Q11 (b)

Criteria	Marks
Provides correct solution	3
Correctly applies integration by parts, or equivalent merit	2
Correctly identifies the two functions to be used in integration by parts, or equivalent merit	1

Sample answer:

$\text{Dot product: }\begin{pmatrix}2\\0\\4\end{pmatrix}\cdot\begin{pmatrix}-3\\1\\2\end{pmatrix} = -6+0+8 = 2$

Magnitudes:

$\left|\begin{pmatrix}2\\0\\4\end{pmatrix}\right|=\sqrt{2^2+0^2+4^2}=\sqrt{20}$

$\left|\begin{pmatrix}-3\\1\\2\end{pmatrix}\right|=\sqrt{(-3)^2+1^2+2^2}=\sqrt{14}$

$\sqrt{20}\sqrt{14}\cos\theta=2$
$\cos\theta=0.1195\dots$
$\theta=83.135^\circ\dots\approx83.1^\circ\text{ (1 d.p.)}$