MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 30(B)

From the official CBSE sample question paper for 2026-27, section C, question 30(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(B) · 3 marks · Short answerThree Dimensional Geometry

Find the angle between the lines: $\frac{x - 1}{2} = \frac{y - 1}{2} = \frac{z - 2}{1}$ and $\frac{x - 5}{2} = y - \frac{1}{2} = \frac{3 - z}{2}$ . Also find the cartesian equation of a line passing through the point $(1, 1, 1)$ and perpendicular to both lines.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Rewrite $\frac{3 - z}{2}$ as $\frac{z - 3}{- 2}$ to read the direction ratios correctly.

Hint 2

A vector perpendicular to both lines is the cross product of their direction vectors.

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.

The second line in standard form is $\frac{x - 5}{2} = \frac{y - \frac{1}{2}}{1} = \frac{z - 3}{- 2}$ . Direction ratios: $b_{1} = ⟨ 2, 2, 1 ⟩$ and $b_{2} = ⟨ 2, 1, - 2 ⟩$ .
½ mark
$cos θ = \frac{2 ( 2 ) + 2 ( 1 ) + 1 ( - 2 )}{9 9} = \frac{4}{9} \Rightarrow θ = cos^{- 1} (\frac{4}{9})$
1 mark
The required line is parallel to $b_{1} \times b_{2} = \hat{i} 22 \hat{j} 21 \hat{k} 1 - 2 = - 5 \hat{i} + 6 \hat{j} - 2 \hat{k}$
1 mark
Required line: $\frac{x - 1}{- 5} = \frac{y - 1}{6} = \frac{z - 1}{- 2}$ , or $\frac{x - 1}{5} = \frac{y - 1}{- 6} = \frac{z - 1}{2}$
½ mark

Answer: $θ = cos^{- 1} \frac{4}{9}$ ; $\frac{x - 1}{- 5} = \frac{y - 1}{6} = \frac{z - 1}{- 2}$

Where the marks are

Rewrites the second line in standard form with direction ratios $⟨ 2, 1, - 2 ⟩$ (½ mark)
Finds $cos θ = \frac{4}{9}$ (1 mark)
Computes the cross product $- 5 \hat{i} + 6 \hat{j} - 2 \hat{k}$ (1 mark)
Writes the cartesian equation of the line (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 paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 30(B)

Hints

Step-by-step solution

Revise the chapter

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