Directions: Each of the following questions consists of a question followed by three statements I, II and III. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What was the total number of candidates appeared at the examination? I. $30\%$ of appeared candidates succeeded in the examination. II. The number of unsuccessful candidates was 1000 more than the successful candidates. III. 1750 candidates were unsuccessful.

### Concept & Logic This percentage problem revolves around a population split into two complementary groups: Successful ($S$) and Unsuccessful ($U$). The total ($T$) is simply $S + U$. We need to find pairs of statements that can uniquely solve for $T$. ### Step-by-Step Solution * Let Total Candidates = $T$, Successful Candidates = $S$, and Unsuccessful Candidates = $U$. * We know universally that $T = S + U$. * Let's evaluate the statements algebraically: * Statement I: $S = 0.30T$ (which implies $U = 0.70T$) * Statement II: $U = S + 1000$ * Statement III: $U = 1750$ * Test Combination (I and II): * Substitute I into II: $0.70T = 0.30T + 1000$ * $0.40T = 1000 \implies T = 2500$. (Sufficient) * Test Combination (I and III): * Substitute III into the implication of I: $0.70T = 1750$ * $T = \frac{1750}{0.70} = 2500$. (Sufficient) * Test Combination (II and III): * From III, $U = 1750$. Substitute into II: $1750 = S + 1000 \implies S = 750$. * $T = S + U = 750 + 1750 = 2500$. (Sufficient) * Since any pair of the three statements leads to the same unique total, any two statements are sufficient. ### Exam Strategy & Shortcut Recognize the components of a simple linear system. You have two variables ($S$ and $U$) that dictate the Total. Each statement gives you one independent linear equation: I gives a ratio, II gives a difference, III gives an absolute value. Any two distinct linear equations involving these two variables will allow you to solve the system. ### Common Pitfall Stopping after finding that I and II work, without testing the other combinations. In options like "Any two of the three", you must quickly verify that the remaining pairings (I & III, II & III) also lead to a solution. ### Final Answer **Therefore, the correct answer is Any two of the three.**