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Real Numbers
RD Sharma
CBSE
Grade 10

Question:

Which of the following is NOT an irrational number?
(a) $(2 - \sqrt{3})^2$
(b) $(\sqrt{2} + \sqrt{3})^2$
(c) $(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3})$
(d) $\dfrac{2}{2\sqrt{7}}$

Step-by-Step Solution

Key Concept: $(a - b)(a + b) = a^2 - b^2$.
$(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1$. [0.5 Mark]
Since $-1$ is a rational number, option 3 is NOT irrational. [0.5 Mark]

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🎯 Official CBSE Marking Scheme:
Evaluating option 3 to $-1$: 0.5 Mark
Identifying $-1$ as rational: 0.5 Mark

Correct Answer: $(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3})$
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