Consider the following data sufficiency question about simple and compound interest: Question: What is the annual rate of interest (per cent per year)? Statements: (1) The principal amount invested is Rs. 6000. (2) The amount becomes Rs. 6741.60 in 2 years at compound interest. (3) The difference between the compound interest and simple interest for 2 years is Rs. 21.60. Which option correctly describes the sufficiency and redundancy of these statements?

Introduction / Context:
This is a higher level data sufficiency problem involving both simple interest and compound interest. The core question is to determine the annual rate of interest. Three separate pieces of information are provided. Instead of just computing the rate once, we must decide how many and which of these statements are needed, and whether any statement is redundant because the answer can be found without it.

Given Data / Assumptions:

• We are working with interest calculated annually.
• Statement (1): Principal P is Rs. 6000.
• Statement (2): Amount after 2 years at compound interest is Rs. 6741.60.
• Statement (3): Difference between compound interest and simple interest for 2 years is Rs. 21.60.
• We must decide sufficiency and redundancy, not just calculate the rate once.

Concept / Approach:
For two years, compound interest has a standard formula. If the annual rate is r per cent, then amount A = P * (1 + r/100)^2. Simple interest for two years is P * r * 2 / 100. The difference between compound interest and simple interest over two years is P * (r/100)^2. By forming equations from different combinations of statements, we can see whether they determine r uniquely.

Step-by-Step Solution:
Step 1: Using statements (1) and (2). P = 6000, A = 6741.60 and A = P * (1 + r/100)^2 for 2 years of compound interest.Step 2: Substitute values: 6741.60 = 6000 * (1 + r/100)^2, so (1 + r/100)^2 = 6741.60 / 6000 = 1.1236.Step 3: Solve: 1 + r/100 = 1.06, so r = 6 per cent. Thus statements (1) and (2) together are sufficient.Step 4: Using statements (1) and (3). Difference between compound and simple interest for 2 years is P * (r/100)^2.Step 5: So 21.60 = 6000 * (r/100)^2. Hence (r/100)^2 = 21.60 / 6000 = 0.0036, giving r/100 = 0.06 and r = 6 per cent. So statements (1) and (3) are also sufficient.Step 6: Using statements (2) and (3) without statement (1). Let P be unknown. From (2): A = P * (1 + r/100)^2 = 6741.60. From (3): difference between compound and simple interest for 2 years is P * (r/100)^2 = 21.60.Step 7: These form two equations in two unknowns P and r. Solving them simultaneously also yields a unique value of r (again 6 per cent). Thus statements (2) and (3) together are sufficient.Step 8: Therefore any two of the three statements are enough to determine the rate of interest uniquely.

Verification / Alternative check:
After finding r = 6 per cent once, you can back substitute to recover P from statements (2) and (3) alone, confirming that the system is fully determined.No alternative value of r will satisfy all the equations simultaneously, which confirms uniqueness.Thus there is no single statement that is entirely redundant; each one can be left out if the other two are used.

Why Other Options Are Wrong:
Option a treats statement (3) as redundant, but we have shown that (1) and (3) alone are sufficient.Option b focuses only on one pair, (2) and (3), and incorrectly labels (1) as redundant instead of recognising full symmetry.Option d, that none of the statements is useful, is clearly false because each pair gives a complete solution.Option e ignores the sufficiency of (1) and (2), and also ignores the power of (2) and (3).

Common Pitfalls:
Confusing the task of data sufficiency with simply computing the rate once.Forgetting the standard result that, over two years, the difference between compound and simple interest equals P * (r/100)^2.Assuming that if one pair is sufficient, the other statements must be redundant, without checking all combinations.

Final Answer:
The correct interpretation is that Any two of the three statements (1), (2) and (3) are sufficient to find the rate of interest, so there is no single permanently redundant statement.