theta =b\], where \[\theta \] is the parameter, is |
| A. | A straight line |
| B. | A circle |
| C. | An ellipse |
| D. | A hyperbola |
| Answer» E. | |
| 4378. |
If the centre, vertex and focus of a hyperbola be (0, 0), (4, 0) and (6, 0) respectively, then the equation of the hyperbola is |
| A. | \[4{{x}^{2}}-5{{y}^{2}}=8\] |
| B. | \[4{{x}^{2}}-5{{y}^{2}}=80\] |
| C. | \[5{{x}^{2}}-4{{y}^{2}}=80\] |
| D. | \[5{{x}^{2}}-4{{y}^{2}}=8\] |
| Answer» D. \[5{{x}^{2}}-4{{y}^{2}}=8\] | |
| 4379. |
If \[(0,\ \pm 4)\] and \[(0,\ \pm 2)\] be the foci and vertices of a hyperbola, then its equation is |
| A. | \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{12}=1\] |
| B. | \[\frac{{{x}^{2}}}{12}-\frac{{{y}^{2}}}{4}=1\] |
| C. | \[\frac{{{y}^{2}}}{4}-\frac{{{x}^{2}}}{12}=1\] |
| D. | \[\frac{{{y}^{2}}}{12}-\frac{{{x}^{2}}}{4}=1\] |
| Answer» D. \[\frac{{{y}^{2}}}{12}-\frac{{{x}^{2}}}{4}=1\] | |
| 4380. |
The locus of the point of intersection of the lines \[bxt-ayt=ab\] and \[bx+ay=abt\] is |
| A. | A parabola |
| B. | An ellipse |
| C. | A hyperbola |
| D. | None of these |
| Answer» D. None of these | |
| 4381. |
The equation of the transverse and conjugate axis of the hyperbola \[16{{x}^{2}}-{{y}^{2}}+64x+4y+44=0\] are |
| A. | \[x=2,\ y+2=0\] |
| B. | \[x=2,\ y=2\] |
| C. | \[y=2,\ x+2=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 4382. |
Find the equation of axis of the given hyperbola \[\frac{{{x}^{2}}}{3}-\frac{{{y}^{2}}}{2}=1\] which is equally inclined to the axes [DCE 2005] |
| A. | \[y=x+1\] |
| B. | \[y=x-1\] |
| C. | \[y=x+2\] |
| D. | \[y=x-2\] |
| Answer» B. \[y=x-1\] | |
| 4383. |
If \[4{{x}^{2}}+p{{y}^{2}}=45\] and \[{{x}^{2}}-4{{y}^{2}}=5\] cut orthogonally, then the value of p is [Kerala (Engg.) 2005] |
| A. | 1/9 |
| B. | 1/3 |
| C. | 3 |
| D. | 18 |
| E. | 9 |
| Answer» F. | |
| 4384. |
If q is the acute angle of intersection at a real point of intersection of the circle \[{{x}^{2}}+{{y}^{2}}=5\] and the parabola \[{{y}^{2}}=4x\] then tanq is equal to [Karnataka CET 2005] |
| A. | 1 |
| B. | \[\sqrt{3}\] |
| C. | 3 |
| D. | \[\frac{1}{\sqrt{3}}\] |
| Answer» D. \[\frac{1}{\sqrt{3}}\] | |
| 4385. |
The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8 [Karnataka CET 2005] |
| A. | \[\frac{{{x}^{2}}}{12}-\frac{{{y}^{2}}}{4}=1\] |
| B. | \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{12}=1\] |
| C. | \[\frac{{{x}^{2}}}{8}-\frac{{{y}^{2}}}{2}=1\] |
| D. | \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1\] |
| Answer» C. \[\frac{{{x}^{2}}}{8}-\frac{{{y}^{2}}}{2}=1\] | |
| 4386. |
The eccentricity of the hyperbola \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{25}=1\] is [Karnataka CET 2005] |
| A. | 3/4 |
| B. | 3/5 |
| C. | \[\sqrt{41}/4\] |
| D. | \[\sqrt{41/5}\] |
| Answer» D. \[\sqrt{41/5}\] | |
| 4387. |
The locus of a point \[P(\alpha ,\,\beta )\] moving under the condition that the line \[y=\alpha x+\beta \] is a tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is [AIEEE 2005] |
| A. | A parabola |
| B. | A hyperbola |
| C. | An ellipse |
| D. | A circle |
| Answer» C. An ellipse | |
| 4388. |
Eccentricity of rectangular hyperbola is [UPSEAT 2002] |
| A. | \[\frac{1}{\sqrt{2}}\] |
| B. | \[\frac{-1}{\sqrt{2}}\] |
| C. | \[\sqrt{2}\] |
| D. | > 2 |
| Answer» D. > 2 | |
| 4389. |
The eccentricity of the hyperbola \[{{x}^{2}}-{{y}^{2}}=25\] is [MP PET 1987] |
| A. | \[\sqrt{2}\] |
| B. | \[1/\sqrt{2}\] |
| C. | 2 |
| D. | \[1+\sqrt{2}\] |
| Answer» B. \[1/\sqrt{2}\] | |
| 4390. |
A point on the curve \[\frac{{{x}^{2}}}{{{A}^{2}}}-\frac{{{y}^{2}}}{{{B}^{2}}}=1\] is [MP PET 1988] |
| A. | \[(A\cos \theta ,\ B\sin \theta )\] |
| B. | \[(A\sec \theta ,\ B\tan \theta )\] |
| C. | \[(A{{\cos }^{2}}\theta ,\ B{{\sin }^{2}}\theta )\] |
| D. | None of these |
| Answer» C. \[(A{{\cos }^{2}}\theta ,\ B{{\sin }^{2}}\theta )\] | |
| 4391. |
The solution of the differential equation \[x+y\frac{dy}{dx}=2y\] is |
| A. | \[\log (y-x)=c+\frac{y-x}{x}\] |
| B. | \[\log (y-x)=c+\frac{x}{y-x}\] |
| C. | \[y-x=c+\log \frac{x}{y-x}\] |
| D. | \[y-x=c+\frac{x}{y-x}\] |
| Answer» C. \[y-x=c+\log \frac{x}{y-x}\] | |
| 4392. |
The solution of the differential equation \[\frac{dy}{dx}=\frac{xy}{{{x}^{2}}+{{y}^{2}}}\]is |
| A. | \[a{{y}^{2}}={{e}^{{{x}^{2}}/{{y}^{2}}}}\] |
| B. | \[ay={{e}^{x/y}}\] |
| C. | \[y={{e}^{{{x}^{2}}}}+{{e}^{{{y}^{2}}}}+c\] |
| D. | \[y={{e}^{{{x}^{2}}}}+{{y}^{2}}+c\] |
| Answer» B. \[ay={{e}^{x/y}}\] | |
| 4393. |
The general solution of the differential equation \[(x+y)dx+xdy=0\] is [MP PET 1994, 95] |
| A. | \[{{x}^{2}}+{{y}^{2}}=c\] |
| B. | \[2{{x}^{2}}-{{y}^{2}}=c\] |
| C. | \[{{x}^{2}}+2xy=c\] |
| D. | \[{{y}^{2}}+2xy=c\] |
| Answer» D. \[{{y}^{2}}+2xy=c\] | |
| 4394. |
The solution of the differential equation \[x\,dy-y\,dx=(\sqrt{{{x}^{2}}+{{y}^{2}})}dx\]is |
| A. | \[y-\sqrt{{{x}^{2}}+{{y}^{2}}}=c{{x}^{2}}\] |
| B. | \[y+\sqrt{{{x}^{2}}+{{y}^{2}}}=c{{x}^{2}}\] |
| C. | \[y+\sqrt{{{x}^{2}}+{{y}^{2}}}+c{{x}^{2}}=0\] |
| D. | None of these |
| Answer» C. \[y+\sqrt{{{x}^{2}}+{{y}^{2}}}+c{{x}^{2}}=0\] | |
| 4395. |
The solution of the differential equation \[(3xy+{{y}^{2}})dx+({{x}^{2}}+xy)dy=0\] is [AISSE 1990] |
| A. | \[{{x}^{2}}(2xy+{{y}^{2}})={{c}^{2}}\] |
| B. | \[{{x}^{2}}(2xy-{{y}^{2}})={{c}^{2}}\] |
| C. | \[{{x}^{2}}({{y}^{2}}-2xy)={{c}^{2}}\] |
| D. | None of these |
| Answer» B. \[{{x}^{2}}(2xy-{{y}^{2}})={{c}^{2}}\] | |
| 4396. |
The solution of the equation \[\frac{dy}{dx}=\frac{x+y}{x-y}\]is [AI CBSE 1990] |
| A. | \[c{{({{x}^{2}}+{{y}^{2}})}^{1/2}}+{{e}^{{{\tan }^{-1}}(y/x)}}=0\] |
| B. | \[c{{({{x}^{2}}+{{y}^{2}})}^{1/2}}={{e}^{{{\tan }^{-1}}(y/x)}}\] |
| C. | \[c({{x}^{2}}-{{y}^{2}})={{e}^{{{\tan }^{-1}}(y/x)}}\] |
| D. | None of these |
| Answer» C. \[c({{x}^{2}}-{{y}^{2}})={{e}^{{{\tan }^{-1}}(y/x)}}\] | |
| 4397. |
The solution of the differential equation \[({{x}^{2}}+{{y}^{2}})dx=2xydy\] is [MP PET 2003; Orissa JEE 2005] |
| A. | \[x=c({{x}^{2}}+{{y}^{2}})\] |
| B. | \[x=c({{x}^{2}}-{{y}^{2}})\] |
| C. | \[x+c({{x}^{2}}-{{y}^{2}})=0\] |
| D. | None of these |
| Answer» C. \[x+c({{x}^{2}}-{{y}^{2}})=0\] | |
| 4398. |
The general solution of the differential equation \[(2x-y+1)dx+(2y-x+1)dy=0\] is [Karnataka CET 2005] |
| A. | \[{{x}^{2}}+{{y}^{2}}+xy-x+y=c\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-xy+x+y=c\] |
| C. | \[{{x}^{2}}-{{y}^{2}}+2xy-x+y=c\] |
| D. | \[{{x}^{2}}-{{y}^{2}}-2xy+x-y=c\] |
| Answer» C. \[{{x}^{2}}-{{y}^{2}}+2xy-x+y=c\] | |
| 4399. |
Solution of differential equation \[2xy\frac{dy}{dx}={{x}^{2}}+3{{y}^{2}}\] is [MP PET 1993] |
| A. | \[{{x}^{3}}+{{y}^{2}}=p{{x}^{2}}\] |
| B. | \[\frac{{{x}^{2}}}{2}+\frac{{{y}^{3}}}{x}={{y}^{2}}+p\] |
| C. | \[{{x}^{2}}+{{y}^{3}}=p{{x}^{2}}\] |
| D. | \[{{x}^{2}}+{{y}^{2}}=p{{x}^{3}}\] |
| Answer» E. | |
| 4400. |
If \[{y}'=\frac{x-y}{x+y}\], then its solution is [MP PET 2000] |
| A. | \[{{y}^{2}}+2xy-{{x}^{2}}=c\] |
| B. | \[{{y}^{2}}+2xy+{{x}^{2}}=c\] |
| C. | \[{{y}^{2}}-2xy-{{x}^{2}}=c\] |
| D. | \[{{y}^{2}}-2xy+{{x}^{2}}=c\] |
| Answer» B. \[{{y}^{2}}+2xy+{{x}^{2}}=c\] | |