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Question #40698

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = 1+\sin x$ and $g(x) = 1-\sin x$ . Let $H(x)$ be the harmonic mean of $f(x)$ and $g(x)$ . Let $y(x)$ be a twice differentiable function on $[0, \pi]$ satisfying the differential equation $H(x) y'' - H'(x) y' = x \cos x$ If $y(\pi) = \pi$ , then the value of $y''\left(\frac{\pi}{2}\right) + y(0)$ is:

Options

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

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

Harmonic Mean Constraints|Linear Differential Equation Structure|Second Order Derivative Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of differential equations, harmonic mean constraints, and removable singularities.

The primary trap lies in the integration constant of the first-order derivative.

Students typically expect an initial condition to find this constant; however, the structural constraint that $y(x)$ is 'twice differentiable' on the entire interval forces the numerator to vanish at the harmonic mean's boundary point ( $x=\pi/2$ ), uniquely determining the constant.

Furthermore, evaluating the second derivative at this boundary requires Taylor series expansion or L'Hopital's rule, punishing mechanical differentiation that leads to an indeterminate form.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Harmonic Mean Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free