Real Numbers
CBSE 2026 Board Exam Set 1 (Code 30/1/1)
CBSE_BOARD_PYQ_2026_30_1_1
Grade 10
Question:
[Section B]
Prove that $\sqrt{3}$ is an irrational number.
OR
Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
Step-by-Step Solution
Key Concept: Main: Proof by contradiction. OR: Factoring out common 13 and 5.
[Main Question Solution]
Assume $\sqrt{3} = p/q$. $3q^2 = p^2 \Rightarrow 3 \mid p \Rightarrow p = 3a$. [1.0 Mark]
$9a^2 = 3q^2 \Rightarrow q^2 = 3a^2 \Rightarrow 3 \mid q$. Contradiction! Hence $\sqrt{3}$ is irrational. [1.0 Mark]
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[OR Choice Question Solution]
$13(7 \times 11 + 1) = 13 \times 78$ (composite). [1.0 Mark]
$5(7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1) = 5 \times 1009$ (composite). [1.0 Mark]
Correct Answer: Proof shown for both choices.
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