Mathematics - Functions Question with Solution | TestHub

(a) Let $f(x)=x{e}^{\sin x}-\cos x$ $\overrightarrow{f^{\prime }}(x)=e^{\sin x}+x{e}^{\sin x}\cos x+\sin x\ge 0$ For interval $x\in \left(0,\frac{\pi }{2}\right)$, $f$ is strictly increasing. $\therefore f(0)=-1$ and $f\left(\frac{\pi }{2}\right)=\frac{\pi }{2}e\Rightarrow$ one solution (b) Since, $\left|\begin{array}{lll}k& 4& 1\\ 4& k& 2\\ 2& 2& 1\end{array}\right|=0$ $\Rightarrow k(k-4)-4(0)+1(8-2k)=0$ $\Rightarrow k^2-6k+8=0$ $\Rightarrow k=2,4$ (c) Let $y=|x-1|+|x-2|+|x+1|$ $+|x+2|$ For solutions, $4k\ge 6\Rightarrow k\ge 3/2$
$\therefore$ Integer values of $k$ are $2,3,4,5$. (d) Given, $\frac{dy}{dx}=y+1$ $\Rightarrow \ln |(y+1)|=x+C$ $\Rightarrow \ln 2=C\Rightarrow \ln |y+1|=x+\ln 2$ Put $x=\ln 2$ $\therefore \ln (y+1)=\ln 2+\ln 2=\ln 4$ $y+1=4\Rightarrow y=3$