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JEE Main 4-April-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 4-April-2025 Paper - Shift 24 Marks
MCQ
If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $\left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5, x \geq 1$, then at $x=2$, the value of cosy is:
  • A
    $\frac{2 e^2-e}{64}$
  • B
    $\frac{2 e^2+e}{64}$
  • ✓
    $\frac{2 e ^2- e }{128}$
  • D
    $\frac{2 e^2+e}{128}$
✓

Answer

Correct option: C.
$\frac{2 e ^2- e }{128}$
(C) $\frac{2 e ^2- e }{128}$
$\frac{d y}{d x}=\frac{7 \cot y}{x}-\frac{e^x \operatorname{cosec} y}{x^5} $
$\frac{d y}{d x}=\frac{7 \cot y}{\sin y \cdot x}-\frac{e^x}{\sin y x^5} $
$\sin y \frac{d y}{d x}-\cos y \cdot \frac{7}{x}=\frac{-e^x}{x^5} $
$\operatorname{let}-\cos y=t $
$\sin y \frac{d y}{d x}=\frac{d t}{d x}$
$\frac{d t}{d x}+\frac{7 t}{x}=\frac{-e^x}{x^5} $
$\text { I.F. }=x^7 $
$\text { t. } x^7=-\int x^2 e^x d x $
$\cos y$ $x^7=x^2 e^x-2 \int x^x d x $
$\cos y$ $x^7=x^2 e^x-2 x e^x+2 e^x+c $
$x=1, y=\frac{\pi}{2}, c=-e $
$\cos y=\frac{2 e^2-e}{128}$

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