JEE Main Mathematics: Direction cosines and direction ratios of a line Practice Questions
15 free multiple-choice questions on Direction cosines and direction ratios of a line (Analytical Geometry) for JEE Main Mathematics, each with the correct answer and a step-by-step explanation. Attempt a question, then reveal the solution to check your reasoning.
Practice thousands more JEE Main questions in the app →Q1. A line makes angles 90°, 60° and 30° with the positive x, y and z axes respectively. Its direction cosines are
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Correct answer: D — 0, 1/2, √3/2
Direction cosines are cos 90°, cos 60°, cos 30°. = 0, 1/2, √3/2 (check: 0 + 1/4 + 3/4 = 1).
Q2. If a line makes angles α, β, γ with the coordinate axes, then sin²α + sin²β + sin²γ equals
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Correct answer: B — 2
sin²α = 1 − cos²α, and similarly for β and γ. Sum = 3 − (cos²α + cos²β + cos²γ) = 3 − 1 = 2.
Q3. The projection of the segment joining P(1, 2, 3) and Q(4, 5, 7) on a line with direction cosines 2/7, 3/7, 6/7 is
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Correct answer: B — 39/7
PQ = (3, 3, 4). Projection = 3·(2/7) + 3·(3/7) + 4·(6/7) = (6 + 9 + 24)/7 = 39/7.
Q4. A line makes the same angle θ with each of the x and y axes, and an angle φ with the z-axis, where sin²φ = 2 sin²θ. Then cos²θ equals
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Correct answer: B — 1/2
2cos²θ + cos²φ = 1. sin²φ = 2 sin²θ gives 1 − cos²φ = 2 − 2cos²θ, so cos²φ = 2cos²θ − 1. 2cos²θ + 2cos²θ − 1 = 1, so cos²θ = 1/2 (and φ = 90°).
Q5. If a line makes angles α, β, γ with the coordinate axes, then cos 2α + cos 2β + cos 2γ equals
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Correct answer: B — −1
cos 2α = 2cos²α − 1. Sum = 2(cos²α + cos²β + cos²γ) − 3 = 2 − 3 = −1.
Q6. If 1/√2, 1/2 and n are the direction cosines of a line, then n equals
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Correct answer: C — ±1/2
1/2 + 1/4 + n² = 1. n² = 1/4, so n = ±1/2.
Q7. The direction cosines of a line whose direction ratios are 2, −1, −2 are
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Correct answer: D — 2/3, −1/3, −2/3
√(4 + 1 + 4) = 3. Divide each direction ratio by 3: 2/3, −1/3, −2/3.
Q8. The acute angle between any two body diagonals of a cube is
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Correct answer: B — cos⁻¹(1/3)
Take diagonals along (1, 1, 1) and (1, 1, −1) for a unit cube. cos θ = |1 + 1 − 1|/(√3·√3) = 1/3. θ = cos⁻¹(1/3).
Q9. A line makes angles α, β, γ, δ with the four body diagonals of a cube. Then cos²α + cos²β + cos²γ + cos²δ equals
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Correct answer: D — 4/3
Take the diagonals along (1, 1, 1), (−1, 1, 1), (1, −1, 1), (1, 1, −1), each of length √3. For a line with DCs l, m, n: cos α = (l + m + n)/√3, etc. Sum of squares = [(l+m+n)² + (−l+m+n)² + (l−m+n)² + (l+m−n)²]/3 = 4(l² + m² + n²)/3 = 4/3.
Q10. A line makes equal angles with the three coordinate axes. The cosine of each angle is
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Correct answer: B — ±1/√3
l = m = n, and l² + m² + n² = 1. 3l² = 1, so l = ±1/√3.
Q11. The direction cosines l, m, n of two lines satisfy l + m + n = 0 and l² + m² − n² = 0. The angle between the two lines is
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Correct answer: C — π/3
From the first relation n = −(l + m); substituting: l² + m² − (l + m)² = 0, i.e. −2lm = 0. l = 0 gives direction (0, 1, −1); m = 0 gives (1, 0, −1). cos θ = (0 + 0 + 1)/(√2·√2) = 1/2, so θ = π/3.
Q12. The direction cosines of the y-axis are
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Correct answer: A — 0, 1, 0
The y-axis makes angles 90°, 0°, 90° with the x, y, z axes. DCs = cos 90°, cos 0°, cos 90° = 0, 1, 0.
Q13. The direction ratios of a line perpendicular to both the lines with direction ratios (1, 2, 3) and (2, −1, 1) are proportional to
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Correct answer: A — 1, 1, −1
Required direction = (1, 2, 3) × (2, −1, 1). = (2·1 − 3·(−1), 3·2 − 1·1, 1·(−1) − 2·2) = (5, 5, −5). Proportional to (1, 1, −1).
Q14. The direction cosines of the line joining A(2, 3, −4) to B(1, −2, 3), taken in the direction from A to B, are
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Correct answer: D — −1/(5√3), −1/√3, 7/(5√3)
Direction ratios = B − A = (−1, −5, 7). Magnitude = √(1 + 25 + 49) = √75 = 5√3. DCs = −1/(5√3), −5/(5√3) = −1/√3, 7/(5√3).
Q15. A line makes an angle of 45° with the x-axis and 60° with the y-axis. The angle it makes with the z-axis is
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Correct answer: D — 60° or 120°
cos²γ = 1 − cos²45° − cos²60° = 1 − 1/2 − 1/4 = 1/4. cos γ = ±1/2, so γ = 60° or 120°.
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