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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 −√(a² + b²) ≤ c ≤ √(a² + b²).

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 does not correctly explain Statement-1

Solution

Writing a cosx + b sinx = R cos(x - phi), the two roots are phi +/- t, so their average is phi, independent of c; Statement-1 is true. Statement-2 (solution exists iff |c| <= sqrt(a^2+b^2)) is also true, but it concerns existence, not the sum of roots, so it does not explain Statement-1.

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