If $f(x) = \begin{vmatrix} (\alpha + 1)\cos^2 x & x & 1-x \\ \beta\sin x & x^2 & 2x \\ (\gamma^2 + 1)\tan x & x & 1-x^2 \end{vmatrix}$
Step-by-Step Solution
Key Concept: Evaluate the determinant f(x) by expanding along rows/columns, then use L'Hôpital's rule repeatedly on limits involving f(x)/x and f(x)/x² as x→0 to determine coefficients α, β, γ. Finally, apply L'Hôpital's rule to the limit of (1/x⁵)∫₀ˣ f(t)dt by differentiating numerator and denominator successively.
Given $f(x)/x$ and $f(x)/x^2$ have specific limits as $x \to 0$, we determine $f(0) = 0$ and $f'(0) = 0$. Computing the matrix determinant and applying L'Hôpital's rule repeatedly, the limit of $\frac{1}{x^5}\int_0^x f(x)dx$ is evaluated by differentiating both numerator and denominator, ultimately yielding $A = \lim_{x \to 0} \frac{3f(x^3) + 9x^3f'(x^3) - 4xf'(x^2)}{24x^3}$.
Correct Answer: [A-q] [B-r, s] [C-p] [D-p]