JEE Main 2026 — 5 April, Morning Shift — Question 48

$\sin \frac{\mathrm{y}}{\mathrm{x}} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}} \sin \frac{\mathrm{y}}{\mathrm{x}}-1$ put $y=t x \Rightarrow \frac{d y}{d x}=t+x \frac{d t}{d x}$ $\sin t\left(t+x \frac{d t}{d x}\right)=t \sin t-1$ $\mathrm{x} \sin \mathrm{t} \frac{\mathrm{dt}}{\mathrm{dx}}+1=0$ $\sin t d t+\frac{d x}{x}=0$ $\Rightarrow-$ cost $+\ell \mathrm{nx}=\mathrm{C}$ $\Rightarrow-\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)+\ell \mathrm{nx}=\mathrm{C}$ $\mathrm{y}(1)=\frac{\pi}{2} \Rightarrow \mathrm{C}=0$ $\Rightarrow \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\ln \mathrm{x}$ $\operatorname{cosec}\left(\frac{\mathrm{y}\left(\mathrm{e}^{12}\right)}{\mathrm{e}^{12}}\right)=12 \Rightarrow \alpha=12$ $x^{2}+y^{2}-2 p x+2 p y+14=0$ $\mathrm{r}=\sqrt{\mathrm{p}^{2}+\mathrm{p}^{2}-14}$ $\Rightarrow \mathrm{r} \leq 6 \Rightarrow \mathrm{r}^{2} \leq 36$ $2 \mathrm{p}^{2}-14 \leq 36$ $\mathrm{p}^{2} \leq 25$ $\mathrm{p} \in[-5,5]$ number of integral value of $p=11$