Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is / are sufficient to answer the given question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. $P, Q, R$ and $S$ are four consecutive even numbers. What is the value of the largest integer among these? I. The average of the four numbers is the first prime number greater than 10. II. The ratio between the largest and the smallest of the numbers is less than 10.

### Concept & Logic Consecutive even numbers can be represented algebraically with a fixed gap of 2. If we can find the exact value of the starting variable, we can determine the entire sequence. $$ \text{Sequence:} \quad x, x+2, x+4, x+6 $$ ### Step-by-Step Solution * **Analyze the Core Question:** Let the four consecutive even numbers $P, Q, R, S$ be $x, x+2, x+4$, and $x+6$ respectively. The largest integer is $S = x+6$. * **From Statement I:** The first prime number greater than 10 is 11. The average of the sequence is given as 11. $$ \frac{x + (x+2) + (x+4) + (x+6)}{4} = 11 $$ $$ 4x + 12 = 44 \implies 4x = 32 \implies x = 8 $$ The sequence is 8, 10, 12, 14. The largest number is 14. We can find a unique integer. Thus, Statement I alone is sufficient. * **From Statement II:** The ratio of the largest to the smallest is less than 10. $$ \frac{x+6}{x} < 10 $$ This is a boundary inequality, not a fixed equation. If $x = 2$, the ratio is $8/2 = 4 < 10$. If $x = 4$, the ratio is $10/4 = 2.5 < 10$. There are multiple valid sets of numbers that satisfy this condition. We cannot find a unique largest integer. Thus, Statement II alone is not sufficient. ### Exam Strategy & Shortcut For any defined sequence (like consecutive evens), knowing the arithmetic mean is algebraically identical to knowing the sum. Knowing the sum of a strictly defined sequence always allows you to solve for the specific numbers. Therefore, Statement I is an instant yes. Conversely, open-ended inequalities (Statement II) almost never yield a single specific integer value in Data Sufficiency unless they create a rigid sandwich constraint. ### Common Pitfall Students might waste valuable time trying to solve the inequality in Statement II to see if it somehow restricts $x$ to exactly one integer. Recognizing that an upper-bound ratio for positive even integers allows infinite valid solutions (as numbers get larger, the ratio of $(x+6)/x$ simply approaches 1) immediately invalidates Statement II. ### Final Answer **Therefore, the correct answer is Data in Statement I alone are sufficient.**