Assertion (A) : Let Z be the set of integers. A function f: Z→ Z defined as f(x)=3x-5, x Z is a bijective. Reason (R) : A function is a bijective if it is both surjective and injective. (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, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics 2025: question 19

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section A, question 19, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q19 · 1 mark · Assertion-reasonCompetency-basedRelations and Functions

Assertion (A) : Let Z be the set of integers. A function $f : Z \to Z$ defined as $f (x) = 3 x - 5$ , $\forall x \in Z$ is a bijective.
Reason (R) : A function is a bijective if it is both surjective and injective.

(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, but Reason (R) is false.
(D)Assertion (A) is false, but Reason (R) is true.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

Check whether every integer has an integer preimage.

Hint 2

Try solving $3 x - 5 = 0$ in integers.

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.

$f$ is one-one, but it is not onto: for $f (x) = 0$ we need $x = \frac{5}{3} \in / Z$ . So the Assertion is false.
The Reason is the correct definition of a bijection, so it is true. Hence option (D).
1 mark

Answer: (D)

Where the marks are

Shows $f$ is not onto (e.g. 0 has no preimage) and that R is true (1 mark)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 19

Hints

Step-by-step solution

Revise the chapter

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