Real Numbers
CBSE 2026 Board Exam Set 1 (Code 30/7/1)
CBSE_BOARD_PYQ_2026_30_7_1
Grade 10
Question:
[Section C]
Prove that $\sqrt{5}$ is an irrational number.
OR
Prove that $\dfrac{2 + 3\sqrt{2}}{5}$ is an irrational number, given that $\sqrt{2}$ is an irrational number.
Step-by-Step Solution
Key Concept: Main: Proof by contradiction. OR: Equate to rational $m$ and show contradiction.
[Main Question Solution]
Assume $\sqrt{5} = p/q$ (co-prime $p, q$). $5q^2 = p^2 \Rightarrow 5 \mid p \Rightarrow p = 5a$. [1.0 Mark]
$25a^2 = 5q^2 \Rightarrow q^2 = 5a^2 \Rightarrow 5 \mid q$. Contradiction! Hence $\sqrt{5}$ is irrational. [2.0 Marks]
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[OR Choice Question Solution]
Let $\dfrac{2 + 3\sqrt{2}}{5} = m$ (rational). Then $3\sqrt{2} = 5m - 2 \Rightarrow \sqrt{2} = \dfrac{5m - 2}{3}$. [1.5 Marks]
RHS is rational (since $m$ is rational), but LHS $\sqrt{2}$ is irrational. Contradiction! Hence irrational. [1.5 Marks]
Correct Answer: Proof by contradiction for both choices.
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