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Question #40413

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \sin x & \cos x & \sin^3 x \\ -\sin x & -\cos x & 6\sin x - 9\sin^3 x \\ \cos x & -\sin x & 3\sin^2 x \cos x \end{vmatrix}$ . Find the value of $f''\left(\frac{\pi}{12}\right)$ .

Options

A.
$-9\sqrt{2}$
B.
$9\sqrt{2}$
C.
$-18\sqrt{2}$
D.
$18\sqrt{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Compound Angle Expansion|Differentiation of Determinants|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation and differentiation of determinants involving compound trigonometric functions, specifically targeting the 'Row Exchange Parity' trap.

The determinant is structured as a Wronskian with its second and third rows swapped ( $R_2$ contains second derivatives, $R_3$ contains first derivatives).

Students who recognize the Wronskian form often swap the rows to apply standard properties but fail to multiply the determinant by $-1$ , leading to an inverted sign in $f(x)$ and consequently $f''(x)$ .

Alternatively, direct row operations ( $R_2 \to R_2 + R_1$ ) elegantly bypass the trap, rewarding algebraic observation over blind formula application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Compound Angle Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free