MCQOPTIONS
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MCQOPTIONS
This section includes 30 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
For the curve \(\rm \dfrac{a^2}{x^2}-\dfrac{b^2}{y^2}=1\), the asymptotes parallel to y-axis are - |
| A. | x = ± b |
| B. | y = ± a |
| C. | x = ± a |
| D. | x = a + b |
| Answer» C. x = ± a | |
| 2. |
If y + b = m1(x + a) and y + b = m2(x + a) are the equations of the two tangents to the parabola y2 = 4ax, then |
| A. | m1m2 = 1 |
| B. | m1m2 = -1 |
| C. | m1 + m2 = 0 |
| D. | m1 + m2 = -1 |
| Answer» C. m1 + m2 = 0 | |
| 3. |
Find the number of point(s) of intersection of the ellipse \(\rm \dfrac{x^2}{4} + \dfrac{(y-1)^2}{9}=1\) and the circle x2 + y2 = 4. |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 1 |
| Answer» C. 2 | |
| 4. |
If the parabola y2 = 4ax passes through the point (2, 4), then the focus of parabola is ? |
| A. | (1,1) |
| B. | (0,1) |
| C. | (2,0) |
| D. | (0,0) |
| Answer» D. (0,0) | |
| 5. |
Find the value of k so that the line 2x + y + k = 0 may touch the hyperbola 3x2 -y2 = 3 |
| A. | k = 1, -1 |
| B. | k = 1 |
| C. | k = 0 |
| D. | k = -2 |
| Answer» B. k = 1 | |
| 6. |
If S and S' are foci of the ellipse \(\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1,\) B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is |
| A. | \(\frac 1 2\) |
| B. | \(\frac 1 3\) |
| C. | \(\frac 1 4\) |
| D. | \(\frac 1 5\) |
| Answer» B. \(\frac 1 3\) | |
| 7. |
If the graph of y = (x - 2)2 - 3 is shifted by 5 units up along y-axis and 2 unit to the right along the x-axis, then the equation of the resultant graph is ? |
| A. | y = x2 + 2 |
| B. | y = (x + 2)2 + 5 |
| C. | y = (x + 2)2 + 2 |
| D. | y = (x - 4)2 + 2 |
| Answer» E. | |
| 8. |
If the straight line x cosα + y sinα = p is tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). then |
| A. | \(p^2 = \frac{a^2b^2}{4}\) |
| B. | p2 = a2 cos2 α - b2 sin2 α |
| C. | p2 = a2 cos2 α + b2 sin2 |
| D. | None of these |
| Answer» D. None of these | |
| 9. |
If the line x - 1 = 0 is the directrix of the parabola y2 - kx + 8 = 0, then one of the values of k is |
| A. | \(\frac{1}{6}\) |
| B. | 8 |
| C. | 4 |
| D. | \(\frac{1}{4}\) |
| Answer» D. \(\frac{1}{4}\) | |
| 10. |
In the parabola, y2 = x, what is the length of the chord passing through the vertex and inclined to the x-axis at an angle θ? |
| A. | sin θ ⋅ sec2 θ |
| B. | cos θ . cosec2 θ |
| C. | cot θ ⋅ sec2 θ |
| D. | 2 tan θ ⋅ cosec2 θ |
| Answer» C. cot θ ⋅ sec2 θ | |
| 11. |
If the focus of a parabola is (-8, -2) and the directrix is y = 2x - 9, then the equation of the parabola is: |
| A. | x2 + 4y2 + 4xy + 111x + 2y + 260 = 0 |
| B. | x2 + 4y2 + 4xy + 116x + 2y + 260 = 0 |
| C. | x2 + 4y2 + 4xy + 116x + 2y + 259 = 0 |
| D. | 4x2 + y2 + 4xy + 116x + 2y + 259 = 0 |
| Answer» D. 4x2 + y2 + 4xy + 116x + 2y + 259 = 0 | |
| 12. |
An equilateral triangle is inscribed in the parabola y2 = 4ax, such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is: |
| A. | \(\rm a\sqrt{3}\) |
| B. | \(\rm 2a\sqrt{3}\) |
| C. | \(\rm 4a\sqrt{3}\) |
| D. | \(\rm 8a\sqrt{3}\) |
| Answer» E. | |
| 13. |
Intersecting of a solid cone with a plane parallel to axis of the cone will generate ___. |
| A. | Ellipse |
| B. | Triangle |
| C. | Parabola |
| D. | Hyperbola |
| Answer» E. | |
| 14. |
If the angle between the lines joining the end points of minor axis of the ellipse with one of its foci is \(\frac{\pi }{2},\) then what is the eccentricity of the ellipse? |
| A. | 1/2 |
| B. | \(\frac{1}{\sqrt{2}}\) |
| C. | \(\frac{\sqrt{3}}{2}\) |
| D. | \(\frac{1}{2\sqrt{2}}\) |
| Answer» C. \(\frac{\sqrt{3}}{2}\) | |
| 15. |
Find the equation of tangents to the hyperbola 3x2 - y2 = 3 which are perpendicular to x + 3y = 2 ? |
| A. | y = 3x ± √6 |
| B. | y = 3x ± 6 |
| C. | y = x ± √6 |
| D. | None of these |
| Answer» B. y = 3x ± 6 | |
| 16. |
Equation of the tangent from the point (3, -1) to the ellipse 2x2 + 9y2 = 3 is |
| A. | 2x - 3y - 3 = 0 |
| B. | 2x + 3y - 3 = 0 |
| C. | 2x + y - 3 = 0 |
| D. | None of these |
| Answer» E. | |
| 17. |
Consider any point P on the ellipse \(\frac{{{{\rm{x}}^2}}}{{25}} + \frac{{{{\rm{y}}^2}}}{9} = 1\) in the first quadrant. Let r and s represent its distance from (4, 0) and (-4, 0) respectively, then (r + s) is equal to |
| A. | 10 unit |
| B. | 9 unit |
| C. | 8 unit |
| D. | 6 unit |
| Answer» B. 9 unit | |
| 18. |
If any point on a hyperbola is (3tan θ, 2sec θ), then what is the eccentricity of the hyperbola? |
| A. | \(\dfrac{3}{2}\) |
| B. | \(\dfrac{5}{2}\) |
| C. | \(\dfrac{\sqrt{11}}{2}\) |
| D. | \(\dfrac{\sqrt{13}}{2}\) |
| Answer» E. | |
| 19. |
Equation of hyperbola whose asymptotes are 3x ± 5y = 0 and vertices are (± 5, 0) is |
| A. | 9x2 - 25y2 = 225 |
| B. | 25x2 - 9y2 = 225 |
| C. | 5x2 - 3y2 = 225 |
| D. | 3x2 - 5y2 = 225 |
| Answer» B. 25x2 - 9y2 = 225 | |
| 20. |
Find the equation of the tangent to the ellipse x2 + 2y2 = 4 at the point where ordinate is 1 such that point lies in the first quadrant? |
| A. | √2x - 4y - 4 = 0 |
| B. | √2x + 4y - 4 = 0 |
| C. | √2x + 4y + 4 = 0 |
| D. | None of these |
| Answer» E. | |
| 21. |
Consider the following with regard to eccentricity (e) of a conic section:1. e = 0 for circle2. e = 1 for parabola3. e < 1 for ellipseWhich of the above statements is/are correct? |
| A. | 1 and 2 only |
| B. | 2 and 3 only |
| C. | 1 and 3 only |
| D. | 1, 2 and 3 |
| Answer» E. | |
| 22. |
If P is a point on the ellipse \(\rm \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\), where foci are s and s', then the value of (SP + S'P) is equal to |
| A. | 2a |
| B. | 2b |
| C. | (a+b)/2 |
| D. | (a-b)/2 |
| Answer» B. 2b | |
| 23. |
Let P(x, y) be any point on the ellipse 25x2 + 16y2 = 400. If Q(0, 3) and R(0, -3) are two points, then what is (PQ + PR) equal to? |
| A. | 12 |
| B. | 10 |
| C. | 8 |
| D. | 6 |
| Answer» C. 8 | |
| 24. |
A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is |
| A. | a parabola |
| B. | a hyperbola |
| C. | a circle |
| D. | an ellipse |
| Answer» B. a hyperbola | |
| 25. |
If 3x + 4y + k = 0 is a tangent to the hyperbola 9x2 - 16y2 = 144, then the value of k is |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | -3 |
| Answer» B. 1 | |
| 26. |
Area of the greatest rectangle that can be inscribed in the ellipse is |
| A. | \(\sqrt {ab}\) |
| B. | 2ab |
| C. | ab |
| D. | a / b |
| Answer» C. ab | |
| 27. |
Find the equation of the parabola with focus at F(3, 0) and directrix x = - 3 ? |
| A. | y2 = 6x |
| B. | y2 = 12x |
| C. | y2 = 8x |
| D. | None of these |
| Answer» C. y2 = 8x | |
| 28. |
If PQ is a double ordinate of the hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}=1\) such that OPQ is an equilateral triangle, where O is the centre of the hyperbola, then which of the following is true? |
| A. | \(b^2 > \dfrac{-a^2}{\sqrt{3}}\) |
| B. | \(b^2 > \dfrac{a^2}{3}\) |
| C. | \(b^2 < \dfrac{a^2}{3}\) |
| D. | \(b^2 < \dfrac{-a^2}{3}\) |
| Answer» C. \(b^2 < \dfrac{a^2}{3}\) | |
| 29. |
A man running round a racecourse notes that the sum of the distance of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path he encloses is |
| A. | 18π square metres |
| B. | 15π square metres |
| C. | 12π square metres |
| D. | 8π square metres |
| Answer» C. 12π square metres | |
| 30. |
If x = 1 is the directrix of the parabola y2 = kx - 8, then k is: |
| A. | \(\frac{1}{8}\) |
| B. | 8 |
| C. | 4 |
| D. | \(\frac{1}{4}\) |
| Answer» D. \(\frac{1}{4}\) | |