# Q. Two Statements S1 and S2 are given below with regard to two numbers followed by a Question:

S1: Their product is 21.

S2: Their sum is 10.

Question: What are the two numbers?

Which one of the following is correct in respect of the above Statements and the Question?

(a) S1 alone is sufficient to answer the Question.

(b) S2 alone is sufficient to answer the Question.

(c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.

(d) S1 and S2 together are not sufficient to answer the Question.

Correct Answer: (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.

#### Question from UPSC Prelims 2020 CSAT Paper

**Explanation :**

## Product & Sum of two numbers

Let the two numbers be x and y. Then, we have:

S1: xy = 21

S2: x + y = 10

From statement S1, we can express one of the variables in terms of the other. For example, we can write y = 21/x. Substituting this value of y in statement S2, we get:

x + 21/x = 10

Multiplying both sides by x, we get:

x^2 + 21 = 10x

Bringing all the terms to one side, we get:

x^2 – 10x + 21 = 0

This is a quadratic equation that can be factored as (x – 7)(x – 3) = 0. Therefore, the two values of x that satisfy this equation are 7 and 3.

Using statement S1, we can find the corresponding values of y as follows:

If x = 7, then y = 21/x = 3.

If x = 3, then y = 21/x = 7.

Hence, the two numbers that satisfy both statements are 3 and 7. Therefore, the correct answer is (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.