Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
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:
$\pm 1$
$\frac{1}{2}$
$\pm 2$
None of these
Step-by-Step Solution
Key Concept: Row operations on determinants preserve their value and can reveal structure; the integral of a constant multiple gives a direct relationship to find unknown parameters.
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