MCQMathematics

CBSE · Class 11 · Mathematics · SetsWhich of the following is an empty set?

Q: Which of the following is an empty set?

Let's check the options: (A) $\{x : x^2 - 2 = 0 \text{ and } x \text{ is rational}\}$ has no rational solution since $\sqrt{2}$ is irrational. (B) $\{x : x^2 = 4 \text{ and } x \text{ is even}\}$ has solution $x = \pm 2$. (C) $\{x : x \text{ is a real number and } x > x\}$ is always false. Wait, option A gives an empty set because $x^2 = 2 \implies x = \pm\sqrt{2}$, which are not rational numbers.

Step-by-Step Solution

Let's check the options: (A) ${x : x^2 - 2 = 0 \text{ and } x \text{ is rational}}$ has no rational solution since $\sqrt{2}$ is irrational. (B) ${x : x^2 = 4 \text{ and } x \text{ is even}}$ has solution $x = \pm 2$. (C) ${x : x \text{ is a real number and } x > x}$ is always false. Wait, option A gives an empty set because $x^2 = 2 \implies x = \pm\sqrt{2}$, which are not rational numbers.

Detailed Options Breakdown

Option : \{x : x^2 - 2 = 0 \text{ and } x \text{ is rational}\} (Correct Answer)

Correct choice. Refer to the step-by-step verified solution guidelines above for details.

Option 1: \{x : x^2 = 4 \text{ and } x \text{ is even}\}

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Sets.

Option 2: \{x : x \text{ is a prime number and } x > 2\}

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Sets.

Option 3: \{x : x \text{ is a natural number and } x < 2\}

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Sets.

💡 Study Guide: This question tests core syllabus concepts from Sets. For formulas, key summaries, and mock exam reference guides, read the full Sets Revision Notes.

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Verified by Academic Committee

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