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JEE Main Mathematics: Angle between two lines Practice Questions

12 free multiple-choice questions on Angle between two lines (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.

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  1. Q1. Statement I: For any real a, b, c (not all equal), the lines with direction ratios a, b, c and b − c, c − a, a − b are perpendicular. Statement II: The lines with direction ratios 1, 2, 3 and 3, 2, 1 are parallel.

    • ABoth Statement I and Statement II are true
    • BStatement I is true but Statement II is false
    • CStatement I is false but Statement II is true
    • DBoth Statement I and Statement II are false
    Show answer & explanation

    Correct answer: B — Statement I is true but Statement II is false

    a(b − c) + b(c − a) + c(a − b) = ab − ac + bc − ab + ac − bc = 0, so Statement I is true. 1/3 ≠ 2/2, so the second pair is not proportional: Statement II is false.

  2. Q2. O is the origin, P is (1, 2, 2) and Q is (2, −1, 2). The angle between the lines OP and OQ is

    • A90°
    • Bcos⁻¹(2/9)
    • Ccos⁻¹(4/3)
    • Dcos⁻¹(4/9)
    Show answer & explanation

    Correct answer: D — cos⁻¹(4/9)

    Direction ratios: (1, 2, 2) and (2, −1, 2), each of magnitude 3. cos θ = (2 − 2 + 4)/(3·3) = 4/9. θ = cos⁻¹(4/9).

  3. Q3. A is (1, −1, 2), B is (3, 4, −2), C is (0, 3, 2) and D is (3, 5, 6). The angle between the lines AB and CD is

    • A60°
    • B0°
    • C90°
    • Dcos⁻¹(16/45)
    Show answer & explanation

    Correct answer: C — 90°

    AB = (2, 5, −4), CD = (3, 2, 4). AB·CD = 6 + 10 − 16 = 0. So AB ⊥ CD.

  4. Q4. The acute angle between the line (x + 1)/2 = y/3 = (z − 3)/6 and the line through (1, −1, 2) and (3, 4, −2) is

    • Acos⁻¹(5/21)
    • Bcos⁻¹(√5/7)
    • C90°
    • Dcos⁻¹(√5/21)
    Show answer & explanation

    Correct answer: D — cos⁻¹(√5/21)

    Direction ratios: (2, 3, 6) with magnitude 7 and (2, 5, −4) with magnitude √45 = 3√5. Dot product = 4 + 15 − 24 = −5. cos θ = 5/(7·3√5) = 5/(21√5) = √5/21.

  5. Q5. The angle between the lines (x − 1)/2 = (y + 1)/3 = z/6 and x/3 = (y − 2)/(−6) = (z + 3)/2 is

    • A90°
    • Bcos⁻¹(1/7)
    • C60°
    • D0°
    Show answer & explanation

    Correct answer: A — 90°

    Direction ratios: (2, 3, 6) and (3, −6, 2). Dot product = 6 − 18 + 12 = 0. So the lines are perpendicular: θ = 90°.

  6. Q6. The lines (x − 2)/1 = (y + 1)/k = z/2 and (x + 1)/(2k) = y/3 = (z − 4)/(−4) are perpendicular. Then k equals

    • A8/5
    • B5/8
    • C8
    • D−8/5
    Show answer & explanation

    Correct answer: A — 8/5

    Direction ratios: (1, k, 2) and (2k, 3, −4). Perpendicular: 2k + 3k − 8 = 0. k = 8/5.

  7. Q7. The angle between the lines r = (î + ĵ) + λ(2î − 2ĵ + k̂) and r = (3î − k̂) + μ(6î + 3ĵ + 2k̂) is

    • Acos⁻¹(8/21)
    • Bcos⁻¹(8/7)
    • Ccos⁻¹(20/21)
    • Dcos⁻¹(4/21)
    Show answer & explanation

    Correct answer: A — cos⁻¹(8/21)

    b₁ = (2, −2, 1), |b₁| = 3; b₂ = (6, 3, 2), |b₂| = 7. b₁·b₂ = 12 − 6 + 2 = 8. cos θ = 8/21.

  8. Q8. The lines x = ay + b, z = cy + d and x = a′y + b′, z = c′y + d′ are perpendicular if and only if

    • Aaa′ + cc′ − 1 = 0
    • Baa′ + cc′ + 1 = 0
    • Caa′ + cc′ = 1
    • Da/a′ = c/c′
    Show answer & explanation

    Correct answer: B — aa′ + cc′ + 1 = 0

    First line: (x − b)/a = y/1 = (z − d)/c, so direction ratios (a, 1, c). Second line: direction ratios (a′, 1, c′). Perpendicular: aa′ + 1·1 + cc′ = 0.

  9. Q9. Assertion (A): The lines with direction ratios 1, −2, 3 and −2, 4, −6 are parallel. Reason (R): Two lines are parallel if their direction ratios are proportional.

    • ABoth Assertion (A) and Reason (R) are true and R is the correct explanation of A
    • BBoth A and R are true but R is not the correct explanation of A
    • CA is true but R is false
    • DA is false but R is true
    Show answer & explanation

    Correct answer: A — Both Assertion (A) and Reason (R) are true and R is the correct explanation of A

    −2/1 = 4/(−2) = −6/3 = −2, so the direction ratios are proportional. R is the correct test, so both are true and R explains A.

  10. Q10. For the points A(2, 3, 3), B(1, 2, 3) and C(1, 3, 4), the angle ABC is

    • A60°
    • B120°
    • C30°
    • D90°
    Show answer & explanation

    Correct answer: A — 60°

    BA = (1, 1, 0), BC = (0, 1, 1). cos B = (0 + 1 + 0)/(√2·√2) = 1/2. ∠ABC = 60°.

  11. Q11. The angle between the line r = λ(î + ĵ + k̂) and the x-axis is

    • A45°
    • Bcos⁻¹(1/3)
    • Ccos⁻¹(1/√3)
    • D60°
    Show answer & explanation

    Correct answer: C — cos⁻¹(1/√3)

    Line direction (1, 1, 1), x-axis direction (1, 0, 0). cos θ = 1/(√3·1) = 1/√3.

  12. Q12. The lines (x − 1)/2 = (y − 3)/a = (z + 2)/(−4) and x/(−3) = (y + 1)/6 = (z − 5)/b are parallel. Then

    • Aa = 4, b = −6
    • Ba = 4, b = 6
    • Ca = −4, b = −6
    • Da = −4, b = 6
    Show answer & explanation

    Correct answer: D — a = −4, b = 6

    Parallel lines have proportional direction ratios: −3/2 = 6/a = b/(−4). 6/a = −3/2 gives a = −4. b/(−4) = −3/2 gives b = 6.

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