The function $\mathrm{f} $is defined as follows: $\mathrm{f}:x\mapsto 3\cos x-2\sin x,x\in \mathbb{R

Question

The function $\mathrm{f} $is defined as follows: $\mathrm{f}:x\mapsto 3\cos x-2\sin x,x\in \mathbb{R},-\pi \le x\le \pi $. Write $\mathrm{f}\left( x \right)$ as $R\cos \,\left( x+\alpha \right)$, where $R$ and $\alpha $ are constants to be found. Hence, or otherwise, find the range of $\mathrm{f}$ and sketch the curve. [4] The function $\mathrm{g}$ is defined as follows: $\mathrm{g}:x\mapsto 3\cos x-2\sin x,x\in \mathbb{R},-\alpha \le x\le b$. Given that the function ${{\mathrm{g}}^{-1}}$ exists, write down the largest value of $b$. Find ${{\mathrm{g}}^{-1}}\left( x \right)$. [3]

Accepted Answer

(i) $R=\sqrt{13}$, $\alpha=\tan^{-1}\left(\dfrac23\right)=0.588$ rad, $R_{\mathrm f}=[-\sqrt{13},\sqrt{13}]$ (ii) $b=\pi-\tan^{-1}\left(\dfrac23\right)=2.55$ rad, $\mathrm g^{-1}(x)=\cos^{-1}\left(\dfrac x{\sqrt{13}}\right)-\tan^{-1}\left(\dfrac23\right)$, $D_{\mathrm g^{-1}}=[-\sqrt{13},\sqrt{13}]$

2017 Specimen Paper P2 Q1

2017 Specimen Paper P2 Q1

Video Solution:

Solution:

Solution locked