Three Dimensional Geometry in past papers.

CBSE Board Examination 2026

• Board 2026 Q16Perpendicular from $(2, 5, 7)$ to the line $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$
  Perpendicular distance of a point from a line1 mark
• Board 2026 Q20Perpendicular lines $x = py + q$⁠, $z = ry + s$
  Perpendicular lines in space1 mark
• Board 2026 Q35Are the lines parallel? If not, find their point of intersection
  Two lines: parallel or intersecting5 marks

CBSE Board Examination 2025

• Board 2025 Q29Skew lines and their shortest distance, or the ratio in which the wicketkeeper divides bowler to fielder
  Skew lines and shortest distance; section formula3 marks
• Board 2025 Q34Image of A(1, 6, 3) in a line and the line AA′, or a point P at distance 7 from Q(2, 4, $-1$⁠)
  Image of a point in a line; a point at distance 75 marks

CBSE Board Examination 2024

• Board 2024 Q9Find $k$ when the direction cosines are $\sqrt{3}k, \sqrt{3}k, \sqrt{3}k$
  Direction cosines of a line1 mark
• Board 2024 Q16Distance of $P(a, b, c)$ from the y-axis
  Distance of a point from a coordinate axis1 mark
• Board 2024 Q20No line is perpendicular to all three axes
  Direction cosines: l² + m² + n² = 11 mark
• Board 2024 Q35Find P when its image in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ is $P'(1, 0, 7)$
  Image of a point in a line5 marks

CBSE Board Examination 2023

• Board 2023 Q11Direction cosines of $\frac{x - 1}{2} = \frac{1 - y}{3} = \frac{2z - 1}{12}$
  Direction cosines of a line1 mark
• Board 2023 Q20Line through $(1, 2, 3)$ and $(3, -1, 3)$
  Equation of a line through two points1 mark
• Board 2023 Q34Line through $(1, 2, -4)$ parallel to AB and the distance between them, or points on AB at 14 units from B
  Line equations; distance between parallel lines5 marks

CBSE Board Examination 2022 (Term 2)

• Board 2022 T2 Q6Values of $\lambda$ for which $(2, 1, \lambda)$ is $2\sqrt2$ units from $3x + 5y + 4z = 11$
  Distance of a point from a plane2 marks
• Board 2022 T2 Q8Plane through the intersection of two planes and the point $(2, 1, 3)$
  Plane through the intersection of two planes3 marks
• Board 2022 T2 Q14Case study: two transmission wires along lines $l_1$ and $l_2$
  Two lines: coplanar and point of intersection4 marks

CBSE Sample Paper 2026-27

• Sample 2026-27 Q23Parallel lines through given points: find $\frac{q}{p}$⁠; or the sides of a rectangle from its diagonals
  Parallel lines; sides of a rectangle as vectors2 marks
• Sample 2026-27 Q30Foot of the perpendicular from $(2, 1, 4)$ to a line; or the angle between two lines and a common perpendicular
  Foot of a perpendicular; angle between two lines3 marks
• Sample 2026-27 Q36Case study: laser beams in a mall, parallel and skew lines
  Lines in space: parallel lines, shortest distance4 marks

CBSE Sample Paper 2025-26

• Sample 2025-26 Q14Distance of $3\hat i + 4\hat j + 5\hat k$ from the y-axis
  Distance of a point from a coordinate axis1 mark
• Sample 2025-26 Q29Distance from $(2, -1, 3)$ to a line parallel to the z-axis, or a point of intersection and its distance from the z-axis
  Lines in space: intersection and distance3 marks
• Sample 2025-26 Q35Point of intersection of two lines and the line perpendicular to both
  Intersecting lines; a line perpendicular to both5 marks

CBSE Sample Paper 2024-25

• Sample 2024-25 Q35Shortest distance between two lines, or the image of $(1, 2, 1)$ in a line
  Shortest distance; image of a point in a line5 marks

CBSE Sample Paper 2023-24

• Sample 2023-24 Q5Lines with directions $2\hat i + 3\hat j - 6\hat k$ and $6\hat i + 9\hat j - 18\hat k$
  Parallel and coincident lines (vector form)1 mark
• Sample 2023-24 Q18$c$ when the direction cosines are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$
  Direction cosines: l² + m² + n² = 11 mark
• Sample 2023-24 Q35Image of $(1, 6, 3)$ in a line, or where two aeroplanes are closest
  Image of a point in a line; shortest distance5 marks

CBSE Sample Paper 2022-23

• Sample 2022-23 Q18z-coordinate of the point with $x = 6$ on line AB
  Point on the line through two points1 mark
• Sample 2022-23 Q20Acute angle between a line and the x-axis
  Angle between two lines1 mark
• Sample 2022-23 Q34Two insects on skew lines, or the path of a rocket
  Shortest distance of skew lines; point to a line5 marks

CBSE Sample Paper 2021-22 (Term 2)

• Sample 2021-22 T2 Q4Direction cosines of the line $\frac{3 - x}{-1} = \frac{2y - 1}{2} = \frac{z}{4}$
  Direction cosines of a line2 marks
• Sample 2021-22 T2 Q10Shortest distance between two parallel lines, or the plane through $\hat i + 2\hat j - \hat k$ parallel to two lines
  Shortest distance between parallel lines3 marks
• Sample 2021-22 T2 Q13Foot of the perpendicular from $(1, 2, 0)$ to $x - 3y + 2z = 9$⁠, and the distance
  Foot of the perpendicular to a plane; distance4 marks

CBSE Sample Paper 2020-21

• Sample 2020-21 Q13Direction cosines of the normal to the YZ plane
  Direction cosines of the normal to the YZ-plane1 mark
• Sample 2020-21 Q14Where the line $\frac{x + 3}{3} = \frac{y - 1}{-1} = \frac{z - 5}{-5}$ cuts the XY plane
  Point where a line meets the XY-plane1 mark
• Sample 2020-21 Q27Plane through $(1, 0, 0)$ containing the line $\vec r = \lambda\hat j$
  Vector equation of a plane containing a line2 marks
• Sample 2020-21 Q37Shortest distance between two lines and their point of intersection, or the foot of a perpendicular to a plane
  Shortest distance; point of intersection of lines5 marks

Questions that also use Three Dimensional Geometry

Their main topic is in another chapter.

• Board 2023 Q23$\vec c$ from $\overrightarrow{AC} = \frac54\overrightarrow{AB}$⁠, or are two lines perpendicular?
  Position vectors on a line; perpendicular lines2 marks
• Sample 2025-26 Q13Projected length of a bird's flight on a metro track
  Projection of a vector on a line1 mark
• Sample 2024-25 Q28Angle between $\vec{l_1}$ and $\vec{l_2}$ and the projection, or the line perpendicular to two given lines
  Angle, projection; line perpendicular to two lines3 marks
• Sample 2022-23 Q23$\lambda$ for $\vec a + \vec b \perp \vec a - \vec b$⁠, or direction cosines from $6x - 12 = 3y + 9 = 2z - 2$
  Orthogonal vectors; direction ratios of a line2 marks