Which of the following statements comparing sums of square roots is or are true? I. √5 + √5 > √7 + √3 II. √6 + √7 > √8 + √5 III. √3 + √9 > √6 + √6

Introduction / Context:
This question checks your intuition and computational skill for comparing sums of square roots without a calculator. Each statement compares two sums, and you must decide which inequalities hold. Such problems are common in simplification and number sense sections of aptitude tests.

Given Data / Assumptions:

• Statement I: √5 + √5 > √7 + √3.
• Statement II: √6 + √7 > √8 + √5.
• Statement III: √3 + √9 > √6 + √6.
• We must determine which subset of these statements is true.

Concept / Approach:
We approximate each square root to a reasonable number of decimal places and then add. Because all values are positive and reasonably spaced, approximations with standard values like √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √8 ≈ 2.828, and √9 = 3 give reliable comparisons. We then check whether each inequality sign is consistent with the calculated sums.

Step-by-Step Solution:
1. For Statement I, compute left and right sums. 2. Left side: √5 + √5 = 2√5 ≈ 2 * 2.236 = 4.472. 3. Right side: √7 + √3 ≈ 2.646 + 1.732 = 4.378. 4. Since 4.472 > 4.378, Statement I is true. 5. For Statement II, compute both sums. 6. Left side: √6 + √7 ≈ 2.449 + 2.646 = 5.095. 7. Right side: √8 + √5 ≈ 2.828 + 2.236 = 5.064. 8. Since 5.095 > 5.064, Statement II is also true. 9. For Statement III, evaluate the two sums. 10. Left side: √3 + √9 = √3 + 3 ≈ 1.732 + 3 = 4.732. 11. Right side: √6 + √6 = 2√6 ≈ 2 * 2.449 = 4.898. 12. Here 4.732 < 4.898, so the inequality √3 + √9 > √6 + √6 is false. 13. Therefore Statements I and II are true, but Statement III is false.

Verification / Alternative check:
For Statement III, another way is to square both sides carefully. Left side squared is (√3 + 3)^2 = 3 + 9 + 6√3 = 12 + 6√3. Right side squared is (2√6)^2 = 4 * 6 = 24. Since √3 is about 1.732, 6√3 is about 10.392, so 12 + 6√3 ≈ 22.392, which is still less than 24. So the right side remains larger, confirming Statement III is false.

Why Other Options Are Wrong:
Option 1 (Only I) ignores the fact that Statement II is true.
Option 3 (Only II and III) incorrectly includes Statement III, which is false.
Option 4 (Only I and III) similarly includes the false third statement and excludes true Statement II.

Common Pitfalls:
Learners sometimes over approximate roots or misremember standard values, which can reverse inequalities if the sums are very close. Another mistake is squaring both sides without expanding correctly, especially with expressions like (a + b)^2. Using consistent approximate values and carefully recomputing avoids such errors.

Final Answer:
The correct choice is that only Statements I and II are true.