Class 12 Math Ch-11 3D Geometry MCQs Exam 2027

Topic: Vector Algebra

1. $\vec{j} \cdot (\vec{k} \times \vec{i}) =$ [BSEB, 2026 A]

(A) $1$

(B) $0$

(C) $-1$

(D) $\vec{j}$

2. $\vec{a} \cdot \vec{b} =$ [BSEB, 2026 A]

(A) $\vec{b} \cdot \vec{a}$

(B) $-\vec{b} \cdot \vec{a}$

(C) $1$

(D) $0$

3. $(\vec{i} – 2\vec{j} + 5\vec{k}) \cdot (-2\vec{i} + 4\vec{j} + 2\vec{k}) =$ [BSEB, 2026 A]

(A) $0$

(B) $20$

(C) $-20$

(D) $10$

4. $\vec{i} \cdot \vec{i} =$ [BSEB, 2026 A]

(A) $1$

(B) $0$

(C) $\vec{j}$

(D) $\vec{k}$

5. $\vec{k} \times \vec{k} =$ [BSEB, 2026 A]

(A) $\vec{0}$

(B) $1$

(C) $\vec{i}$

(D) $\vec{j}$

6. $\vec{i} \times \vec{j} =$ [BSEB, 2026 A]

(A) $\vec{k}$

(B) $-\vec{k}$

(C) $0$

(D) $1$

7. The position vector of point $(1, 0, 2)$ is: [BSEB, 2026 A]

(A) $\vec{i} + 2\vec{k}$

(B) $\vec{i} + 2\vec{j}$

(C) $\vec{j} + 2\vec{k}$

(D) $\vec{i} + \vec{j} + 2\vec{k}$

8. If $\vec{a} = 2\vec{i} + \vec{j} + 2\vec{k}$, then $|\vec{a}| =$ [BSEB, 2026 A]

(A) $3$

(B) $9$

(C) $\sqrt{5}$

(D) $5$

9. Which of the following is a unit vector? [BSEB, 2026 A]

(A) $\frac{\vec{i}+\vec{j}+\vec{k}}{\sqrt{3}}$

(B) $\vec{i}+\vec{j}+\vec{k}$

(C) $\frac{\vec{i}+\vec{j}}{2}$

(D) $\vec{k}$

10. The position vector of point $(1, 0, 2)$ is: [BSEB, 2026 A]

(A) $\hat{i} + \hat{j} + 2\hat{k}$

(B) $\hat{i} + 2\hat{k}$

(C) $2\hat{i} + \hat{k}$

(D) $\hat{i} + 2\hat{j}$

Topic: Direction Cosines and Ratios

11. The direction cosines of $z$-axis are: [BSEB, 2026 A]

(A) $(0, 0, 0)$

(B) $(1, 0, 0)$

(C) $(0, 0, 1)$

(D) $(0, 1, 0)$

12. The direction cosines of $y$-axis are: [BSEB, 2026 A]

(A) $(0, 0, 0)$

(B) $(1, 0, 0)$

(C) $(0, 1, 0)$

(D) $(0, 0, 1)$

13. The direction cosines of $x$-axis are: [BSEB, 2026 A]

(A) $(0, 0, 0)$

(B) $(1, 0, 0)$

(C) $(0, 1, 0)$

(D) $(0, 0, 1)$

14. If $l, m, n$ are direction cosines of a line, then the value of $l^2 + m^2 + n^2$ is: [BSEB, 2026 A]

(A) 0

(B) 1

(C) -1

(D) 2

15. If $l, m, n$ are direction cosines of a straight line, then: [BSEB, 2026 A]

(A) $l^2 + m^2 – n^2 = 1$

(B) $l^2 – m^2 + n^2 = 1$

(C) $l^2 – m^2 – n^2 = -1$

(D) $l^2 + m^2 + n^2 = 1$

16. Direction cosines of the line joining the points $(1, 2, 3)$ and $(4, 5, 6)$ are: [BSEB, 2026 A]

(A) $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

(B) $3, 3, 3$

(C) $\frac{1}{3}, \frac{1}{3}, \frac{1}{3}$

(D) None of these

17. If the direction ratios of a straight line are $1, 3, 5$, then the direction cosines of the line are: [BSEB, 2026 A]

(A) $\frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}}$

(B) $\frac{1}{9}, \frac{1}{3}, \frac{5}{9}$

(C) $\frac{5}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{1}{\sqrt{35}}$

(D) None

18. The direction cosines of a line with direction ratios $2, -1, -2$ are: [BSEB, 2026 A]

(A) $\frac{2}{3}, \frac{-1}{3}, \frac{-2}{3}$

(B) $\frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{-2}{\sqrt{14}}$

(C) $\frac{2}{5}, \frac{-1}{5}, \frac{-2}{5}$

(D) None of these

19. Sum of direction cosines of $x$-axis is: [BSEB, 2026 A]

(A) 1

(B) 2

(C) 3

(D) 4

20. Direction ratios of $z$-axis are: [BSEB, 2026 A]

(A) $1, 0, 0$

(B) $0, 1, 0$

(C) $0, 0, 1$

(D) $0, 0, 0$

21. If $a, b, c$ are direction ratios of a line, then its direction cosines will be: [BSEB, 2026 A]

(A) $\frac{a}{\sqrt{\sum a^2}}, \frac{b}{\sqrt{\sum a^2}}, \frac{c}{\sqrt{\sum a^2}}$

(B) $\frac{1}{\sqrt{\sum a^2}}, \frac{1}{\sqrt{\sum a^2}}, \frac{1}{\sqrt{\sum a^2}}$

(C) $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$

(D) $\frac{a}{\sqrt{\sum b^2}}, \frac{b}{\sqrt{\sum c^2}}, \frac{c}{\sqrt{\sum a^2}}$

22. If $\frac{4}{\sqrt{77}}, \frac{5}{\sqrt{77}}$ and $\frac{x}{\sqrt{77}}$ are direction cosines of a line, then the value of $x$ is: [BSEB, 2026 A]

(A) 6

(B) 7

(C) 8

(D) 9

23. If the direction cosines of a straight line are $\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, x$, then the value of $x$ is: [BSEB, 2026 A]

(A) $\frac{6}{\sqrt{77}}$

(B) $\frac{8}{\sqrt{77}}$

(C) $\frac{9}{\sqrt{77}}$

(D) $\frac{1}{\sqrt{77}}$

24. If $l, m, n$ are direction cosines of $PQ$, then the direction cosines of $QP$ will be: [BSEB, 2026 A]

(A) $l, m, n$

(B) $-l, -m, n$

(C) $-l, -m, -n$

(D) None of these

25. Direction cosines of a line equally inclined to the coordinate axes are: [BSEB, 2026 A]

(A) $\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

(B) $\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$

(C) $\frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

(D) $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

26. Direction ratios are $1, 2, 3$. Its direction cosines are: [BSEB, 2026 A]

(A) $\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}$

(B) $\frac{1}{6}, \frac{2}{6}, \frac{3}{6}$

(C) $1, 2, 3$

(D) None

27. Direction cosines of the $z$-axis are: [BSEB, 2026 A]

(A) $(1, 0, 0)$

(B) $(0, 1, 0)$

(C) $(0, 0, 1)$

(D) $(1, 1, 1)$

28. Direction cosines of $x$-axis are: [BSEB, 2026 A]

(A) $(1, 0, 0)$

(B) $(0, 1, 0)$

(C) $(0, 0, 1)$

(D) $(1, 1, 1)$

29. The direction cosines of the z-axis are: [BSEB, 2026 A]

(A) $(0, 0, 0)$

(B) $(1, 0, 0)$

(C) $(0, 1, 0)$

(D) $(0, 0, 1)$

30. The direction cosines of a line having direction ratios $1, 1, 1$ are: [BSEB, 2026 A]

(A) $1, 1, 1$

(B) $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

(C) $\frac{1}{3}, \frac{1}{3}, \frac{1}{3}$

(D) $0, 0, 0$

31. Direction cosines of the $z$-axis are: [BSEB, 2026 A]

(A) $(0, 0, 1)$

(B) $(1, 0, 0)$

(C) $(0, 1, 0)$

(D) $(1, 1, 1)$

32. Direction cosines of $y$-axis are: [BSEB, 2026 A]

(A) $(1, 0, 0)$

(B) $(0, 1, 0)$

(C) $(0, 0, 1)$

(D) $(0, 0, 0)$

Topic: Distance Formula and 3D Points

33. The distance of point $(x, y, z)$ from the origin is: [BSEB, 2026 A]

(A) $\sqrt{x^2 + y^2 + z^2}$

(B) $x^2 + y^2 + z^2$

(C) $x+y+z$

(D) $\sqrt{x+y+z}$

34. Distance between $(4, 3, 7)$ and $(1, -1, -5) = $ …….. [BSEB, 2026 A]

(A) 7

(B) 12

(C) 13

(D) 25

35. Direction ratios of the line joining points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are: [BSEB, 2026 A]

(A) $x_1+x_2, y_1+y_2, z_1+z_2$

(B) $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$

(C) $\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}$

(D) $x_2-x_1, y_2-y_1, z_2-z_1$

36. Distance of point $(-3, 4, 5)$ from the origin is: [BSEB, 2026 A]

(A) 50

(B) $5\sqrt{2}$

(C) 6

(D) None

37. Coordinates of the mid-point of the line segment joining points $A(2, 3, 4)$ and $B(4, 5, 8)$ are: [BSEB, 2026 A]

(A) $(3, 4, 6)$

(B) $(4, 3, 6)$

(C) $(2, 4, 6)$

(D) $(4, 3, 2)$

38. Every point on the $x$-axis is of the form: [BSEB, 2026 A]

(A) $(x, 0, 0)$

(B) $(0, y, 0)$

(C) $(0, 0, z)$

(D) $(x, y, z)$

39. Distance between points $(-4, -3, 7)$ and $(-1, 1, -5)$ is: [BSEB, 2026 A]

(A) 12

(B) 13

(C) 14

(D) None of these

40. Distance of plane $3x – 4y + 6z = 11$ from point $(0, 0, 0)$ is: [BSEB, 2026 A]

(A) $\frac{3}{\sqrt{61}}$

(B) $\frac{11}{\sqrt{61}}$

(C) $\frac{6}{\sqrt{61}}$

(D) $\frac{6}{\sqrt{61}}$

41. Distance of point $(-3, -4, -5)$ from origin is: [BSEB, 2026 A]

(A) 6

(B) $5\sqrt{2}$

(C) 50

(D) None of these

42. Distance between points $(4, 3, 7)$ and $(1, -1, -5)$ is: [BSEB, 2026 A]

(A) 7

(B) 12

(C) 13

(D) 25

43. Distance of $(x, y, z)$ from origin is: [BSEB, 2026 A]

(A) $\sqrt{x^2 + y^2 + z^2}$

(B) $x+y+z$

(C) $x^2 + y^2 + z^2$

(D) $\sqrt{x+y+z}$

44. Distance between $(4, 3, 7)$ and $(1, -1, -5)$ is: [BSEB, 2026 A]

(A) 13

(B) 12

(C) 5

(D) 25

45. The distance of the point $(x, y, z)$ from the origin is: [BSEB, 2026 A]

(A) $\sqrt{x^2 + y^2 + z^2}$

(B) $x + y + z$

(C) $x^2 + y^2 + z^2$

(D) $\sqrt{x+y+z}$

46. The direction ratios of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are: [BSEB, 2026 A]

(A) $x_1+x_2, y_1+y_2, z_1+z_2$

(B) $x_2-x_1, y_2-y_1, z_2-z_1$

(C) $x_1x_2, y_1y_2, z_1z_2$

(D) None of these

47. The distance between the points $(4, 3, 7)$ and $(1, -1, -5)$ is: [BSEB, 2026 A]

(A) 7

(B) 12

(C) 13

(D) 15

48. Distance of point $(x, y, z)$ from origin is: [BSEB, 2026 A]

(A) $\sqrt{x^2+y^2+z^2}$

(B) $x+y+z$

(C) $x^2+y^2+z^2$

(D) $\sqrt{x+y+z}$

49. Distance of point $(3, 4, 5)$ from $x$-axis is: [BSEB, 2026 A]

(A) 3

(B) 5

(C) $\sqrt{41}$

(D) None of these

Topic: Parallel and Perpendicular Conditions

50. If a straight line makes angles $\alpha, \beta, \gamma$ with the positive direction of $x, y, z$ axes respectively, then: [BSEB, 2026 A]

(A) $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + 1 = 0$

(B) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$

(C) $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

(D) $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 2$

51. If direction ratios of two parallel lines are $a, b, c$ and $x, y, z$, then $az = $ [BSEB, 2026 A]

(A) $cy$

(B) $cx$

(C) $bz$

(D) $ax$

52. If direction ratios of two parallel lines are $x, 5, 3$ and $20, 10, 6$, then value of $x$ is: [BSEB, 2026 A]

(A) 10

(B) 5

(C) 3

(D) 40

53. If the direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ of two lines are parallel, then: [BSEB, 2026 A]

(A) $l_1l_2 + m_1m_2 + n_1n_2 = 0$

(B) $\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}$

(C) $l_1 = l_2, m_1 = m_2, n_1 = n_2$

(D) None of these

54. If $\theta$ is the angle between two lines with direction cosines $l_1, m_1, n_1$ and $l_2, m_2, n_2$, then $\cos \theta = $ [BSEB, 2026 A]

(A) $l_1m_2 + m_1n_2 + n_1l_2$

(B) $l_1m_1 + m_1n_1 + n_1l_1$

(C) $l_1l_2 + m_1m_2 + n_1n_2$

(D) $l_1n_1 + m_1m_1 + n_1l_1$

55. If the direction cosines of two straight lines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$, then the cosine of the angle between them will be: [BSEB, 2026 A]

(A) $(l_1 + m_1 + n_1)(l_2 + m_2 + n_2)$

(B) $\frac{l_1}{l_2} + \frac{m_1}{m_2} + \frac{n_1}{n_2}$

(C) $l_1l_2 + m_1m_2 + n_1n_2$

(D) None of these

56. Two straight lines with direction ratios $l, m, n$ and $l_1, m_1, n_1$ are parallel if: [BSEB, 2026 A]

(A) $ll_1 + mm_1 + nn_1 = 0$

(B) $\frac{l}{l_1} = \frac{m}{m_1} = \frac{n}{n_1}$

(C) $\frac{l+l_1}{l_1} = \frac{m+m_1}{m_1} = \frac{n+n_1}{n_1}$

(D) $l_1 + mm_1 + nn_1 = 1$

57. If a line makes angles $\alpha, \beta, \gamma$ with positive coordinate axes, then: [BSEB, 2026 A]

(A) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 0$

(B) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$

(C) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$

(D) $\sin^2 \theta = \sin^2 \beta = \sin^2 \gamma$

58. Two straight lines with direction ratios $l, m, n$ and $l’, m’, n’$ are perpendicular if: [BSEB, 2026 A]

(A) $\frac{l}{l’} = \frac{m}{m’} = \frac{n}{n’}$

(B) $l \cdot l’ + m \cdot m’ + n \cdot n’ = 0$

(C) $l^2 + m^2 + n^2 = l’^2 + m’^2 + n’^2$

(D) $ll’ + mm’ + nn’ = 0$

59. If a line makes angles $\alpha, \beta, \gamma$ with the axes, then: [BSEB, 2026 A]

(A) $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

(B) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$

(C) $\cos \alpha + \cos \beta + \cos \gamma = 1$

(D) $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$

60. If the direction cosines of two straight lines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$ then the cosine of the angle will be: [BSEB, 2026 A]

(A) $(l_1 + m_1 + n_1)(l_2 + m_2 + n_2)$

(B) $\frac{l_1}{l_2} + \frac{m_1}{m_2} + \frac{n_1}{n_2}$

(C) $l_1l_2 + m_1m_2 + n_1n_2$

(D) none of these

61. If direction ratios of two parallel lines are $2, 7, 9$ and $6, 21, x$, then the value of $x$ is: [BSEB, 2026 A]

(A) 9

(B) 18

(C) 27

(D) 3

62. If direction ratios of two parallel lines are $x, 5, 3$ and $20, 10, 6$, then value of $x$ is: [BSEB, 2026 A]

(A) 10

(B) 5

(C) 3

(D) 40

63. If direction ratios of two perpendicular lines are $5, 2, 4$ and $4, 8, x$, then the value of $x$ is: [BSEB, 2026 A]

(A) 9

(B) -9

(C) 8

(D) -8

64. If direction ratios of two perpendicular lines are $2, 3, 5$ and $x, y, 4$, then $2x + 3y = $ [BSEB, 2026 A]

(A) 20

(B) -20

(C) 30

(D) -30

Topic: Equation of Plane

65. The length of perpendicular from point $(0, -1, 3)$ to the plane $2x + y – 2z + 1 = 0$ is: [BSEB, 2026 A]

(A) 4

(B) $2\sqrt{3}$

(C) $\frac{2}{3}$

(D) 2

66. The equation of the plane perpendicular to the plane $4x + 3y – z + 1 = 0$ is: [BSEB, 2026 A]

(A) $x – 5y – 11z + 7 = 0$

(B) $x – y – z = 2$

(C) $3x – 11y + 9z = 1$

(D) None of these

67. The intercept cut on $z$-axis by the plane $x + y – z = 7$ is: [BSEB, 2026 A]

(A) 7

(B) -7

(C) 1

(D) 0

68. The equation of $xy$-plane is: [BSEB, 2026 A]

(A) $x=0, y=0$

(B) $z=0$

(C) $x \neq 0, y \neq 0$

(D) None of these

69. Equation of a plane parallel to the plane $3x – 5y + 4z = 11$ is: [BSEB, 2026 A]

(A) $3x – 5y + 4z = 21$

(B) $3x + 5y + 4z = 25$

(C) $3x + 5y + 4z = 35$

(D) None of these

70. The standard equation of a plane in intercept form is: [BSEB, 2026 A]

(A) $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 0$

(B) $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

(C) $\frac{x}{a} – \frac{y}{b} – \frac{z}{c} = 0$

(D) None of these

71. Distance of plane $2x – 3y + 4z = 6$ from origin is: [BSEB, 2026 A]

(A) $\frac{6}{\sqrt{35}}$

(B) $\frac{6}{\sqrt{37}}$

(C) $\frac{6}{\sqrt{29}}$

(D) $\frac{6}{\sqrt{31}}$

72. Equation of a plane parallel to the plane $2x + 3y – 4z + 8 = 0$ is: [BSEB, 2026 A]

(A) $2x + 3y + 4z + 8 = 0$

(B) $3x + 2y – 4z – 8 = 0$

(C) $2x + 3y – 4z + k = 0$

(D) $2x + 3y – 4z + 15 = 0$

73. The planes $2x – y + 4z = 5$ and $5x – 2.5y + 10z = 6$ are: [BSEB, 2026 A]

(A) Perpendicular

(B) Parallel

(C) Intersecting at $y$-axis

(D) Passing through point $(0, 0, \frac{5}{4})$

74. If two planes $2x – 4y + 3z = 5$ and $x + 2y + \lambda z = 12$ are perpendicular, then $\lambda = $ [BSEB, 2026 A]

(A) -2

(B) 2

(C) 3

(D) None of these

75. The intercept cut on $x$-axis by the plane $3x + 4y + 5z = 13$ is: [BSEB, 2026 A]

(A) $\frac{3}{13}$

(B) $\frac{13}{3}$

(C) $\frac{13}{4}$

(D) $\frac{13}{5}$

76. Distance of plane $x – 2y + 4z = 9$ from point $(2, 1, -1)$ is: [BSEB, 2026 A]

(A) $\frac{13}{21}$

(B) $\frac{13\sqrt{21}}{21}$

(C) $\frac{21}{13}$

(D) None of these

77. The equation of $yz$-plane is: [BSEB, 2026 A]

(A) $y=0, z=0$

(B) $x=0$

(C) $y=0$

(D) $x=1$

78. Direction ratios of the normal to the plane $3x + 4y + 5z – 6 = 0$ are: [BSEB, 2026 A]

(A) $3, 4, 5$

(B) $-3, 4, 5$

(C) $3, -4, 5$

(D) None of these

79. Distance of plane $3x – 4y + 6z = 11$ from origin is: [BSEB, 2026 A]

(A) $\frac{3}{\sqrt{61}}$

(B) $\frac{11}{\sqrt{61}}$

(C) $\frac{6}{\sqrt{61}}$

(D) $\frac{6}{\sqrt{61}}$

80. Direction cosines of the normal to the plane $2x – 3y + 6z – 3 = 0$ are: [BSEB, 2026 A]

(A) $\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}$

(B) $\frac{2}{7}, \frac{3}{7}, \frac{6}{7}$

(C) $\frac{-2}{7}, \frac{3}{7}, \frac{-6}{7}$

(D) None of these

81. Equation of a plane parallel to the plane $x – 8y – 9z = 12$ is: [BSEB, 2026 A]

(A) $x + 8y + 9z = 12$

(B) $x – 8y – 9z = 2023$

(C) $8x – y – 9z = 12$

(D) $x – 9y – 8z = 12$

82. If two planes $2x + 4y + 3z = 5$ and $x + 2y + kz = 1$ are parallel, then the value of $k$ is: [BSEB, 2026 A]

(A) 3

(B) $\frac{3}{2}$

(C) 6

(D) 1

83. Direction ratios of the normal to the plane $7x + 4y – 2z + 5 = 0$ are: [BSEB, 2026 A]

(A) $7, 4, -2$

(B) $7, 4, 5$

(C) $7, 4, 2$

(D) $4, -2, 5$

84. Equation of a plane parallel to the plane $x – 5y – 11z + 7 = 0$ is: [BSEB, 2026 A]

(A) $2x – 3y + 4z = 11$

(B) $2x + 3y + 4z = 7$

(C) $3x – 2y + 4z = 7$

(D) $4x – 3y + 2z = 7$

85. Equation of a plane parallel to the plane $5x – 6y + 7z – 8 = 0$ is: [BSEB, 2026 A]

(A) $5x – 6y + 7z + 5 = 0$

(B) $6x – 7y + 7z – 8 = 0$

(C) $5x + 6y – 7z – 8 = 0$

(D) None of these

86. Distance of plane $2x – 3y + 4z = 6$ from origin is: [BSEB, 2026 A]

(A) $\frac{|d|}{\sqrt{a^2+b^2+c^2}}$

(B) $\frac{d}{\sqrt{a^2+b^2+c^2}}$

(C) $\frac{|a+b+c|}{\sqrt{d}}$

(D) None of these

87. Direction cosines of the normal to the plane $2x – 3y + 6z + 11 = 0$ are: [BSEB, 2026 A]

(A) $2, -3, 6$

(B) $\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}$

(C) $\frac{2}{11}, \frac{-3}{11}, \frac{6}{11}$

(D) None of these

88. Equation of a plane parallel to plane $z = 0$ is: [BSEB, 2026 A]

(A) $x = 0$

(B) $y = 0$

(C) $z = k$

(D) $x + y = 0$

89. Direction ratios of the normal to the plane $x + 2y – 3z + 15 = 0$ are: [BSEB, 2026 A]

(A) $1, 2, 3$

(B) $1, -2, 3$

(C) $1, 2, -3$

(D) $1, 2, 15$

90. Distance of plane $2x – 3y + 6z + 7 = 0$ from point $(2, -3, -1)$ is: [BSEB, 2026 A]

(A) 4

(B) 3

(C) 2

(D) $\frac{1}{5}$

91. $PQ$ is the perpendicular dropped from point $P(1, 2, 3)$ to the plane $x + y + z = 3$, where $Q$ is the foot of the perpendicular, then: [BSEB, 2026 A]

(A) $PQ = 3$

(B) $PQ = \sqrt{3}$

(C) $Q = (0, 1, 2)$

(D) $Q = (2, 1, 0)$

92. If two planes $a_1x + b_1y + c_1z + d_1 = 0$ and $a_2x + b_2y + c_2z + d_2 = 0$ are perpendicular, then: [BSEB, 2026 A]

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(B) $\frac{a_1}{a_2} + \frac{b_1}{b_2} + \frac{c_1}{c_2} = 0$

(C) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(D) None of these

93. Distance between two parallel planes $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is: [BSEB, 2026 A]

(A) 2 units

(B) 8 units

(C) $\frac{2}{\sqrt{29}}$ units

(D) 4 units

94. Distance of plane $3x – 4y + 6z = 11$ from point $(0, 0, 0)$ is: [BSEB, 2026 A]

(A) $\frac{3}{\sqrt{61}}$

(B) $\frac{11}{\sqrt{61}}$

(C) $\frac{6}{\sqrt{61}}$

(D) $\frac{6}{\sqrt{61}}$

95. Which of the following planes is parallel to the $zx$-plane? [BSEB, 2026 A]

(A) $z=k$

(B) $y=k$

(C) $x=k$

(D) None of these

96. Distance of $(2, -3, -1)$ from plane $2x – 3y + 6z + 7 = 0$ is: [BSEB, 2026 A]

(A) 2

(B) 3

(C) $\frac{1}{7}$

(D) $\frac{14}{7} = 2$

97. Angle between planes $2x + y – 2z = 5$ and $3x – 6y – 2z = 7$ is: [BSEB, 2026 A]

(A) $\cos^{-1}(\frac{4}{21})$

(B) $\cos^{-1}(\frac{1}{3})$

(C) $\cos^{-1}(\frac{2}{3})$

(D) $\frac{\pi}{2}$

98. Direction ratios of normal to the plane $2x – 3y + 4z = 7$ are: [BSEB, 2026 A]

(A) $2, -3, 4$

(B) $2, 3, 4$

(C) $2, -3, 7$

(D) $4, -3, 2$

99. Equation of a plane parallel to $x = 0$ is: [BSEB, 2026 A]

(A) $x = k$

(B) $y = 0$

(C) $z = 0$

(D) $x + y = 0$

100. Plane parallel to $2x – 3y + 5z + 7 = 0$ is: [BSEB, 2026 A]

(A) $2x – 3y + 5z + 11 = 0$

(B) $2x + 3y – 5z + 7 = 0$

(C) $3x – 2y + 5z + 7 = 0$

(D) $2x – 3y + 5 = 0$

101. The distance of the plane $x + 2y – 2z = 9$ from the point $(2, 3, -5)$ is: [BSEB, 2026 A]

(A) 1

(B) 2

(C) 3

(D) 4

102. If two planes $x – 4y + \lambda z + 3 = 0$ and $2x + 2y + 3z = 5$ are perpendicular, then $\lambda = $ [BSEB, 2026 A]

(A) 1

(B) 2

(C) 3

(D) 4

103. The equation of the plane whose intercepts on the $x, y$ and $z$-axes are $-2, 3$ and $4$ will be: [BSEB, 2026 A]

(A) $6x – 4y – 3z + 12 = 0$

(B) $6x + 4y + 3z + 12 = 0$

(C) $6x – 4y – 3z = 0$

(D) none of these

104. The plane $2x – 3y + 4z = 7$ is parallel to which of the following planes? [BSEB, 2026 A]

(A) $2x – 3y + 4z = 0$

(B) $4x – 6y + 8z = 10$

(C) Both (A) and (B)

(D) None of these

105. The equation of the plane perpendicular to the plane $4x + 3y – z + 1 = 0$ is: [BSEB, 2026 A]

(A) $x – 5y – 11z + 7 = 0$

(B) $x – y – z = 2$

(C) $3x – 11y + 9z = 1$

(D) None of these

Topic: Equation of Straight Line

106. Through which point does the line $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ pass? [BSEB, 2026 A]

(A) $(3, 7, 2)$

(B) $(5, 4, 6)$

(C) $(5, -4, 6)$

(D) $(-5, 4, -6)$

107. If the line $\frac{x-3}{a} = \frac{y-4}{b} = \frac{z-5}{c}$ is parallel to the line $\frac{x}{5} = \frac{y}{3} = \frac{z}{2}$, then: [BSEB, 2026 A]

(A) $5a + 3b + 2c = 0$

(B) $\frac{a}{5} = \frac{b}{3} = \frac{c}{2}$

(C) $5a = 3b = 2c$

(D) None of these

108. Direction ratios of the line perpendicular to the plane $x + 2y + 3z + 4 = 0$ are: [BSEB, 2026 A]

(A) $1, 2, 3$

(B) $4, 2, 3$

(C) $1, 2, 4$

(D) None of these

109. Direction ratios of the straight line $\frac{x+1}{3} = \frac{y-2}{3} = \frac{z-5}{6}$ are: [BSEB, 2026 A]

(A) $1, -2, 5$

(B) $3, 2, 5$

(C) $3, 3, 6$

(D) $1, 3, 5$

110. If the line $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ is parallel to the plane $a_2x + b_2y + c_2z + d = 0$, then: [BSEB, 2026 A]

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(B) $a_1x + b_1y + c_1z = 0$

(C) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(D) None of these

111. If the line $\frac{x-2}{a} = \frac{y-3}{b} = \frac{z-4}{c}$ is parallel to the line $\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$, then: [BSEB, 2026 A]

(A) $\frac{a}{4} = \frac{b}{2} = \frac{c}{3}$

(B) $\frac{a}{5} = \frac{b}{3} = \frac{c}{2}$

(C) $5a = 3b = 2c$

(D) None of these

112. If the line $\frac{x-2}{3} = \frac{y-3}{4} = \frac{z-5}{6}$ is parallel to the plane $ax + by + cz + d = 0$, then: [BSEB, 2026 A]

(A) $2a + 3b + 5c = 0$

(B) $3a + 4b + 5c = 0$

(C) $3a + 4b + 6c = 0$

(D) None of these

113. Direction ratios of a line parallel to the line $\frac{x-10}{1} = \frac{y-11}{2} = \frac{z-12}{3}$ are: [BSEB, 2026 A]

(A) $10, 11, 12$

(B) $1, 2, 3$

(C) $1, 1, 1$

(D) $3, 2, 1$

114. A straight line passes through $(\alpha, \beta, \gamma)$ and its direction cosines are $l, m, n$. The equations of this straight line are: [BSEB, 2026 A]

(A) $\frac{x}{l} = \frac{y}{m} = \frac{z}{n}$

(B) $\frac{x-\alpha}{l} = \frac{y-\beta}{m} = \frac{z-\gamma}{n}$

(C) $\frac{x+\alpha}{l} = \frac{y+\beta}{m} = \frac{z+\gamma}{n}$

(D) $\frac{x-\alpha}{l} = \frac{y+\beta}{m} = \frac{z-\gamma}{n}$

115. The angle between the straight lines $\frac{x-2}{2} = \frac{y-1}{7} = \frac{z+3}{-3}$ and $\frac{x+2}{-1} = \frac{y-4}{2} = \frac{z-5}{4}$ is: [BSEB, 2026 A]

(A) $\pi/2$

(B) 0

(C) $\pi/6$

(D) $\pi/4$

116. If the line $\frac{x – x_1}{a_1} = \frac{y – y_1}{b_1} = \frac{z – z_1}{c_1}$ is parallel to the plane $a_2x + b_2y + c_2z + d = 0$ then: [BSEB, 2026 A]

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(B) $a_1x + b_1y + c_1z + d = 0$

(C) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(D) none of these

117. Direction ratios of a line parallel to the $z$-axis can be: [BSEB, 2026 A]

(A) $1, 0, 0$

(B) $0, 1, 0$

(C) $0, 0, 5$

(D) $1, 1, 1$