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Q1
Answer
  1. Show that $$1 + \mathrm{e}^{i2\theta} = 2\cos\theta(\cos\theta + i\sin\theta)$$
    1. Given $$\mathrm{C} = 1 + \begin{pmatrix}n\\1\end{pmatrix}\cos 2\theta + \begin{pmatrix}n\\2\end{pmatrix}\cos 4\theta + ... + \cos 2n\theta$$ $$\mathrm{S} = \begin{pmatrix}n\\1\end{pmatrix}\sin 2\theta + \begin{pmatrix}n\\1\end{pmatrix}\sin 4\theta + ... + \sin 2n\theta$$ and by considering $\mathrm{C} + i\mathrm{S}$, find an expression for $\mathrm{S}$.
      1. Factor out $\mathrm{e}^{i\theta}$
      2. $2^n\cos^n\theta\sin n\theta$
      Q2
      Answer
      Find the general solution of the differential equation $$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3$$
        $y = A + B\mathrm{e}^{2x} - \dfrac{3}{2}x$
        Q3
        Answer
        With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a} = \mathbf{i} + 2\mathbf{j}$, $\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ and $\mathbf{c} = 3\mathbf{i} - \mathbf{j} + \mathbf{k}$.
        1. Find $\overrightarrow{AB} \times \overrightarrow{AC}$.
          1. Hence find the area of triangle $ABC$.
            1. Find the equation of the plane containing $A$, $B$ and $C$.
              (a) $-4\mathbf{i} + 7\mathbf{j} - 8\mathbf{k}$ $\quad$ (b) $\dfrac{\sqrt{129}}{2}$ $\quad$ (c) $4x - 7y + 8z = -10$
              1. $\overrightarrow{AB} \times \overrightarrow{AC} = -4\mathbf{i} + 7\mathbf{j} - 8\mathbf{k}$
              2. Area $= \dfrac{\sqrt{129}}{2}$
              3. $4x - 7y + 8z = -10$
              Q4
              Answer
              Two lines $L_1$ and $L_2$ have vector equations $\mathbf{r} = \begin{pmatrix}1\\2\\3\end{pmatrix} + \lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix}3\\0\\1\end{pmatrix} + \mu\begin{pmatrix}1\\-1\\2\end{pmatrix}$.
              1. Show that $L_1$ and $L_2$ intersect and find the point of intersection.
                1. Find the acute angle between $L_1$ and $L_2$.
                  (a) Intersection at $(3, 3, 2)$ $\quad$ (b) $\approx 79.1°$
                  1. Point of intersection: $(3, 3, 2)$
                  2. Acute angle $\approx 79.1°$