MCQOPTIONS
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MCQOPTIONS
This section includes 66 Mcqs, each offering curated multiple-choice questions to sharpen your BITSAT knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
All the pairs (x, y) that satisfy the inequality \({2^{\sqrt {{\rm{si}}{{\rm{n}}^2}{\rm{x}} - 2{\rm{sinx}} + 5} }}\cdot\frac{1}{{{4^{{\rm{si}}{{\rm{n}}^2}{\rm{y}}}}}} \le 1\) also satisfy the equation: |
| A. | 2|sin x| = 3sin y |
| B. | 2sin x = sin y |
| C. | sin x = 2sin y |
| D. | sin x = |sin y| |
| Answer» E. | |
| 2. |
If the line \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\) lies on the plane 2x - 4y + z = 7, then what is the value of k? |
| A. | 2 |
| B. | 3 |
| C. | 5 |
| D. | 7 |
| Answer» E. | |
| 3. |
Equation of the plane parallel to the x axis and passes through the point (4, 6, 2) and (4, - 5, 3) |
| A. | y + 11z + 28 = 0 |
| B. | y – 11z + 28 = 0 |
| C. | y + 11z – 28 = 0 |
| D. | y – 11z – 28 = 0 |
| Answer» D. y – 11z – 28 = 0 | |
| 4. |
Find the radius of the sphere x2 + y2 + z2 = 22 ? |
| A. | √12 |
| B. | √17 |
| C. | √19 |
| D. | √22 |
| Answer» E. | |
| 5. |
If the line ax + y = c, touches both the curves x2 + y2 = 1 and \({y^2} = 4\sqrt 2 x,{\rm{\;}}\)then |c| is equal to: |
| A. | 2 |
| B. | \(\frac{1}{{\sqrt 2 }}\) |
| C. | \(\frac{1}{2}\) |
| D. | \(\sqrt 2 \) |
| Answer» E. | |
| 6. |
If the plane 2x - y + 2z + 3 = 0 has the distances \(\frac{1}{3}{\rm{\;and\;}}\frac{2}{3}\) units from the planes 4x - 2y + 4z + λ = 0 and 2x - y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to: |
| A. | 9 |
| B. | 15 |
| C. | 5 |
| D. | 13 |
| Answer» E. | |
| 7. |
If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is: |
| A. | x – 2y + 8 = 0 |
| B. | 2x – 3y + 10 = 0 |
| C. | 2x – y – 2 = 0 |
| D. | 3x – 2y = 0 |
| Answer» D. 3x – 2y = 0 | |
| 8. |
Consider the following statements:1) The angle between the planes 2x – y + z = 1 and x + y + 2z = 3 is \(\frac{\pi }{3}.\) 2) The distance between the planes6x – 3y + 6z + 2 = 0 and2x – y + 2z + 4 = 0 is \(\frac{{10}}{9}\)Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 9. |
In spherical polar coordinates (P, θ, α), θ denotes the polar angle around z-axis and α denotes the azimuthal angle raised from x-axis. Then the y-component of \(\vec P\) is given by _______. |
| A. | P sin θ sin α |
| B. | P sin θ cos α |
| C. | P cos θ sin α |
| D. | P cos θ cos α |
| Answer» B. P sin θ cos α | |
| 10. |
Lines are drawn parallel to the line 4x - 3y + 2 = 0, at a distance \(\frac{3}{5}\) from the origin. Then which one of the following points lies on any of these lines? |
| A. | \(\left( { - \frac{1}{4},\frac{2}{3}} \right)\) |
| B. | \(\left( {\frac{1}{4}, - \frac{1}{3}} \right)\) |
| C. | \(\left( {\frac{1}{4},\frac{1}{3}} \right)\) |
| D. | \(\left( { - \frac{1}{4}, - \frac{2}{3}} \right)\) |
| Answer» B. \(\left( {\frac{1}{4}, - \frac{1}{3}} \right)\) | |
| 11. |
Consider the following for the next two (02) items that follow:Let Q be the image of the point P (-2, 1, - 5) in the plan 3x – 2y + 2z + 1 = 0.Consider the following:1. The coordinates of Q are (4, -3, -1).2. PQ is of length more than 8 units.3. The point (1, -1, -3) is the mid-point of the line segment PQ and lies on the given plane.Which of the above statements are correct? |
| A. | 1 and 2 only |
| B. | 2 and 3 only |
| C. | 1 and 3 only |
| D. | 1, 2 and 3 |
| Answer» E. | |
| 12. |
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line? |
| A. | \(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\) |
| B. | \(\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}\) |
| C. | \(\frac{1}{\sqrt{2}},\frac{1}{2},-\frac{1}{2}\) |
| D. | Cannot be determined due to insufficient data |
| Answer» B. \(\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}\) | |
| 13. |
A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is: |
| A. | 6 |
| B. | \(\sqrt {137}\) |
| C. | \(\sqrt {41}\) |
| D. | 13 |
| Answer» D. 13 | |
| 14. |
If \({\rm{co}}{{\rm{s}}^{ - 1}}x - {\rm{co}}{{\rm{s}}^{ - 1}}\frac{y}{2} = \alpha ,{\rm{\;}}\)where -1 ≤ x ≤ 1, -2 ≤ y ≤ 2, \(x \le \frac{y}{2},{\rm{\;}}\)then for all x, y, 4x2 - 4xy cos α + y2 is equal to: |
| A. | 4 sin2 α |
| B. | 2 sin2 α |
| C. | 4 sin2 α - 2x2y2 |
| D. | 4 cos2 α + 2x2y2 |
| Answer» B. 2 sin2 α | |
| 15. |
If the line \({\rm{\;}}y = mx + 7\sqrt 3\) is normal to the hyperbola \({\rm{\;}}\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,\) then a value of m is: |
| A. | \(\frac{{\sqrt 5 }}{2}\) |
| B. | \(\frac{{\sqrt {15} }}{2}\) |
| C. | \(\frac{2}{{\sqrt 5 }}\) |
| D. | \(\frac{3}{{\sqrt 5 }}\) |
| Answer» D. \(\frac{3}{{\sqrt 5 }}\) | |
| 16. |
If the functionf : R - {1,-1} →Adefined by \(f\left( x \right) = \frac{{{x^2}}}{{1 - {x^2}}},{\rm{\;}}\) is surjective, then A is equal to: |
| A. | R - {-1} |
| B. | [0, ∞) |
| C. | R - [-1, 0) |
| D. | R - (-1, 0) |
| Answer» D. R - (-1, 0) | |
| 17. |
For \(x \in \left( {0,\frac{3}{2}} \right),{\rm{\;let\;}}f\left( x \right) = \sqrt x\), g(x) = tanx and \(h\left( x \right) = \frac{{1 - {x^2}}}{{1 + {x^2}}}\). If ϕ(x) = ((hof)og)(x), then \(\phi \left( {\frac{\pi }{3}} \right)\) is equal to: |
| A. | \(tan\frac{\pi }{{12}}\) |
| B. | \(tan\frac{{11\pi }}{{12}}\) |
| C. | \(tan\frac{{7\pi }}{{12}}\) |
| D. | \(tan\frac{{5\pi }}{{12}}\) |
| Answer» C. \(tan\frac{{7\pi }}{{12}}\) | |
| 18. |
If the plane P touches the sphere x2 + y2 + z2 = r2, then what is r equal to? |
| A. | \(\frac{2}{{\sqrt {29} }}\) |
| B. | \(\frac{4}{{\sqrt {29} }}\) |
| C. | \(\frac{5}{{\sqrt {29} }}\) |
| D. | 1 |
| Answer» D. 1 | |
| 19. |
Find the foot of the perpendicular of the point (2, 2, 2) on the plane x - y - z – 1 = 0. |
| A. | (3, 1, 1) |
| B. | (1, 3, 3) |
| C. | (1, 3, 1) |
| D. | (3, 3, 1) |
| Answer» B. (1, 3, 3) | |
| 20. |
Let A(3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2:1, then cos (∠GOA) (O being the origin) is equal to: |
| A. | \(\frac{1}{{2\sqrt {15} }}\) |
| B. | \(\frac{1}{{\sqrt {15} }}\) |
| C. | \(\frac{1}{{6\sqrt {10} }}\) |
| D. | \(\frac{1}{{\sqrt {30} }}\) |
| Answer» C. \(\frac{1}{{6\sqrt {10} }}\) | |
| 21. |
A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the centroid of this triangle is: |
| A. | \(\left( {1,\;\frac{7}{3}} \right)\) |
| B. | \(\left( {\frac{1}{3},{\rm{\;}}2} \right)\) |
| C. | \(\left( {\frac{1}{3},{\rm{\;}}1} \right)\) |
| D. | \(\left( {\frac{1}{3},{\rm{\;}}\frac{5}{3}} \right)\) |
| Answer» C. \(\left( {\frac{1}{3},{\rm{\;}}1} \right)\) | |
| 22. |
Find the equation of the tangent plane to the sphere x2 + y2 + z2 - 4x + 2y - 6z + 5 = 0 which is parallel to the plane 3x + 2y - 2z = 0. |
| A. | 3x + 2y - 2z - 11 = 0 |
| B. | 3x + 2y - 2z + 7 = 0 |
| C. | 3x + 2y - 2z - 7 = 0 |
| D. | 3x - 2y - 2z - 7 = 0 |
| Answer» D. 3x - 2y - 2z - 7 = 0 | |
| 23. |
Every homogeneous equation f(x, y, z) = 0 represents a: |
| A. | Sphere. |
| B. | Cone with vertex at origin. |
| C. | Cylinder. |
| D. | None of these. |
| Answer» B. Cone with vertex at origin. | |
| 24. |
For what value of α + β, the three points (α, β), (5, 0), (0, 5) will be collinear? |
| A. | 5 |
| B. | -25 |
| C. | 0 |
| D. | 25 |
| Answer» B. -25 | |
| 25. |
In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 – c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is: |
| A. | \(\frac{3}{2}y\) |
| B. | \(\frac{c}{{\sqrt 3 }}\) |
| C. | c/3 |
| D. | \(\frac{y}{{\sqrt 3 }}\) |
| Answer» C. c/3 | |
| 26. |
If the two lines x + (a – 1)y = 1 and 2x + a2 y = 1 (a ϵ R – {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is: |
| A. | \(\sqrt {\frac{2}{5}} \) |
| B. | \(\frac{2}{5}\) |
| C. | \(\frac{2}{{\sqrt 5 }}\) |
| D. | \(\frac{{\sqrt 2 }}{5}\) |
| Answer» B. \(\frac{2}{5}\) | |
| 27. |
A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point: |
| A. | (1, -4, 1) |
| B. | (1, 4, -1) |
| C. | (2, 4, 1) |
| D. | (2, -4, 1) |
| Answer» E. | |
| 28. |
A line makes equal angle with coordinate axis. Direction cosines of this line are |
| A. | ± (1, 1, 1) |
| B. | \(\pm \left(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}\right)\) |
| C. | \(\pm \left(\dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}\right)\) |
| D. | \(\pm \left(\dfrac{1}{3}, \dfrac{1}{3}, \dfrac{1}{3}\right)\) |
| Answer» C. \(\pm \left(\dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}\right)\) | |
| 29. |
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is: |
| A. | 25 |
| B. | 22 |
| C. | 24 |
| D. | 20 |
| Answer» B. 22 | |
| 30. |
Let S and S’ be the foci of an ellipse and B be any one of the extremities of its minor axis. If ΔS' BS is a right angled triangle with right angle at B and area (ΔS'BS) = 8 sq units, then the length of a latus rectum of the ellipse is |
| A. | \(2\sqrt 2\) |
| B. | \({\rm{\;}}4\sqrt 2\) |
| C. | 2 |
| D. | 4 |
| Answer» E. | |
| 31. |
If the tangent to the parabola y2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse x2 + 2y2 = 1, then α is equal to: |
| A. | \(\sqrt 2 - 1\) |
| B. | \(2\sqrt 2 - 1\) |
| C. | \(2\sqrt 2 + 1\) |
| D. | \(\sqrt 2 + 1\) |
| Answer» E. | |
| 32. |
If lines \(\frac {x-1}{-3} = \frac {y - 2}{2k} = \frac {z - 3}{2}\) and \(\frac {x-1}{3k} = \frac {y-5}{1}=\frac {z-6}{-5}\) are mutually perpendicular, then k is equal to- |
| A. | \(\frac {-10}{7}\) |
| B. | \(-\frac {7}{10}\) |
| C. | -10 |
| D. | -7 |
| Answer» B. \(-\frac {7}{10}\) | |
| 33. |
If \({\rm{a}} > 0{\rm{\;and\;z}} = \frac{{{{(1 + {\rm{i}})}^2}}}{{{\rm{a}} - {\rm{i}}}}\), has magnitude \(\sqrt {\frac{2}{5}} ,{\rm{\;then\;\bar z}}\) is equal to: |
| A. | \(- \frac{1}{5} - \frac{3}{5}{\rm{i}}\) |
| B. | \(- \frac{3}{5} - \frac{1}{5}{\rm{i}}\) |
| C. | \(\frac{1}{5} - \frac{3}{5}{\rm{i}}\) |
| D. | \(- \frac{1}{5} + \frac{3}{5}{\rm{i}}\) |
| Answer» B. \(- \frac{3}{5} - \frac{1}{5}{\rm{i}}\) | |
| 34. |
If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4), and (2, 5), then the equation of the diagonal AD is: |
| A. | 5x - 3y + 1 = 0 |
| B. | 5x + 3y - 11 = 0 |
| C. | 3x - 5y + 7 = 0 |
| D. | 3x + 5y - 13 = 0 |
| Answer» B. 5x + 3y - 11 = 0 | |
| 35. |
A perpendicular is drawn from a point on the line \(\frac{{x - 1}}{2} = \frac{{y + 1}}{{ - 1}} = \frac{z}{1}\) to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x - y + z = 3. Then the co-ordinates of Q are: |
| A. | (1, 0, 2) |
| B. | (2, 0, 1) |
| C. | (-1, 0, 4) |
| D. | (4, 0, -1) |
| Answer» C. (-1, 0, 4) | |
| 36. |
If the angle θ between the line \(\rm \frac {x+1}{1} = \frac {y-1}{2} = \frac {z-2}{2}\) and the plane 2x - y + √λ z + 4 = 0 is such that sin \(\theta = \frac 1 3\). The value of λ is- |
| A. | \(- \frac 4 3\) |
| B. | \(\frac 3 4\) |
| C. | \(- \frac 3 5\) |
| D. | \(\frac 5 3\) |
| Answer» E. | |
| 37. |
Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC? |
| A. | (3, 4, -2) |
| B. | (4, -2, 5) |
| C. | (4, 5, -2) |
| D. | (2, 4, 5) |
| Answer» D. (2, 4, 5) | |
| 38. |
A sphere of constant radius r through the origin intersects the coordinate axes in A, B, and C. What is the locus of the centroid of the triangle ABC? |
| A. | x2 + y2 + z2 = r2 |
| B. | x2 + y2 + z2 = 4r2 |
| C. | 9(x2 + y2 +z2) = 4r2 |
| D. | 3(x2 + y2 + z2) = 2r2 |
| Answer» D. 3(x2 + y2 + z2) = 2r2 | |
| 39. |
A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (-8, 5) and (6, 5), then the area of the rectangle (in sq. units) is: |
| A. | 84 |
| B. | 98 |
| C. | 72 |
| D. | 56 |
| Answer» B. 98 | |
| 40. |
A straight line with direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\) is |
| A. | Parallel to x-axis |
| B. | parallel to y-axis |
| C. | parallel to z-axis |
| D. | Equally inclined to all the axes |
| Answer» C. parallel to z-axis | |
| 41. |
Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is: |
| A. | 9 |
| B. | 18 |
| C. | 36 |
| D. | 32 |
| Answer» D. 32 | |
| 42. |
A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B and C respectively. The locus of the center of the sphere OABC, O being the origin, is |
| A. | \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} + \frac{{\rm{z}}}{{\rm{c}}} = 1\) |
| B. | \(\frac{{\rm{a}}}{{\rm{x}}} + \frac{{\rm{b}}}{{\rm{y}}} + \frac{{\rm{c}}}{{\rm{z}}} = 1\) |
| C. | \(\frac{{\rm{a}}}{{\rm{x}}} + \frac{{\rm{b}}}{{\rm{y}}} + \frac{{\rm{c}}}{{\rm{z}}} = 2\) |
| D. | \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} + \frac{{\rm{z}}}{{\rm{c}}} = 2\) |
| Answer» D. \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} + \frac{{\rm{z}}}{{\rm{c}}} = 2\) | |
| 43. |
If the line, \(\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 2}}{4}{\rm{\;}}\) meets the plane, \(x + 2y + 3z = 15{\rm{\;}}\) at a point P, then the distance of P from the origin is: |
| A. | \(\frac{{\sqrt 5 }}{2}\) |
| B. | \(2\sqrt 5\) |
| C. | \(\frac{9}{2}\) |
| D. | \(\frac{7}{2}\) |
| Answer» D. \(\frac{7}{2}\) | |
| 44. |
If the tangent to the curve, y = x3 + ax – b at the point (1, -5) is perpendicular to the line, - x + y + 4 = 0, then which one of the following points lies on the curve? |
| A. | (-2, 1) |
| B. | (-2, 2) |
| C. | (2, -1) |
| D. | (2, -2) |
| Answer» E. | |
| 45. |
Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the midpoint D of AC subtends an angle 30° at B. The height (in m) of the lamp-post is: |
| A. | \(\frac{3}{2}\sqrt{21}\) |
| B. | \(\frac{2}{3}\sqrt{21}\) |
| C. | \({2}\sqrt{21}\) |
| D. | \({7}\sqrt{3}\) |
| Answer» C. \({2}\sqrt{21}\) | |
| 46. |
If Q(0, -1, -3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, -1, -2), then the area (in sq. units) of ΔPQR is: |
| A. | 2√13 |
| B. | \(\frac{{\sqrt {91} }}{4}\) |
| C. | \(\frac{{\sqrt {91} }}{2}\) |
| D. | \(\frac{{\sqrt {65} }}{2}\) |
| Answer» D. \(\frac{{\sqrt {65} }}{2}\) | |
| 47. |
If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is: |
| A. | (5, 0) |
| B. | \(\left( { - \frac{5}{3},{\rm{\;}}0} \right)\) |
| C. | \(\left( {\frac{5}{3},0} \right)\) |
| D. | (-5, 0) |
| Answer» E. | |
| 48. |
If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line\(\frac{\text{x}}{1}=\frac{\text{y}-1}{0}=\frac{\text{z}+1}{-1}\text{ }\!\!~\!\!\text{ is }\!\!~\!\!\text{ }\sqrt{\frac{3}{2}}\) , then β is equal to: |
| A. | 1 |
| B. | 2 |
| C. | -1 |
| D. | -2 |
| Answer» D. -2 | |
| 49. |
A straight line ‘L’ at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an equation of the line ‘L’ is: |
| A. | \(x + \sqrt 3 y = 8\) |
| B. | \(\left( {\sqrt 3 + 1} \right)x + \left( {\sqrt 3 - 1} \right)y = 8\sqrt 2\) |
| C. | \(\sqrt 3 x + y = 8\) |
| D. | \(\left( {\sqrt 3 - 1} \right)x + \left( {\sqrt 3 + 1} \right)y = 8\sqrt 2\) |
| Answer» C. \(\sqrt 3 x + y = 8\) | |
| 50. |
P is a point on the line segment joining the points (3, 2, -1) and (6, -4, -2). If x coordinate of P is 5, then its y coordinate is: |
| A. | 2 |
| B. | 1 |
| C. | -1 |
| D. | -2 |
| Answer» E. | |