In this aptitude based question, you are given two mathematical inequalities involving surds and are asked to identify which of the statements is true. You must compare the values of 2 times square root of 3 and 3 times square root of 2, and also compare 4 times square root of 2 with 2 times square root of 8, using basic algebraic manipulation rather than a calculator.

Introduction / Context:
This question checks your understanding of how to compare expressions containing square roots, also called surds. Instead of approximating the values with decimals, you are expected to manipulate the expressions algebraically. Both statements involve positive quantities, so squaring is a safe and powerful method to compare their magnitudes. Mastering such comparisons is very useful in many competitive exams where calculators are not allowed.

Given Data / Assumptions:
- Statement I: 2 times square root of 3 is greater than 3 times square root of 2 (written as 2√3 > 3√2).
- Statement II: 4 times square root of 2 is greater than 2 times square root of 8 (written as 4√2 > 2√8).
- All roots are positive real numbers, and usual rules of arithmetic apply.

Concept / Approach:
To compare expressions like a√b and c√d, you can either square both sides or simplify the surds so that they share a common base. Squaring is valid when both sides are non negative, because if x and y are non negative, then x > y is equivalent to x^2 > y^2. For the second statement, simplifying √8 using factorization is efficient, because 8 can be written as 4 times 2 and the square root of 4 is easy to handle.

Step-by-Step Solution:
Step 1: Consider Statement I, 2√3 and 3√2. Both are positive, so square both expressions. Step 2: (2√3)^2 = 4 * 3 = 12. Step 3: (3√2)^2 = 9 * 2 = 18. Step 4: Since 12 is less than 18, we have 2√3 < 3√2, which means Statement I is false. Step 5: Now consider Statement II, 4√2 and 2√8. First simplify √8. Step 6: 8 = 4 * 2, so √8 = √4 * √2 = 2√2. Step 7: Substitute this back: 2√8 = 2 * 2√2 = 4√2. Step 8: Therefore 4√2 and 2√8 are exactly equal, not greater than or less than each other, so Statement II is also false.

Verification / Alternative check:
You can also approximate numerically to confirm. Square root of 2 is about 1.414 and square root of 3 is about 1.732. Then 2√3 is roughly 3.464 and 3√2 is roughly 4.242, so the first inequality is incorrect. For the second, once we see both sides simplify to 4√2, equality is obvious, so the strict greater than sign cannot be correct.

Why Other Options Are Wrong:
Option A claims only Statement I is true, which is incorrect because we just showed 2√3 is less than 3√2. Option B claims only Statement II is true, but Statement II leads to equality, not a strict inequality. Option D says both statements are true, which contradicts the analysis. Option E suggests both statements are equal, which misinterprets the situation because the first statement compares two different values and the second compares two equal values, and both comparisons are false as written.

Common Pitfalls:
Many students forget that they must be careful when squaring inequalities and must check that all quantities involved are non negative, which they are in this case. Another common error is to assume √8 is 8^(1/2) and then miscalculate it directly without factorizing. Memorizing that 8 = 4 * 2 and that √4 = 2 makes such problems much easier.

Final Answer:
The correct option is Neither I nor II, because both statements are false when the surds are compared correctly.