Q37 / 75 · JEE Main 2025 (23 Jan Shift 2) · Mathematics

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :

  • A
    $1-2\left(\log _e 2\right)^2$
  • B
    $1-2\left(\log _{\mathrm{e}} 2\right)$
  • C
    $2\left(\log _e 2\right)-1$
  • Correct answer
    D
    $2\left(\log _e 2\right)^2-1$

Solution

$\begin{aligned} & y d y=(x d y-y d x) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\ & \rightarrow \ell n y=\cos \frac{x}{y}+C\end{aligned}$ $\begin{aligned} & x(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+C \Rightarrow C=0 \\ & \text { lny }=\cos \frac{x}{y} \\ & \text { but } y=2 \Rightarrow \cos \frac{x}{2}=\ell n 2 \\ & \cos x=2 \cos ^2 \frac{x}{2}-1 \\ & =2(\ln 2)^2-1\end{aligned}$

Hence, the correct answer is option: 'D'.

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