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Three-dimensional geometry — KCSE Mathematics

KCSE Mathematics · 101 practice questions · 3 syllabus objectives

28 easy37 medium36 hard

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

State the geometric properties of common solids and identify skew lines

Calculate the length between two points in three-dimensional geometry

Calculate the angle between two lines, a line and a plane, and two planes

Sample Questions

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1
easySHORT ANSWER3 marks

Given the line defined by the direction ratios (2, -3, 1) and the plane given by the equation 3x + 4y - 2z = 12, calculate the angle between the line and the plane. (3 marks)

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Part (a) — 3 marks
Normal vector of the plane = (3, 4, -2) (1 mk)
Dot product = 2*3 + (-3)*4 + 1*(-2) (1 mk)
Use formula cos θ = (|dot product|)/(magnitude of line * magnitude of normal) (1 mk)
2
easySHORT ANSWER3 marks

Given points P(1, 2, 3) and Q(4, 6, 8), find the distance PQ in three-dimensional space. (3 marks)

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Part (b) — 3 marks
PQ = √((4 - 1)² + (6 - 2)² + (8 - 3)²) (1 mk)
PQ = √(3² + 4² + 5²) (1 mk)
PQ = √(9 + 16 + 25) = √50 (1 mk)
3
easySHORT ANSWER3 marks

The coordinates of points A and B are (2, 3, 5) and (4, 7, 1) respectively. Calculate the length of the line segment AB. (3 marks)

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Part (a) — 3 marks
AB = √((4 - 2)² + (7 - 3)² + (1 - 5)²) (1 mk)
AB = √(2² + 4² + (-4)²) (1 mk)
AB = √(4 + 16 + 16) = √36 (1 mk)
4

Identify two pairs of skew lines from the edges of a rectangular prism with vertices A, B, C, D, E, F, G, and H. (2 marks)

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Why Practise Three-dimensional geometry?

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