If $p, q, r, s$ are in arithmetic progression and $f(x) = \begin{vmatrix} p + \sin x & q + \sin x & p - r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s - q + \sin x \end{vmatrix}$ such that $\int_0^2 f(x) dx = -4$, then the common difference of the progression is:
Step-by-Step Solution
Key Concept: Since p, q, r, s are in AP with common difference d, express them as p, p+d, p+2d, p+3d. Apply row operations on the determinant (specifically R₁ → R₁ + R₃ - 2R₂) to eliminate the sin x terms and reduce f(x) to a constant multiple of d². Use the integral condition ∫₀² f(x)dx = -4 to solve for d².
Given $q = p + d$, $r = p + 2d$, $s = p + 3d$, we construct the determinant $f(x)$ and apply row operations $R_1 \to R_1 + R_3 - 2R_2$ to obtain a simplified form with a zero first row. Using the given condition $\int_0^2 f(x)dx = -4$, we deduce that $\int_0^2 (-2d^2)dx = -4$, which yields $d^2 = 1$, so $d = \pm 1$.
Correct Answer: 1