- A. Value-I < Value-II
- B. Value-II < Value-I
- C. Value-I = Value-II
- D. Cannot be determined due to insufficient data
Answer: C
Explanation
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.