MCQOPTIONS
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MCQOPTIONS
This section includes 27 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Let P (4, -4) and Q (9, 6) be two points on the parabola, y2 = 4x and let X be any point on the POQ of this parabola, where O is the vertex of this parabola, such that the area of ΔPXQ is maximum. Then, this maximum area (in sq units) is |
| A. | 125/2 |
| B. | 75/2 |
| C. | 625/4 |
| D. | 125/4 |
| Answer» E. | |
| 2. |
Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it? |
| A. | \(\left( {4\sqrt 2 ,2\sqrt 2 } \right)\) |
| B. | \(\left( {4\sqrt 3 ,2\sqrt 2 } \right)\) |
| C. | \(\left( {4\sqrt 3 ,2\sqrt 3 } \right)\) |
| D. | \(\left( {4\sqrt 2 ,2\sqrt 3 } \right)\) |
| Answer» C. \(\left( {4\sqrt 3 ,2\sqrt 3 } \right)\) | |
| 3. |
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x - a2) = 0 and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of ‘a’ is: |
| A. | \(5\sqrt 5 \) |
| B. | 5(21⁄3) |
| C. | 102⁄3 |
| D. | 5 |
| Answer» E. | |
| 4. |
If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve: |
| A. | \(\frac{1}{{4{x^2}}} + \frac{1}{{2{y^2}}} = 1\) |
| B. | \(\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1\) |
| C. | \(\frac{1}{{2{x^2}}} + \frac{1}{{4{y^2}}} = 1\) |
| D. | \(\frac{{{x^2}}}{2} + \frac{{{y^2}}}{4} = 1\) |
| Answer» D. \(\frac{{{x^2}}}{2} + \frac{{{y^2}}}{4} = 1\) | |
| 5. |
If the foci of the ellipse \(b^2x^2 +16y^2 = 16b^2\) and the hyperbola \(81x^2 -144y^2 = \dfrac{81 \times 144}{25}\) coincide, then the value of b, is |
| A. | 1 |
| B. | \(\sqrt{5}\) |
| C. | \(\sqrt{7}\) |
| D. | 3 |
| Answer» D. 3 | |
| 6. |
A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Find the eccentricity of the hyperbola. |
| A. | 3/2 |
| B. | √3 |
| C. | 2 |
| D. | \(\frac{2}{{\sqrt 3 }}\) |
| Answer» E. | |
| 7. |
Let \(0 < {\rm{\theta }} < \frac{{\rm{\pi }}}{2}\). If the eccentricity of the hyperbola \(\frac{{{{\rm{x}}^2}}}{{{\rm{co}}{{\rm{s}}^2}{\rm{\theta }}}} - \frac{{{{\rm{y}}^2}}}{{{\rm{si}}{{\rm{n}}^2}{\rm{\theta }}}} = 1\) is greater than 2, then the length of its latus rectum lies in the interval: |
| A. | (3, ∞) |
| B. | \(\left( {\frac{3}{2},{\rm{\;}}2} \right]\) |
| C. | (2, 3] |
| D. | \(\left( {1,{\rm{\;}}\frac{3}{2}} \right]\) |
| Answer» B. \(\left( {\frac{3}{2},{\rm{\;}}2} \right]\) | |
| 8. |
If the area (in sq. units) of the region {(x, y) : y2 ≤ 4x, x + y ≤ 1, x ≥ 0, y ≥ 0} is \({\rm{a}}\sqrt 2 + {\rm{b}}\), then a – b is equal to: |
| A. | 10/3 |
| B. | 6 |
| C. | 8/3 |
| D. | \(- \frac{2}{3}\) |
| Answer» C. 8/3 | |
| 9. |
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at \(\left( {0,\;5\sqrt 3 } \right),\) then the length of its latus rectum is: |
| A. | 10 |
| B. | 5 |
| C. | 8 |
| D. | 6 |
| Answer» C. 8 | |
| 10. |
If the vertices of a hyperbola be at (-2, 0) and (2, 0) and one of its foci be at (-3, 0), then which one of the following points does not lie on this hyperbola? |
| A. | \(\left( 2\sqrt{6},5 \right)\) |
| B. | \(\left( 6,~5\sqrt{2} \right)\) |
| C. | \(\left( 4,~\sqrt{15} \right)\) |
| D. | \(\left( -6,~2\sqrt{10} \right)\) |
| Answer» C. \(\left( 4,~\sqrt{15} \right)\) | |
| 11. |
If a directrix of a hyperbola centered at the origin and passing through the point \(\left( {4, - 2\sqrt 3 } \right){\rm{\;is\;}}5x = 4\sqrt 5\) and its eccentricity is e, then: |
| A. | 4e4 – 24e2 + 27 = 0 |
| B. | 4e4 – 12e2 – 27 = 0 |
| C. | 4e4 – 24e2 + 35 = 0 |
| D. | 4e4 + 8e2 – 35 = 0 |
| Answer» D. 4e4 + 8e2 – 35 = 0 | |
| 12. |
Let \({\rm{S}} = \left\{ {\left( {x,y} \right) \in {{\rm{R}}^2}:\frac{{{y^2}}}{{1 + {\rm{r}}}} - \frac{{{x^2}}}{{1 - {\rm{r}}}} = 1} \right\}\), where r ≠ ±1. Then s represents: |
| A. | A hyperbola whose eccentricity is \(\frac{2}{{\sqrt {1 - r} }}\), when 0 < r < 1. |
| B. | An ellipse whose eccentricity is \(\sqrt {\frac{2}{{r + 1}}} \), when r > 1. |
| C. | A hyperbola whose eccentricity is \(\frac{2}{{\sqrt {1 + r} }}\), when 0 < r < 1. |
| D. | An ellipse whose eccentricity is \(\frac{1}{{\sqrt {r + 1} }}\), when r > 1. |
| Answer» C. A hyperbola whose eccentricity is \(\frac{2}{{\sqrt {1 + r} }}\), when 0 < r < 1. | |
| 13. |
If the ellipse 9x2 + 16y2 = 144 intercepts the line 3x + 4y = 12, then what is the length of the chord so formed? |
| A. | 5 units |
| B. | 6 units |
| C. | 8 unit s |
| D. | 10 units |
| Answer» B. 6 units | |
| 14. |
An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points? |
| A. | \(\left( {\sqrt 2 ,\;2} \right)\) |
| B. | \(\left( {2,{\rm{\;}}\sqrt 2 } \right)\) |
| C. | \(\left( {2,{\rm{\;}}2\sqrt 2 } \right)\) |
| D. | \(\left( {1,{\rm{\;}}2\sqrt 2 } \right)\) |
| Answer» B. \(\left( {2,{\rm{\;}}\sqrt 2 } \right)\) | |
| 15. |
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola y2 = 4x, is: |
| A. | \(2\sqrt 3 {\rm{y}} = 12{\rm{x}} + 1\) |
| B. | \(\sqrt 3 {\rm{y}} = {\rm{x}} + 3\) |
| C. | \(2\sqrt 3 {\rm{y}} = - {\rm{x}} - 12\) |
| D. | \(\sqrt 3 {\rm{y}} = 3{\rm{x}} + 1\) |
| Answer» C. \(2\sqrt 3 {\rm{y}} = - {\rm{x}} - 12\) | |
| 16. |
If the line x – 2y = 12 is tangent to the ellipse \(\frac{{{\text{x}}^{2}}}{{{\text{a}}^{2}}}+\frac{{{\text{y}}^{2}}}{{{\text{b}}^{2}}}=1\text{ }\!\!~\!\!\text{ at }\!\!~\!\!\text{ the }\!\!~\!\!\text{ point }\!\!~\!\!\text{ }\left( 3,\frac{-9}{2} \right)\), then the length of the latus rectum of the ellipse is: |
| A. | 9 |
| B. | 12√2 |
| C. | 5 |
| D. | 8√3 |
| Answer» B. 12√2 | |
| 17. |
Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it? |
| A. | \(\left( {5,{\rm{\;}}2\sqrt 6 } \right)\) |
| B. | (8, 6) |
| C. | \(\left( {6,{\rm{\;}}4\sqrt 2 } \right)\) |
| D. | (4, -4) |
| Answer» C. \(\left( {6,{\rm{\;}}4\sqrt 2 } \right)\) | |
| 18. |
If a point moves in a plane in such a way that the sum of its distances from two fixed points is constant the curve so traced is called |
| A. | Ellipse |
| B. | Parabola |
| C. | Hyperbola |
| D. | None of these |
| Answer» B. Parabola | |
| 19. |
If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is: |
| A. | \(\frac{{13}}{{12}}\) |
| B. | 2 |
| C. | \(\frac{{13}}{6}\) |
| D. | \(\frac{{13}}{8}\) |
| Answer» B. 2 | |
| 20. |
Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is: |
| A. | x + y + 1 = 0 |
| B. | x – 2y + 4 = 0 |
| C. | x + 2y + 4 = 0 |
| D. | 4x + 2y + 1 = 0 |
| Answer» D. 4x + 2y + 1 = 0 | |
| 21. |
If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is: |
| A. | \(\frac{{\sqrt 3 }}{2}\) |
| B. | \(\frac{1}{{\sqrt 3 }}\) |
| C. | √3 |
| D. | \(\frac{2}{{\sqrt 3 }}\) |
| Answer» C. √3 | |
| 22. |
If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)? |
| A. | \(\left( {\frac{1}{2},2,{\rm{\;}}3} \right)\) |
| B. | (1, 1, 3) |
| C. | \(\left( {\frac{1}{2},2,\;0} \right)\) |
| D. | (1, 1, 0) |
| Answer» C. \(\left( {\frac{1}{2},2,\;0} \right)\) | |
| 23. |
A helicopter is flying along the curve given by y - x3⁄2 = 7, (x ≥ 0). A solider positioned at the point \(\left( {\frac{1}{2},7} \right)\) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is: |
| A. | \(\frac{{\sqrt 5 }}{6}\) |
| B. | \(\frac{1}{3}\sqrt {\frac{7}{3}} \) |
| C. | \(\frac{1}{6}\sqrt {\frac{7}{3}} \) |
| D. | \(\frac{1}{2}\) |
| Answer» D. \(\frac{1}{2}\) | |
| 24. |
If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other, a2 is equal to: |
| A. | \(\frac{{128}}{{17}}\) |
| B. | \(\frac{{64}}{{17}}\) |
| C. | \(\frac{4}{{17}}\) |
| D. | \(\frac{2}{{17}}\) |
| Answer» E. | |
| 25. |
Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of ΔAOP is 4, is: |
| A. | 8x2 – 9y2 + 9y = 18 |
| B. | 9x2 – 8y2 + 8y = 16 |
| C. | 9x2 + 8y2 – 8y = 16 |
| D. | 8x2 + 9y2 – 9y = 18 |
| Answer» D. 8x2 + 9y2 – 9y = 18 | |
| 26. |
If the normal of the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to: |
| A. | \(\frac{5\sqrt{5}}{2}\) |
| B. | \(\frac{\sqrt{61}}{2}\) |
| C. | \(\frac{\sqrt{221}}{2}\) |
| D. | \(\frac{\sqrt{157}}{2}\) |
| Answer» B. \(\frac{\sqrt{61}}{2}\) | |
| 27. |
Let A(4, -4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ∆ACB is maximum. Then, the area (in sq. units) of ∆ACB, is: |
| A. | \(31\frac{1}{4}\) |
| B. | \(30\frac{1}{2}\) |
| C. | 32 |
| D. | \(31\frac{3}{4}\) |
| Answer» B. \(30\frac{1}{2}\) | |