Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What is the principal sum ? I. The sum amounts to ₹ 690 in 3 years at S.I. II. The sum amounts to ₹ 750 in 5 years at S.I. III. The rate of interest is 5% p.a.

### Concept & Finding Principal with Three Variables To find the Principal ($P$), we need to find a way to isolate it using the available Simple Interest variables: Amount ($A$), Rate ($R$), Time ($T$), and Simple Interest ($SI$). $$A = P + \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution * **From the Question:** We need to calculate the exact principal sum ($P$). * **Analyze Statements I & II (Combined):** * Statement I gives the amount after 3 years: $P + 3(SI) = 690$ * Statement II gives the amount after 5 years: $P + 5(SI) = 750$ * Subtracting the two equations gives the interest for 2 years: $2(SI) = 750 - 690 = 60$. So, $SI = 30$ per year. * Substitute back into the first equation: $P + 3(30) = 690 \Rightarrow P = 600$. (I and II are sufficient). * **Analyze Statements I & III (Combined):** * Statement I gives $A = 690$ at $T = 3$. * Statement III gives $R = 5\%$. * Using the amount formula: $690 = P + \frac{P \times 5 \times 3}{100} \Rightarrow 690 = P(1 + 0.15) \Rightarrow 1.15P = 690 \Rightarrow P = 600$. (I and III are sufficient). * **Analyze Statements II & III (Combined):** * Statement II gives $A = 750$ at $T = 5$. * Statement III gives $R = 5\%$. * Using the amount formula: $750 = P + \frac{P \times 5 \times 5}{100} \Rightarrow 750 = P(1 + 0.25) \Rightarrow 1.25P = 750 \Rightarrow P = 600$. (II and III are sufficient). * Since combining any two pairs of statements allows us to uniquely determine the principal, any two of the three are sufficient. ### Exam Strategy & Shortcut Notice the variables provided. Statement I gives $(A, T)$. Statement II gives a second set of $(A, T)$. Statement III gives $(R)$. To solve for $P$, you either need two different $(A, T)$ points to find the linear slope (which gives yearly SI), OR you need one $(A, T)$ point and the rate $(R)$. Therefore, any combination of two statements fulfills the mathematical requirements. ### Common Pitfall A common pitfall is stopping after realizing I and II work, and hastily selecting "I and II only" without checking if pairing the rate (III) with one of the amounts also yields a solvable equation. ### Final Answer Therefore, the correct answer is **Any two of the three**.