MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 26(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, section C, question 26(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(b) · 3 marks · Short answerReal Numbers

Let p, q and r be three distinct prime numbers. Check whether $p \cdot q \cdot r + q$ is a composite number or not. Further, give an example for 3 distinct primes p, q, r such that:

(i) $p \cdot q \cdot r + 1$ is a composite number.

(ii) $p \cdot q \cdot r + 1$ is a prime number.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

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

Hint 1

Is there a common factor in $pq r$ and $q$ ?

Hint 2

For the examples, try small distinct primes like 2, 3, 5, 7 and test $pq r + 1$ .

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.

$p \cdot q \cdot r + q = q (p r + 1)$
½ mark
Here $q > 1$ and $p r + 1 > 1$ , so the number has factors other than 1 and itself; hence $pq r + q$ is a composite number.
½ mark
(i) Take $p = 3, q = 5, r = 7$ : $pq r + 1 = 105 + 1 = 106 = 2 \times 53$ , a composite number.
1 mark
(ii) Take $p = 2, q = 3, r = 5$ : $pq r + 1 = 30 + 1 = 31$ , a prime number.
1 mark

Answer: $pq r + q$ is composite; (i) 3, 5, 7 give 106 (composite); (ii) 2, 3, 5 give 31 (prime)

Where the marks are

Factorises $pq r + q = q (p r + 1)$ (½ mark)
Concludes it is composite (½ mark)
Correct example for composite $pq r + 1$ (1 mark)
Correct example for prime $pq r + 1$ (1 mark)

Revise the chapter

Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
Topic: Fundamental Theorem of Arithmetic

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 30/6/1 (Standard): 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 Standard 2025: question 26(b)

Hints

Step-by-step solution

Revise the chapter

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