Two statements S1 and S2 are given below with regard to two numbers followed by a Question:

Correct Option (3)

Let the two numbers be denoted as x and y.

Statement S1 provides their product: xy = 21.

Statement S2 provides their sum: x + y = 10.

To determine the two numbers, both statements must be considered together. From S2, we can express x in terms of y: x = 10 - y. Substituting this expression for x into S1 yields:

(10 - y)y = 21

Expanding the equation gives:

10y - y² = 21

Rearranging into a standard quadratic form:

y² - 10y + 21 = 0

Factoring the quadratic equation:

y² - 7y - 3y + 21 = 0

y(y - 7) - 3(y - 7) = 0

This leads to the solutions for y:

y = 7,3

If y = 7, then from x = 10 - y, x = 10 - 7 = 3.

If y = 3, then from x = 10 - y, x = 10 - 3 = 7.

Therefore, the two numbers are uniquely determined as 3 and 7. The order of the numbers does not alter the pair. Thus, S1 and S2 together are sufficient to answer the question.

Incorrect Options:

S1 alone is not sufficient: Statement S1 (xy = 21) provides numerous pairs of numbers whose product is 21. Examples include (1, 21), (3, 7), (-1, -21), (0.5, 42), and various non-integer or negative pairs. Without additional information, the specific two numbers cannot be uniquely identified.

S2 alone is not sufficient: Statement S2 (x + y = 10) allows for an infinite number of pairs whose sum is 10. Examples include (1, 9), (2, 8), (4, 6), (5, 5), (0.5, 9.5), and numerous other integer, non-integer, or negative combinations. Consequently, S2 alone is insufficient to uniquely determine the two numbers.

S1 and S2 together are not sufficient: This option is incorrect because, as demonstrated above, combining both statements leads to a unique solution for the pair of numbers.