Assertion (A): Given two non-zero vectors a and b. If r is another non-zero vector such that r×( a+ b)= 0. Then r is perpendicular to a× b. Reason (R): The vector ( a+ b) is perpendicular to the plane of a and b (A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and (R) are true but (R) is not the correct explanation of (A). (C) (A) is true but (R) is false. (D) (A) is false but (R) is true.

MathsCBSE Class 12 · Sample paper, session 2025-26 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics sample paper 2025-26: question 20

From the official CBSE sample question paper for 2025-26, 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-basedVector Algebra

Assertion (A): Given two non-zero vectors $a$ and $b$ . If $r$ is another non-zero vector such that $r \times (a + b) = 0$ . Then $r$ is perpendicular to $a \times b$ .

Reason (R): The vector $(a + b)$ is perpendicular to the plane of $a$ and $b$

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

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

What does a zero cross product say about two vectors?

Hint 2

Where does $a + b$ lie relative to the plane of $a$ and $b$ ?

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.

$r \times (a + b) = 0$ means $r ∥ (a + b)$ . Since $a + b$ lies in the plane of $a, b$ and $a \times b$ is normal to that plane, $r ⊥ a \times b$ . So (A) is true.
$a + b$ lies in the plane of $a$ and $b$ , so it is not perpendicular to that plane; (R) is false. Option (C).
1 mark

Answer: Option (C)

Where the marks are

Shows $r ∥ (a + b)$ , which lies in the plane, so (A) is true and (R) is false (1 mark)

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Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 20

Hints

Step-by-step solution

Revise the chapter

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