Before You Begin

Question #41444

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_n = \int_0^{\pi/2} \Delta_n(x) \, dx$ for $n \ge 1$ , where $\Delta_n(x) = \begin{vmatrix} \sin^n x & \sin^{n+2} x & \sin^{n+4} x \\ \sin^n x & \sin^n x \cos^2 x & \sin^n x \cos^4 x \\ 1 & -\sin^2 x & \sin^4 x \end{vmatrix}$ Which of the following recurrence relations holds true?

Options

A.
$2n(n+2) I_n - (2n+1)(n+1) I_{n-1} = 0$
B.
$(2n+1)(n+2) I_n - 2n(n+1) I_{n-1} = 0$
C.
$n(2n+1) I_n - (n+1)(2n+3) I_{n-1} = 0$
D.
$2n(n+1) I_n - (2n+1)(n+2) I_{n-1} = 0$

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

Reduction Formulae|Determinant as Geometric Area/Volume|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Linear Algebra (Determinant properties) and Calculus (Reduction Formulae).

The primary trap lies in the determinant expansion: students who attempt brute-force expansion or arbitrary row operations will generate an unmanageable, high-degree trigonometric polynomial that is virtually impossible to integrate under exam constraints.

The intended path requires recognizing the hidden Vandermonde structure by factoring out common terms from the rows.

Once simplified, the integration demands the application of Wallis' reduction formula to establish a recurrence relation, filtering out students who lack fluency in advanced integration techniques.

Quick Facts About This Question

Question Information

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