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5. Continuity and Differentiation
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Functions $f(x)$ and $g(x)$ are such that $f(x) + \int\limits_0^x {g(t)dt = 2\,\sin \,x\, - \,\frac{\pi }{2}} $ and $f'(x).g (x) = cos^2\,x$ , then number of solution $(s)$ of equation $f(x) + g(x) = 0$ in $(0,3 \pi$) is-
A
$0$
B
$1$
C
$2$
D
$3$
Solution
$\text { Given } f^{\prime}(x)+g(x)=2 \cos x$ …….$(1)$
and $f^{\prime}(x) . g(x)=\cos ^{2} x$ ………$(2)$
$(2) \Rightarrow f^{\prime}(\mathrm{x})=\frac{\cos ^{2} \mathrm{x}}{\mathrm{g}(\mathrm{x})}$
$\therefore(1) \Rightarrow \frac{\cos ^{2} x}{g(x)}+g(x)=2 \cos x$
$\Rightarrow(g(x)-\cos x)^{2}=0 \Rightarrow g(x)=\cos x$
and $f(\mathrm{x})=-\frac{\pi}{2}+\sin \mathrm{x}$
$\Rightarrow f(\mathrm{x})+\mathrm{g}(\mathrm{x})=0$
$\Rightarrow \cos x+\sin x=\frac{\pi}{2}$ has no solution.
Standard 12
Mathematics
