Q46. 46. Let p + q = 10, where p, q are integers. Value-I = Maximum value of p × q when p, q are positive integers. Value-II = Maximum value of p × q when p ≥- 6, q ≥ -4.

Which one of the following is correct?

Given p + q = 10, where p and q are integers. Value-I: Maximum value of p × q when p, q are positive integers. For a fixed sum, the product of two numbers is maximized when the numbers are as close to each other as possible.

Since p + q = 10, and p, q are positive integers, the closest integers are p=5 and q=5. Their product is 5 * 5 = 25. So, Value-I = 25. Value-II: Maximum value of p × q when p ≥ -6, q ≥ -4.

Again, p + q = 10. The product pq can be written as p(10-p). This is a quadratic function of p, f(p) = 10p - p², which represents a downward-opening parabola. The maximum value occurs at the vertex, where p = -10 / (2 * -1) = 5.

If p=5, then q = 10 - 5 = 5. These values satisfy the constraints p ≥ -6 and q ≥ -4. At p=5, q=5, the product pq = 5 * 5 = 25. Let's check the boundary conditions: If p = -6, then q = 10 - (-6) = 16.

This satisfies q ≥ -4. Product = (-6) * 16 = -96. If q = -4, then p = 10 - (-4) = 14. This satisfies p ≥ -6. Product = 14 * (-4) = -56. The maximum product within the given constraints is 25.

So, Value-II = 25. Comparing Value-I and Value-II: Value-I = 25 Value-II = 25 Therefore, Value-I = Value-II. This question tests understanding of number properties and optimization within given constraints.