Difficulty: Medium
Correct Answer: Neither statement I nor statement II is true
Explanation:
Introduction / Context:
This question tests your ability to compare surds, that is, expressions containing square roots. You must evaluate the relative sizes of 2√3 and 3√2, and of 4√2 and 2√8, and then decide which of the two given statements are true.
Given Data / Assumptions:
We are given the following statements.
Concept / Approach:
To compare expressions involving square roots, a common technique is to square both sides, provided all values are positive. Squaring preserves the inequality direction when both sides are non negative. We can also simplify radicals, such as rewriting √8 as 2√2, to make comparisons simpler.
Step-by-Step Solution:
Verification / Alternative check:
We can approximate numerically to cross check. √2 is about 1.414 and √3 is about 1.732. Then 2√3 is roughly 2 × 1.732 ≈ 3.464, and 3√2 is roughly 3 × 1.414 ≈ 4.242, confirming that 2√3 is less than 3√2. For the second comparison, since we have already simplified 2√8 to 4√2 exactly, equality is clear.
Why Other Options Are Wrong:
Option A says only statement I is true, but we found statement I is false. Option B says only statement II is true, but statement II is also false. Option D says both statements are true, which again is incorrect because both fail.
Common Pitfalls:
Students sometimes miscompare surds by comparing the numbers under the root directly or by relying on rough mental guesses. The safe approach is to square the expressions or simplify them to a common standard form before comparison. Another common mistake is ignoring equality; in statement II many may assume that if two surds look different, one must be greater, but here they are equal.
Final Answer:
Both statements I and II are false. Thus, the correct option is “Neither statement I nor statement II is true.”
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