Question

If \(x \sin \left(\frac{y}{x}\right) d y=\left[y \sin \left(\frac{y}{x}\right)-x\right] d x, x>0\) and \(y(1)=\frac{\pi}{2}\) then the value of \(\cos \left(\frac{y}{x}\right)\) is
(A) 1
(B) \(\log x\)
(C) \(\mathrm{e}\)
(D) 0

Answer 1

Ans: (B)
Hint \(: \operatorname{sinv}\left(v+x \cdot \frac{d v}{d x}\right)=v \sin v-1\)

$$
\begin{aligned}
&\Rightarrow x \sin v \cdot \frac{d v}{d x}=-1 \\
&\Rightarrow \int \sin v d v=-\int \frac{d x}{x} \\
&\Rightarrow-\cos v=-\log x+c
\end{aligned}
$$
$$
\begin{aligned}
&\text { at } x=1 ; y=\frac{\pi}{2} ; c=0 \\
&\cos \left(\frac{y}{x}\right)=\log x
\end{aligned}
$$

Answer 2

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Answer 3

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