Which of the following statements involving square roots is/are true? I. √11 + √7 < √10 + √8 II. √17 + √11 > √15 + √13

Introduction / Context:
This question tests your ability to compare expressions containing square roots without a calculator. Instead of exact decimal values, you are expected to use estimation and inequality reasoning to decide which of two sums of square roots is larger. Such questions are common in aptitude tests because they strengthen number sense and the ability to compare approximate magnitudes.

Given Data / Assumptions:

• Statement I claims: √11 + √7 < √10 + √8.
• Statement II claims: √17 + √11 > √15 + √13.
• All square roots are positive principal square roots.
• We must decide which statements are true.

Concept / Approach:
Direct squaring of sums like (√a + √b)^2 expands into a + b + 2√(ab), which can be compared, but often a simpler approach is to estimate approximate decimal values of each root. For small integers, square roots are easy to estimate using nearby perfect squares. After getting reasonable approximations, we compare the two sums in each statement. If the difference is consistent and not extremely tiny, we can confidently label the inequality as true or false.

Step-by-Step Solution:
Consider Statement I: √11 + √7 versus √10 + √8. Use nearby perfect squares. 9 and 16 are squares of 3 and 4. So √10 is a little more than 3.16 and √11 a little more than 3.31. Similarly, 4 and 9 are squares of 2 and 3, so √7 is around 2.64 and √8 around 2.83. Approximate sums: √11 + √7 ≈ 3.32 + 2.65 ≈ 5.97. √10 + √8 ≈ 3.16 + 2.83 ≈ 5.99. So √10 + √8 is slightly larger. Therefore √11 + √7 < √10 + √8 appears to be true. Now consider Statement II: √17 + √11 versus √15 + √13. Estimate values: √17 is just above 4.12, √11 about 3.32, giving √17 + √11 ≈ 4.12 + 3.32 = 7.44. √15 is about 3.87 and √13 about 3.61, so √15 + √13 ≈ 3.87 + 3.61 = 7.48. Comparing 7.44 and 7.48, the second sum is slightly larger. So √17 + √11 is actually less than √15 + √13, not greater. Hence Statement II is false.

Verification / Alternative check:
For a more algebraic check, you could compare the squares of each sum. However, the approximate calculations already show a clear consistent difference in both cases. For Statement I, even if the approximations are slightly off, the values on the right side remain a bit larger than those on the left, confirming the inequality √11 + √7 < √10 + √8. For Statement II, the left side sum is consistently a little smaller than the right side sum, making the claimed inequality √17 + √11 > √15 + √13 incorrect.

Why Other Options Are Wrong:
Option B claiming that only II is true is wrong because we have shown II is false. Option C claiming both statements are true is wrong because II fails. Option D claiming neither is true is wrong because Statement I is supported by our estimates. Only option A correctly recognises that exactly Statement I is true and Statement II is false.

Common Pitfalls:
A common mistake is to assume that because 11 + 7 equals 10 + 8, the sums of square roots must also be equal or hard to compare, leading to confusion. Another error is to compare each pair separately, such as deciding whether √11 is greater than √10, without considering the effect of the entire sum. Candidates also sometimes reverse inequalities when working too quickly. Carefully estimating each root using nearby perfect squares and then adding them helps avoid these issues.

Final Answer:
Only the first inequality, √11 + √7 < √10 + √8, is true, so the correct choice is Only I, which corresponds to option A.