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STD 12 - 9. differential equations — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 9. differential equations4 Marks
MCQ
Let $y=y(x)$ satisfies the equation $\frac{d y}{d x}-|A|=0$, for all $x>0$, where $A=\left[\begin{array}{ccc}y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{array}\right] .$

If $y(\pi)=\pi+2$, then the value of $y\left(\frac{\pi}{2}\right)$ is:

  • A
    $\frac{\pi}{2}-\frac{4}{\pi}$
  • B
    $\frac{\pi}{2}-\frac{4}{\pi}$
  • C
    $\frac{\pi}{2}-\frac{1}{\pi}$
  • ✓
    $\frac{\pi}{2}+\frac{4}{\pi}$
✓

Answer

Correct option: D.
$\frac{\pi}{2}+\frac{4}{\pi}$
d
$|\mathrm{A}|=\frac{-\mathrm{y}}{\mathrm{x}}+2 \sin x+2$

$\frac{\mathrm{dy}}{\mathrm{dx}}=|\mathrm{A}|$

$\frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}=\frac{\mathrm{y}}{\mathrm{x}}+2 \sin \mathrm{x}+2$

$\frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}+\frac{-\mathrm{y}}{\mathrm{x}}=2 \sin x+2$

$\text { I.F. }=\mathrm{e}^{\int \frac{1}{\mathrm{x}} \mathrm{dx}}=\mathrm{x}$

$\Rightarrow \mathrm{yx}=\int \mathrm{x}(2 \sin \mathrm{x}+2) \mathrm{d} \mathrm{x}$

$\mathrm{xy}=\mathrm{x}^{2}-2 \mathrm{x} \cos x+2 \sin x+\mathrm{c}.....(i)$

Now $x=\pi, y=\pi+2$

Use in $(i)$

$c=0$

Now $(i)$ be comes

$x y=x^{2}-2 x \cos x+2 \sin x$

put $x=\frac{\pi}{2}$

$\frac{\pi}{2} y=\frac{\pi^{2}}{4}+2$

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