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Question #41443

Difficulty: 8/10 | 🟢 Free

The Question

Let $I(x) = \int_{\pi/2}^x \cos^3 t \, dt$ . A curve $C$ is defined by $y = f(x)$ for $x \in (\pi/2, \pi)$ , which satisfies the differential equation: $\frac{dy}{dx} + \frac{3 \cos^3 x}{|I(x)|} y = 4 \cos^3 x |I(x)|$ Given that $y(3\pi/4) = 4 \left( I(3\pi/4) \right)^2$ . A particle moves along the curve $C$ . At the instant when $x = 3\pi/4$ , its $x$ -coordinate is increasing at a rate of $\sqrt{2}$ units/sec. Find the rate of change of its $y$ -coordinate at this instant.

Options

A.
$\frac{8 - 5\sqrt{2}}{3}$
B.
$\frac{5\sqrt{2} - 8}{3}$
C.
$\frac{16 - 10\sqrt{2}}{3}$
D.
$\frac{10\sqrt{2} - 16}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Reduction Formulae|Integrating Factor (IF) Sign Sensitivity|Implicit Differentiation Rate

Why This Question Is Tricky

Common Mistakes

The problem presents a differential equation where the IF is derived through an integral requiring reduction (Wallis-style integral of $\cos^3 t$ ).

The student must use implicit differentiation to link the rates of change.

The trap lies in the Sign Sensitivity of the IF; ignoring the absolute value inside the log-derivative causes a sign error in the integration index (yielding an IF of $I(x)^3$ instead of $-I(x)^{-3}$ ).

This false IF propagates through the initial value problem, creating a non-zero integration constant that perfectly inverts the final differential rate, leading the student to a trap answer.

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