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Question #41503

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} 2 & 2\cos x & 2\cos 2x \\ 2 & 2\cos 2x & 2\cos 4x \\ 2 & 2\cos 3x & 2\cos 6x \end{vmatrix}$ . Evaluate the integral $\displaystyle I = \int_0^\pi f(x) \, dx$ .

Options

A.
$-4\pi$
B.
$-2\pi$
C.
$-\pi$
D.
$4\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Trigonometric Identical Substitutions|Factor Theorem in Determinants|Determinant Scaling Property

Why This Question Is Tricky

Common Mistakes

The integral contains a determinant with trigonometric functions.

By using the Factor Theorem (Vandermonde structure), the determinant reduces to a product of differences of cosines.

The primary trap is 'Determinant Scaling': students often pull the constant 2 out of the matrix as a single factor rather than $2^3=8$ for the three rows, leading to a constant multiplier error.

Furthermore, extracting the constant term from the trigonometric product tests advanced algebraic manipulation, as integrating the full expansion is prohibitively time-consuming.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Trigonometric Identical Substitutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free