Consider a sum of money invested at simple interest. The following information is given: I. The sum amounts to ₹690 in 3 years at simple interest. II. The sum amounts to ₹750 in 5 years at simple interest. III. The rate of interest is 5% per annum. Which of the following sets of statements is sufficient to determine the principal sum?

Introduction:
This is a data-sufficiency style question about finding the principal under simple interest. The key idea is that the amount under simple interest is A = P * (1 + r*t/100). Depending on which statements you have, you may be able to solve for the unknowns P and r. We must check whether each pair of statements removes ambiguity.

Given Data / Assumptions:

• Statement I: A = ₹690 when t = 3 years
• Statement II: A = ₹750 when t = 5 years
• Statement III: r = 5% per annum
• Simple interest amount formula: A = P * (1 + r*t/100)

Concept / Approach:
Test each combination: (I + III), (II + III), and (I + II). If each pair allows solving for P uniquely, then “Any two of the three” is correct. Under simple interest, with known r and t, one amount is enough to find P. With two amounts at two times, you can solve for both P and r even if r is not given.

Step-by-Step Solution:
Check I + III: 690 = P * (1 + 5*3/100) = P * 1.15 P = 690 / 1.15 = 600 Check II + III: 750 = P * (1 + 5*5/100) = P * 1.25 P = 750 / 1.25 = 600 Check I + II (without III): 690 = P * (1 + 3r/100) 750 = P * (1 + 5r/100) Subtracting eliminates P after rearranging, giving a unique r, then P is unique.

Verification / Alternative check:
From I and II, the increase in amount from 3 to 5 years is ₹60 over 2 years, so yearly interest is ₹30. Then amount at 3 years is P + 3*30 = 690, giving P = 600. This confirms uniqueness without needing statement III.

Why Other Options Are Wrong:
I and III only is sufficient (so choosing it alone is not the best statement if the correct choice is broader). II and III only is also sufficient. I and II only is sufficient as well, so the best answer is that any two statements are enough.

Common Pitfalls:
Assuming you must always know the rate, or forgetting that two amounts at two times can determine both rate and principal in simple interest because the growth is linear.

Final Answer:
Any two of the three statements are sufficient to determine the principal.