Directions: In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): (a + √(b))·(a - √(b)) is a rational number, where a and b are positive integers. Reason (R): Product of two irrationals is always rational. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/6/1 · 1 mark · Assertion-reason

CBSE Class 10 Mathematics Basic 2025: question 19

Set in the CBSE Class 10 board exam of 2025, paper 430/6/1, 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-basedReal Numbers

Directions: In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $(a + b) \cdot (a - b)$ is a rational number, where $a$ and $b$ are positive integers.

Reason (R): Product of two irrationals is always rational.

(A)Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
(B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for 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 430/6/1.

Hints

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

Hint 1

Use the identity $(x + y) (x - y) = x^{2} - y^{2}$ .

Hint 2

Test R with $2 \times 3$ .

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.

$(a + b) (a - b) = a^{2} - b$ , which is an integer and hence rational. So A is true.
R is false: for example, $2 \times 3 = 6$ is irrational. Hence option (C).
1 mark

Answer: Option (C)

Where the marks are

Shows A is true and gives a counterexample to R (1 mark)

Revise the chapter

Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
Topic: Proving Numbers Irrational

Practise this chapterBoard-style questions from Real Numbers, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/6/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2025: question 19

Hints

Step-by-step solution

Revise the chapter

More questions from Real Numbers