Oxford University MAT 2015 Solved Papers

E. $\sin \left(\mathit{x}\right)=0$ when $\mathit{x}=0,\mathit{\pi }$, or $2\mathit{\pi }$ in the interval $0\le \mathit{x}\le 2\mathit{\pi }$. Hence $2\cos \left(2\mathit{x}\right)+2=0,\mathit{\pi }$, or $2\mathit{\pi }$. Rearranging gives $\cos \left(2\mathit{x}\right)=-1,\frac{\mathit{\pi }-2}{2}$, or $\mathit{\pi }-1$. $\cos \left(2\mathit{x}\right)$ achieves the first of these values twice for two different values of $\mathit{x}$, achieves the second of these values four times, and never achieves the value of $\mathit{\pi }-1$ and so the answer is $\left(\mathit{d}\right)$.