Correct Option (2)
The problem requires comparing two maximum values, Value-I and Value-II, derived from the equation p + q = 10, where p and q are integers.
Calculation of Value-I:
Value-I represents the maximum value of p × q when p and q are positive integers such that p + q = 10.
The pairs of positive integers (p, q) that sum to 10 are:
- (1, 9) → p × q = 1 × 9 = 9
- (2, 8) → p × q = 2 × 8 = 16
- (3, 7) → p × q = 3 × 7 = 21
- (4, 6) → p × q = 4 × 6 = 24
- (5, 5) → p × q = 5 × 5 = 25
- (6, 4) → p × q = 6 × 4 = 24
- (7, 3) → p × q = 7 × 3 = 21
- (8, 2) → p × q = 8 × 2 = 16
- (9, 1) → p × q = 9 × 1 = 9
The maximum product obtained is 25. Therefore, Value-I = 25.
Calculation of Value-II:
Value-II represents the maximum value of p × q when p and q are integers such that p + q = 10, with additional constraints p ≥ 6 and q ≥ 4.
We examine integer pairs (p, q) that satisfy all given conditions:
- If p = 6: q = 10 - 6 = 4. This pair (6, 4) satisfies p ≥ 6 and q ≥ 4. The product p × q = 6 × 4 = 24.
- If p = 7: q = 10 - 7 = 3. This pair (7, 3) satisfies p ≥ 6 but does not satisfy q ≥ 4. Thus, it is an invalid pair.
- For any p > 7 (e.g., p = 8, 9, 10), the corresponding value of q (10 - p) will be less than 3, failing the condition q ≥ 4. Hence, no further valid pairs exist.
The only valid pair satisfying all conditions is (6, 4), yielding a product of 24. Therefore, Value-II = 24.
Comparison:
Comparing the calculated values:
- Value-I = 25
- Value-II = 24
It is evident that Value-II < Value-I (24 < 25).
Incorrect Options:
Option 1 (Value-I < Value-II): This is incorrect because 25 is not less than 24.
Option 3 (Value-I = Value-II): This is incorrect because 25 is not equal to 24.
Option 4 (Cannot be determined due to insufficient data): This is incorrect as both Value-I and Value-II can be precisely determined from the given information and constraints.
