Find the angle between the following pair of lines : (x - 2)/(3) = (y + 5)/(2) = (1 - z)/(-6) and (x - 7)/(1) = y/2 = (6 - z)/(-2)

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2026: question 25

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section B, question 25, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25 · 2 marks · Very short answerThree Dimensional Geometry

Find the angle between the following pair of lines :

$\frac{x - 2}{3} = \frac{y + 5}{2} = \frac{1 - z}{- 6}$ and

$\frac{x - 7}{1} = \frac{y}{2} = \frac{6 - z}{- 2}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Make the coefficient of $z$ positive in each line before reading off the direction ratios.

Hint 2

Use $cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Write the lines in standard form: $\frac{x - 2}{3} = \frac{y + 5}{2} = \frac{z - 1}{6}$ and $\frac{x - 7}{1} = \frac{y}{2} = \frac{z - 6}{2}$ . Their direction ratios are $⟨ 3, 2, 6 ⟩$ and $⟨ 1, 2, 2 ⟩$ .
½ mark
$cos θ = \frac{∣3 ( 1 ) + 2 ( 2 ) + 6 ( 2 ) ∣}{9 + 4 + 36 1 + 4 + 4} = \frac{19}{7 \times 3} = \frac{19}{21}$
1 mark
$θ = cos^{- 1} (\frac{19}{21})$
½ mark

Answer: $θ = cos^{- 1} (\frac{19}{21})$

Where the marks are

Correct direction ratios $⟨ 3, 2, 6 ⟩$ , $⟨ 1, 2, 2 ⟩$ (½ mark)
Correct substitution in the cosine formula (1 mark)
$θ = cos^{- 1} (19/21)$ (½ mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
Topic: Shortest Distance Between Two Lines

Practise this chapterBoard-style questions from Three Dimensional Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 25

Hints

Step-by-step solution

Revise the chapter

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