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Question #41526

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Delta(x) = \begin{vmatrix} \sin x & \cos x & 1 \\ \cos x & 1 & \sin x \\ 1 & \sin x & \cos x \end{vmatrix}$ .

If $\int \frac{(\cos x - \sin x)(\sin x + \cos x + 1)}{\Delta(x)} \, dx = \frac{1}{\sqrt{k}} \ln \left| \frac{\sin x + \cos x + 1 - \sqrt{k}}{\sin x + \cos x + 1 + \sqrt{k}} \right| + C$ where $C$ is the constant of integration, then the value of $k$ is:

Options

A.
3
B.
5
C.
6
D.
8

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

Logarithmic Integration via Trig Manipulation|Determinant Product Rule|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of determinant properties and logarithmic integration.

The primary trap is related to the 'Determinant Product Rule' and premature expansion: students often mistakenly expand the determinant first, leading to a divergent expression before integration.

By applying the Factor Theorem for cyclic determinants ( $a^3+b^3+c^3-3abc$ ), the determinant factors cleanly, allowing a crucial cancellation with the numerator.

The subsequent integration requires a standard substitution ( $t = \sin x + \cos x$ ) that transforms the integrand into a rational function, culminating in a classic logarithmic integral.

This perfectly mirrors the JEE Advanced pattern of hiding simple integrals behind intimidating algebraic structures.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Logarithmic Integration via Trig Manipulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free