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Statement-1: For two different roots α and β of the equation a cos x + b sin x = c, the quantity (α + β)/2 does not depend on c. Statement-2: The equation a cos x + b sin x = c has a solution whenever −√(a^2 + b^2) ≤ c ≤ √(a^2 + b^2).

1. Statement-1 is correct, Statement-2 is correct; Statement-2 correctly explains Statement-1
2. Statement-1 is correct, Statement-2 is correct; Statement-2 does not correctly explain Statement-1
3. Statement-1 is incorrect, Statement-2 is correct
4. Statement-1 is correct, Statement-2 is incorrect

Correct answer: Statement-1 is correct, Statement-2 is correct; Statement-2 correctly explains Statement-1

Solution

Statement-1 is correct because the average of the roots of the equation is determined by the coefficients of the trigonometric functions, which remain constant regardless of the value of c. Statement-2 is also correct as it defines the range of c for which the equation has real solutions, thereby supporting the validity of Statement-1.

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