Assertion (A) : Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other when pp' + rr' = 1. Reason (R) : Two lines r = a_1 + _1 and r = a_2 + _2 are perpendicular to each other if b_1 · b_2 = 0. (A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true and Reason (R) is false. (D) Assertion (A) is false and Reason (R) is true.

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics 2026: question 20

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section A, question 20, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q20 · 1 mark · Assertion-reasonCompetency-basedThree Dimensional Geometry

Assertion (A) : Lines given by $x = p y + q$ , $z = r y + s$ and $x = p^{'} y + q^{'}$ , $z = r^{'} y + s^{'}$ are perpendicular to each other when $p p^{'} + r r^{'} = 1$ .
Reason (R) : Two lines $r = a_{1} + λ b_{1}$ and $r = a_{2} + μ b_{2}$ are perpendicular to each other if $b_{1} \cdot b_{2} = 0$ .

(A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
(B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C)Assertion (A) is true and Reason (R) is false.
(D)Assertion (A) is false and Reason (R) is true.

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

Hints

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

Hint 1

Express each line in symmetric form with y as parameter.

Hint 2

Apply $b_{1} \cdot b_{2} = 0$ to the direction ratios.

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.

First line: $\frac{x - q}{p} = \frac{y}{1} = \frac{z - s}{r}$ , DRs $p, 1, r$ ; second: $p^{'}, 1, r^{'}$ .
Perpendicular ⇔ $p p^{'} + 1 + r r^{'} = 0$ , i.e. $p p^{'} + r r^{'} = - 1$ , so A is false. R is the standard true condition. Option (D).
1 mark

Answer: Option (D)

Where the marks are

Derives DRs and condition $p p^{'} + r r^{'} = - 1$ ; R true (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)

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 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 20

Hints

Step-by-step solution

Revise the chapter

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