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Question #68625

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a thrice differentiable function on $(0, \pi)$ satisfying the differential equation: $\begin{vmatrix} y & \sin x & \cos x \\ y''' & -\cos x & \sin x \\ y' & \cos x & -\sin x \end{vmatrix} \sin x + \begin{vmatrix} y & \sin x & \cos x \\ y' & \cos x & -\sin x \\ y'' & -\sin x & -\cos x \end{vmatrix} \cos x = 0$ Given that $y\left(\frac{\pi}{2}\right) = 1$ , $y'\left(\frac{\pi}{2}\right) = 1$ , and $y''\left(\frac{\pi}{2}\right) = 0$ , find the value of $y\left(\frac{\pi}{6}\right)$ .

Options

A.
$\frac{1}{2} - \frac{\sqrt{3}}{2} + \frac{\pi \sqrt{3}}{12}$
B.
$\frac{1}{2} - \frac{\sqrt{3}}{2} + \frac{\pi \sqrt{3}}{6} - \frac{1}{2} \ln 2$
C.
$\frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{\pi \sqrt{3}}{12}$
D.
$\frac{1}{2} - \frac{\sqrt{3}}{2} - \frac{\pi \sqrt{3}}{6} + \frac{1}{2} \ln 2$

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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

Integration Constant Trap|Differentiation of Determinants|Row/Column Exchange Parity|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the differentiation of determinants and the row-exchange parity trap.

Students must recognize that differentiating the determinant $D(x)$ row-by-row yields a determinant where the second and third rows are swapped compared to the first term in the given equation.

Failing to account for this parity flip (or incorrectly integrating $\cot x$ with the wrong sign) leads to the erroneous differential equation $D'(x) \sin x + D(x) \cos x = 0 \implies D(x) = K \csc x$ .

This forces them to solve $y'' + y = \csc x$ , leading to a completely different particular solution involving logarithmic terms, perfectly matching the distractor option B.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Integration Constant Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free