Oxford University MAT 2017 Solved Papers

Considering each of the integrals in turn:
(a) note that $\left({\mathit{x}}^2-4\right){\sin }^8\left(\mathit{\pi }\mathit{x}\right)\le 0$ when $0\le \mathit{x}\le 2$ so this integral will evaluate as negative;
(b) $\left(2+\cos \mathit{x}\right)$ will always evaluate positive as it translates the $\mathit{y}=\cos \mathit{x}$ graph positively along the $\mathit{y}$-axis, so the integral will be positive and will be less than $54\mathit{\pi }$ for $0\le \mathit{x}\le 2\mathit{\pi }$;
(c) the maximum value of ${\sin }^{100}\left(\mathit{x}\right)$ is 1 , and so the value of the integral for $0\le \mathit{x}\le \mathit{\pi }$ must be $\le \mathit{\pi }$;
(d) for $0\le \mathit{x}\le \mathit{\pi }$ the smallest ${\left(3-\sin \mathit{x}\right)}^6$ can be is $64\mathit{\pi }$;
(e) $\left({\sin }^3\left(\mathit{x}\right)-1\right)$ will always be $\le 0$, and so this integral will evaluate as negative.