'S' undertook an investment endeavor by investing ₹'a + 1000' into a scheme yielding a simple interest rate of 20% per annum for a duration of three years. Simultaneously, 'A' ventured into a financial enterprise by investing ₹'a - 4000' in a compound interest scheme at a 20% annual rate for a three-year period. If the difference between interest earned from both investments is ₹1,080, then which of the following statement(s) is/are true? I. The variable 'a' can take multiple values. II. The minimal value for 'a' is less than 10,000. III. 'a' is an even integer.

### Concept & Interest Calculation We need to calculate both Simple Interest (SI) and Compound Interest (CI) using the given formulas and find the absolute difference between them to solve for $a$. Since it's an absolute difference, we must evaluate both cases (SI > CI and CI > SI). $$ SI = \frac{P \cdot R \cdot T}{100} $$ $$ CI = P \left(1 + \frac{R}{100}\right)^T - P $$ ### Step-by-Step Solution 1. **Calculate Simple Interest (SI) for 'S':** Principal ($P$) = $a + 1000$, Rate ($R$) = $20\%$, Time ($T$) = 3 years. $SI = (a + 1000) \times 20 \times \frac{3}{100} = 0.6(a + 1000) = 0.6a + 600$ 2. **Calculate Compound Interest (CI) for 'A':** Principal ($P$) = $a - 4000$, Rate ($R$) = $20\%$, Time ($T$) = 3 years. $CI = (a - 4000) \times \left[(1 + 0.2)^3 - 1\right]$ $CI = (a - 4000) \times (1.728 - 1) = (a - 4000) \times 0.728 = 0.728a - 2912$ 3. **Use the given difference (₹1,080) to solve for $a$:** Since we don't know which interest is larger, we have two cases: *Case 1: $SI - CI = 1080$* $(0.6a + 600) - (0.728a - 2912) = 1080$ $-0.128a + 3512 = 1080$ $0.128a = 2432 \implies a = \frac{2432}{0.128} = 19000$ *Case 2: $CI - SI = 1080$* $(0.728a - 2912) - (0.6a + 600) = 1080$ $0.128a - 3512 = 1080$ $0.128a = 4592 \implies a = \frac{4592}{0.128} = 35875$ 4. **Evaluate the Statements:** * I. The variable 'a' can take multiple values: True ($a$ can be 19000 or 35875). * II. The minimal value for 'a' is less than 10,000: False (The minimal value is 19000). * III. 'a' is an even integer: False (35875 is an odd integer). ### Exam Strategy & Shortcut Whenever a question asks about the "difference" between two variables without specifying which is larger, always set up absolute value equations leading to two distinct cases. Evaluating one case only is the most common trap. ### Common Pitfall Students often forget to subtract the principal from the amount when calculating Compound Interest, resulting in an incorrect expression for CI. The formula $P(1+r/n)^{nt}$ gives the Amount, not the Interest. ### Final Answer Therefore, the correct answer is **Only I**.