MCQOPTIONS
Saved Bookmarks
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.
| 8451. |
If the circles \[{{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\] and \[{{x}^{2}}+{{y}^{2}}-3ax+dy-1=0\] intersect in two distinct points P and Q then the line \[5x+by-a=0\]passes through P and Q for |
| A. | Exactly one value of a |
| B. | No value of a |
| C. | Infinitely many values of a |
| D. | Exactly two values of a |
| Answer» C. Infinitely many values of a | |
| 8452. |
If the line \[x\cos \alpha +y\sin \alpha =p\] represents the common chord of the circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] and \[{{x}^{2}}+{{y}^{2}}+{{b}^{2}}(a>b),\] where A and B lie on the first circle and P and Q lie on the second circle, then AP is equal to |
| A. | \[\sqrt{{{a}^{2}}+{{p}^{2}}}+\sqrt{{{b}^{2}}+{{p}^{2}}}\] |
| B. | \[\sqrt{{{a}^{2}}-{{p}^{2}}}+\sqrt{{{b}^{2}}-{{p}^{2}}}\] |
| C. | \[\sqrt{{{a}^{2}}-{{p}^{2}}}-\sqrt{{{b}^{2}}-{{p}^{2}}}\] |
| D. | \[\sqrt{{{a}^{2}}+{{p}^{2}}}-\sqrt{{{b}^{2}}+{{p}^{2}}}\] |
| Answer» D. \[\sqrt{{{a}^{2}}+{{p}^{2}}}-\sqrt{{{b}^{2}}+{{p}^{2}}}\] | |
| 8453. |
Distances form the origin to the centres of the three circles \[{{x}^{2}}+{{y}^{2}}-2{{\lambda }_{i}}x={{c}^{2}}\] (where c is constant and i= 1, 2, 3) are in GP. Then the lengths of tangents drawn from any point on the circle \[{{x}^{2}}+{{y}^{2}}={{c}^{2}}\] to these circles are in |
| A. | A.P. |
| B. | GP. |
| C. | H.P. |
| D. | None |
| Answer» C. H.P. | |
| 8454. |
If the line \[x+y=1\] is a tangent to a circle with centre (2, 3), then its equation is |
| A. | \[{{x}^{2}}+{{y}^{2}}+2x+2y+5=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}-4x-6y+5=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}-x-y+3=0\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+5x+2y=0\] |
| Answer» C. \[{{x}^{2}}+{{y}^{2}}-x-y+3=0\] | |
| 8455. |
A line is drawn through a fixed point \[P(\alpha ,\beta )\] to cut the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] at A and B, then PA.PB is equal to |
| A. | \[{{\alpha }^{2}}+{{\beta }^{2}}\] |
| B. | \[{{\alpha }^{2}}+{{\beta }^{2}}-{{\alpha }^{2}}\] |
| C. | \[{{\alpha }^{2}}\] |
| D. | \[{{\alpha }^{2}}+{{\beta }^{2}}+{{\alpha }^{2}}\] |
| Answer» C. \[{{\alpha }^{2}}\] | |
| 8456. |
Let S is a circle with centre \[(0,\sqrt{2}).\] Then |
| A. | There cannot be any rational point on S |
| B. | There can be infinitely many rational points on S |
| C. | There can be at most two rational points on S |
| D. | There are exactly two rational points on S |
| Answer» D. There are exactly two rational points on S | |
| 8457. |
Let \[{{S}_{1}},{{S}_{2}}\] be the foci of the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1.\]If \[A(x+y)\] is any point on the ellipse, then the maximum area of the triangle \[A{{S}_{1}}{{S}_{2}}\] (in square units) is |
| A. | \[2\sqrt{2}\] |
| B. | \[2\sqrt{3}\] |
| C. | 8 |
| D. | 4 |
| Answer» D. 4 | |
| 8458. |
If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are |
| A. | 3 unit and 4 unit |
| B. | 6 unit and 4 unit |
| C. | 3 unit and 8 unit |
| D. | 6 unit and 8 unit |
| Answer» E. | |
| 8459. |
If the parabola \[{{y}^{2}}=4ax\] passes through the point (1, ?2), then the tangent at this point is [MP PET 1998] |
| A. | \[x+y-1=0\] |
| B. | \[x-y-1=0\] |
| C. | \[x+y+1=0\] |
| D. | \[x-y+1=0\] |
| Answer» D. \[x-y+1=0\] | |
| 8460. |
If line \[x=my+k\] touches the parabola \[{{x}^{2}}=4ay\], then \[k=\] [MP PET 1995] |
| A. | \[\frac{a}{m}\] |
| B. | am |
| C. | \[a{{m}^{2}}\] |
| D. | \[-a{{m}^{2}}\] |
| Answer» B. am | |
| 8461. |
The locus of a foot of perpendicular drawn to the tangent of parabola \[{{y}^{2}}=4ax\] from focus, is [RPET 1989] |
| A. | \[x=0\] |
| B. | \[y=0\] |
| C. | \[{{y}^{2}}=2a(x+a)\] |
| D. | \[{{x}^{2}}+{{y}^{2}}(x+a)=0\] |
| Answer» B. \[y=0\] | |
| 8462. |
The straight line \[y=2x+\lambda \] does not meet the parabola \[{{y}^{2}}=2x\], if [MP PET 1993; MNR 1977] |
| A. | \[\lambda <\frac{1}{4}\] |
| B. | \[\lambda >\frac{1}{4}\] |
| C. | \[\lambda =4\] |
| D. | \[\lambda =1\] |
| Answer» C. \[\lambda =4\] | |
| 8463. |
The point of intersection of the latus rectum and axis of the parabola \[{{y}^{2}}+4x+2y-8=0\] |
| A. | (5/4, ?1) |
| B. | (9/4, ?1) |
| C. | (7/2, 5/2) |
| D. | None of these |
| Answer» B. (9/4, ?1) | |
| 8464. |
Focus of the parabola \[{{(y-2)}^{2}}=20(x+3)\] is [Karnataka CET 1999] |
| A. | (3, -2) |
| B. | (2, -3) |
| C. | (2, 2) |
| D. | (3, 3) |
| Answer» D. (3, 3) | |
| 8465. |
The length of latus rectum of the parabola \[4{{y}^{2}}+2x-20y+17=0\] is [MP PET 1999] |
| A. | 3 |
| B. | 6 |
| C. | \[\frac{1}{2}\] |
| D. | 9 |
| Answer» D. 9 | |
| 8466. |
The equation of the locus of a point which moves so as to be at equal distances from the point (a, 0) and the y-axis is |
| A. | \[{{y}^{2}}-2ax+{{a}^{2}}=0\] |
| B. | \[{{y}^{2}}+2ax+{{a}^{2}}=0\] |
| C. | \[{{x}^{2}}-2ay+{{a}^{2}}=0\] |
| D. | \[{{x}^{2}}+2ay+{{a}^{2}}=0\] |
| Answer» B. \[{{y}^{2}}+2ax+{{a}^{2}}=0\] | |
| 8467. |
Vertex of the parabola \[{{x}^{2}}+4x+2y-7=0\] is [MP PET 1990] |
| A. | (?2, 11/2) |
| B. | (?2, 2) |
| C. | (?2, 11) |
| D. | (2, 11) |
| Answer» B. (?2, 2) | |
| 8468. |
The equation of latus rectum of a parabola is \[x+y=8\] and the equation of the tangent at the vertex is \[x+y=12\], then length of the latus rectum is [MP PET 2002] |
| A. | \[4\sqrt{2}\] |
| B. | \[2\sqrt{2}\] |
| C. | 8 |
| D. | \[8\sqrt{2}\] |
| Answer» E. | |
| 8469. |
Vertex of the parabola \[9{{x}^{2}}-6x+36y+9=0\] is |
| A. | \[(1/3,\ -2/9)\] |
| B. | \[(-1/3,\ -1/2)\] |
| C. | \[(-1/3,\ 1/2)\] |
| D. | \[(1/3,\ 1/2)\] |
| Answer» B. \[(-1/3,\ -1/2)\] | |
| 8470. |
Locus of the poles of focal chords of a parabola is of parabola [EAMCET 2002] |
| A. | The tangent at the vertex |
| B. | The axis |
| C. | A focal chord |
| D. | The directrix |
| Answer» E. | |
| 8471. |
The length intercepted by the curve \[{{y}^{2}}=4x\] on the line satisfying \[dy/dx=1\] and passing through point (0, 1) is given by [Orissa JEE 2005] |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 8472. |
The point on parabola \[2y={{x}^{2}}\], which is nearest to the point (0, 3) is [J & K 2005] |
| A. | (±4, 8) |
| B. | \[(\pm 1,\,1/2)\] |
| C. | (±2, 2) |
| D. | None of these |
| Answer» D. None of these | |
| 8473. |
The polar of focus of parabola [RPET 1999] |
| A. | x-axis |
| B. | y-axis |
| C. | Directrix |
| D. | Latus rectum |
| Answer» D. Latus rectum | |
| 8474. |
The area of triangle formed inside the parabola \[{{y}^{2}}=4x\] and whose ordinates of vertices are 1, 2 and 4 will be [RPET 1990] |
| A. | \[\frac{7}{2}\] |
| B. | \[\frac{5}{2}\] |
| C. | \[\frac{3}{2}\] |
| D. | \[\frac{3}{4}\] |
| Answer» E. | |
| 8475. |
The length of the normal chord to the parabola \[{{y}^{2}}=4x\], which subtends right angle at the vertex is [RPET 1999] |
| A. | \[6\sqrt{3}\] |
| B. | \[3\sqrt{3}\] |
| C. | 2 |
| D. | 1 |
| Answer» B. \[3\sqrt{3}\] | |
| 8476. |
The ends of latus rectum of parabola \[{{x}^{2}}+8y=0\] are [MP PET 1995] |
| A. | (?4, ?2) and (4, 2) |
| B. | (4, ?2) and (?4, 2) |
| C. | (?4, ?2) and (4, ?2) |
| D. | (4, 2) and (?4, 2) |
| Answer» D. (4, 2) and (?4, 2) | |
| 8477. |
Tangents drawn at the ends of any focal chord of a parabola \[{{y}^{2}}=4ax\] intersect in the line |
| A. | \[y-a=0\] |
| B. | \[y+a=0\] |
| C. | \[x-a=0\] |
| D. | \[x+a=0\] |
| Answer» E. | |
| 8478. |
If the normals at two points P and Q of a parabola \[{{y}^{2}}=4ax\] intersect at a third point R on the curve, then the product of ordinates of P and Q is |
| A. | \[4{{a}^{2}}\] |
| B. | \[2{{a}^{2}}\] |
| C. | \[-4{{a}^{2}}\] |
| D. | \[8{{a}^{2}}\] |
| Answer» E. | |
| 8479. |
If the line \[2x+y+k=0\] is normal to the parabola \[{{y}^{2}}=-8x\], then the value of k will be [RPET 1986, 97] |
| A. | \[-16\] |
| B. | \[-8\] |
| C. | \[-24\] |
| D. | 24 |
| Answer» E. | |
| 8480. |
The maximum number of normal that can be drawn from a point to a parabola is [MP PET 1990] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 8481. |
. The angle of intersection between the curves \[{{x}^{2}}=4(y+1)\] and \[{{x}^{2}}=-4(y+1)\] is [UPSEAT 2002] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | 0 |
| D. | \[\frac{\pi }{2}\] |
| Answer» D. \[\frac{\pi }{2}\] | |
| 8482. |
The equation of the tangent to the parabola \[{{y}^{2}}=16x\], which is perpendicular to the line \[y=3x+7\] is [MP PET 1998] |
| A. | \[y-3x+4=0\] |
| B. | \[3y-x+36=0\] |
| C. | \[3y+x-36=0\] |
| D. | \[3y+x+36=0\] |
| Answer» E. | |
| 8483. |
If a double ordinate of the parabola \[{{y}^{2}}=4ax\] be of length \[8a\], then the angle between the lines joining the vertex of the parabola to the ends of this double ordinate is |
| A. | 30o |
| B. | 60o |
| C. | 90o |
| D. | 120o |
| Answer» D. 120o | |
| 8484. |
If the chord y=mx+1 of the circle\[{{x}^{2}}+{{y}^{2}}=1\]subtends an angle of measure \[45{}^\circ \]at the major segment of the circle, then the value of m is |
| A. | \[2\pm \sqrt{2}\] |
| B. | \[-2\pm \sqrt{2}\] |
| C. | \[-1\pm \sqrt{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8485. |
The locus of the feet of the perpendiculars drawn from either focus on a variable tangent to the hyperbola\[16{{y}^{2}}-9{{x}^{2}}=1\]is |
| A. | \[{{x}^{2}}+{{y}^{2}}=9\] |
| B. | \[{{x}^{2}}+{{y}^{2}}=1/9\] |
| C. | \[{{x}^{2}}+{{y}^{2}}=7/144\] |
| D. | \[{{x}^{2}}+{{y}^{2}}=1/16\] |
| Answer» E. | |
| 8486. |
The ellipse \[{{x}^{2}}+4{{y}^{2}}=4\]is inscribed in a rectangle aligned with the coordinate axes, which is in turn inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is |
| A. | \[{{x}^{2}}+16{{y}^{2}}=16\] |
| B. | \[{{x}^{2}}+12{{y}^{2}}=16\] |
| C. | \[4{{x}^{2}}+48{{y}^{2}}=48\] |
| D. | \[4{{x}^{2}}+64{{y}^{2}}=48\] |
| Answer» C. \[4{{x}^{2}}+48{{y}^{2}}=48\] | |
| 8487. |
The straight line \[x+y=\sqrt{2}p\]will touch the hyperbola \[4{{x}^{2}}-9{{y}^{2}}=36\], if [Orissa JEE 2003] |
| A. | \[{{p}^{2}}=2\] |
| B. | \[{{p}^{2}}=5\] |
| C. | \[5{{p}^{2}}=2\] |
| D. | \[2{{p}^{2}}=5\] |
| Answer» E. | |
| 8488. |
The equation of the tangent to the hyperbola \[2{{x}^{2}}-3{{y}^{2}}=6\]which is parallel to the line \[y=3x+4\], is [MNR 1993] |
| A. | \[y=3x+5\] |
| B. | \[y=3x-5\] |
| C. | \[y=3x+5\]and\[y=3x-5\] |
| D. | None of these |
| Answer» D. None of these | |
| 8489. |
In the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], the equation of diameter conjugate to the diameter \[y=\frac{b}{a}x\], is |
| A. | \[y=-\frac{b}{a}x\] |
| B. | \[y=-\frac{a}{b}x\] |
| C. | \[x=-\frac{b}{a}y\] |
| D. | None of these |
| Answer» B. \[y=-\frac{a}{b}x\] | |
| 8490. |
The equation \[{{x}^{2}}-16xy-11{{y}^{2}}-12x+6y+21=0\] represents |
| A. | Parabola |
| B. | Ellipse |
| C. | Hyperbola |
| D. | Two straight lines |
| Answer» D. Two straight lines | |
| 8491. |
The distance of the point \['\theta '\]on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] from a focus is |
| A. | \[a(e+\cos \theta )\] |
| B. | \[a(e-\cos \theta )\] |
| C. | \[a(1+e\cos \theta )\] |
| D. | \[a(1+2e\cos \theta )\] |
| Answer» D. \[a(1+2e\cos \theta )\] | |
| 8492. |
If the centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then its equation is [AMU 1981] |
| A. | \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{25}=1\] |
| B. | \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] |
| C. | \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{25}=1\] |
| D. | None of these |
| Answer» B. \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] | |
| 8493. |
The square root of 3 - 4i is [RPET 1999] |
| A. | \[\pm (2+i)\] |
| B. | \[\pm (2-i)\] |
| C. | \[\pm (1-2i)\] |
| D. | \[\pm (1+2i)\] |
| Answer» B. \[\pm (2-i)\] | |
| 8494. |
If \[-1+\sqrt{-3}=r{{e}^{i\theta }},\]then \[\theta \] is equal to [RPET 1989; MP PET 1999] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[-\frac{\pi }{3}\] |
| C. | \[\frac{2\pi }{3}\] |
| D. | \[-\frac{2\pi }{3}\] |
| Answer» D. \[-\frac{2\pi }{3}\] | |
| 8495. |
The roots of the equation \[{{x}^{4}}-4{{x}^{3}}+6{{x}^{2}}-4x+1=0\] are [MP PET 1986] |
| A. | 1, 1, 1, 1 |
| B. | 2, 2, 2, 2 |
| C. | 3, 1, 3, 1 |
| D. | 1, 2, 1, 2 |
| Answer» B. 2, 2, 2, 2 | |
| 8496. |
The roots of the equation \[{{x}^{4}}-2{{x}^{3}}+x=380\] are [UPSEAT 2004] |
| A. | \[5,-4,\frac{1\pm 5\sqrt{-3}}{2}\] |
| B. | \[-5,4,-\frac{1\pm 5\sqrt{-}3}{2}\] |
| C. | \[5,4,\frac{-1\pm 5\sqrt{-}3}{2}\] |
| D. | \[-5,-4,\frac{1\pm 5\sqrt{-}3}{2}\] |
| Answer» B. \[-5,4,-\frac{1\pm 5\sqrt{-}3}{2}\] | |
| 8497. |
The roots of \[4{{x}^{2}}+6px+1=0\] are equal, then the value of p is [MP PET 2003] |
| A. | \[\frac{4}{5}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | \[\frac{4}{3}\] |
| Answer» D. \[\frac{4}{3}\] | |
| 8498. |
The solution set of the equation \[pq{{x}^{2}}-{{(p+q)}^{2}}x+{{(p+q)}^{2}}=0\] is [Kerala (Engg.) 2005] |
| A. | \[\left\{ \frac{p}{q},\,\frac{q}{p} \right\}\] |
| B. | \[\left\{ pq,\,\frac{p}{q} \right\}\] |
| C. | \[\left\{ \frac{q}{p},\,pq \right\}\] |
| D. | \[\left\{ \frac{p+q}{p},\,\frac{p+q}{q} \right\}\] |
| Answer» E. | |
| 8499. |
The expression \[y=a{{x}^{2}}+bx+c\] has always the same sign as c if |
| A. | \[4ac<{{b}^{2}}\] |
| B. | \[4ac>{{b}^{2}}\] |
| C. | \[ac<{{b}^{2}}\] |
| D. | \[ac>{{b}^{2}}\] |
| Answer» C. \[ac<{{b}^{2}}\] | |
| 8500. |
Roots of the equations \[2{{x}^{2}}-5x+1=0\], \[{{x}^{2}}+5x+2=0\] are |
| A. | Reciprocal and of same sign |
| B. | Reciprocal and of opposite sign |
| C. | Equal in product |
| D. | None of these |
| Answer» C. Equal in product | |