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MP Board Class 12 (MPBSE) Three Dimensional Geometry — practice questions

Explore 54 free Three Dimensional Geometry questions in MP Board Class 12 (MPBSE) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Three Dimensional Geometry questions and exercises with answers

Showing questions 1–30 of 54.

If l, m, n are the direction cosines of a line in space, thenRead the answer and worked solution →The vector equation of a line passing through point a and parallel to vector b isRead the answer and worked solution →The equation of a plane in normal form (with unit normal n̂ and perpendicular distance d from origin) isRead the answer and worked solution →A plane meets the axes at (a,0,0), (0,b,0), (0,0,c). Its equation isRead the answer and worked solution →The direction cosines of the line joining (1, 2, 3) and (4, 5, 6) areRead the answer and worked solution →The Cartesian equation (x-1)/2 = (y-2)/3 = (z-3)/4 representsRead the answer and worked solution →The angle θ between two lines with direction vectors b₁ and b₂ is given byRead the answer and worked solution →The distance of the point (x₀, y₀, z₀) from the plane ax + by + cz + d = 0 isRead the answer and worked solution →The shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ isRead the answer and worked solution →Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if and only ifRead the answer and worked solution →The angle between two planes with normals n₁ and n₂ is given byRead the answer and worked solution →The angle ϕ between a line with direction b and a plane with normal n satisfiesRead the answer and worked solution →The equation of a plane passing through point (x₁, y₁, z₁) with normal vector (a, b, c) isRead the answer and worked solution →Two lines with direction vectors b₁ and b₂ are parallel if and only ifRead the answer and worked solution →The equation of a plane through the intersection of π₁ = 0 and π₂ = 0 has the formRead the answer and worked solution →The perpendicular distance from a point P with position vector p to a line r = a + λb isRead the answer and worked solution →The angle between the lines with direction ratios (1, 2, 2) and (2, 1, 2) satisfiesRead the answer and worked solution →The distance of the point (2, 1, -1) from the plane 2x - y + 2z + 5 = 0 isRead the answer and worked solution →The angle between the planes 2x - y + 2z = 5 and x + 2y - 2z = 4 satisfiesRead the answer and worked solution →The equation of the plane through the points (1,0,0), (0,1,0), (0,0,1) isRead the answer and worked solution →The distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z = 5 isRead the answer and worked solution →The line r = (2, -1, 4) + λ(3, 0, 2), expressed in symmetric Cartesian form, isRead the answer and worked solution →The shortest distance between r = (1,2,3) + λ(2,3,4) and r = (2,4,5) + μ(3,4,5) isRead the answer and worked solution →The equation of a plane parallel to x + 2y - 3z = 5 and passing through (1, 1, 1) isRead the answer and worked solution →The plane through the points (1,1,0), (1,0,1) and (0,1,1) has equationRead the answer and worked solution →A line has direction ratios (1, 1, 2) and meets the plane x - y + z = 0. The angle ϕ between them satisfiesRead the answer and worked solution →The line through the origin with direction ratios (1, 1, 1) meets the plane x + y + z = 6 atRead the answer and worked solution →The foot of the perpendicular from origin to the plane 2x + 3y - 6z = 14 isRead the answer and worked solution →The image of the point (1, 2, 3) in the plane x + y + z = 0 isRead the answer and worked solution →Match each term to its definitionRead the answer and worked solution →

Worked practice questions

If l, m, n are the direction cosines of a line in space, then

  1. l + m + n = 1
  2. l² + m² + n² = 1
  3. lmn = 1
  4. l² + m² + n² = 0
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Correct answer: B. l² + m² + n² = 1

1. Direction cosines are the cosines of angles a line makes with the x, y, z axes. 2. If the line has unit direction vector (l, m, n), its squared length equals one. 3. So l² + m² + n² = 1. 4. This is the standard identity for direction cosines. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.2.1_

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The vector equation of a line passing through point a and parallel to vector b is

  1. r = a × b
  2. r = a · b
  3. r = a + λb
  4. r = λa + b
Show answer and solution

Correct answer: C. r = a + λb

1. A line through point a in the direction of b is the set of all r reachable by moving from a along b. 2. The position vector is r = a + λb, where λ is the scalar parameter. 3. Varying λ traces the entire line. 4. So option C is correct. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.3.1_

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The equation of a plane in normal form (with unit normal n̂ and perpendicular distance d from origin) is

  1. r · n̂ = d
  2. r × n̂ = d
  3. r + n̂ = d
  4. r / n̂ = d
Show answer and solution

Correct answer: A. r · n̂ = d

1. In normal form, n̂ is a unit normal vector and d is the perpendicular distance from the origin. 2. For any point r on the plane, its projection on n̂ equals d. 3. The projection is the dot product r · n̂. 4. So the plane equation is r · n̂ = d. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.6.1_

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