Click here to do these on a digital whiteboard (will not work on small screens).
Q1
Answer
  1. Show that $$1-\mathrm{e}^{i\theta} = -2i\mathrm{e}^{i\tfrac{\theta}{2}}\sin\tfrac{\theta}{2}$$
    1. Given $$\mathrm{C} = 1 - \begin{pmatrix}n\\1\end{pmatrix}\cos\theta + \begin{pmatrix}n\\2\end{pmatrix}\cos2\theta - \begin{pmatrix}n\\3\end{pmatrix}\cos3\theta + ...$$ $$\mathrm{S} = - \begin{pmatrix}n\\1\end{pmatrix}\sin\theta + \begin{pmatrix}n\\2\end{pmatrix}\sin2\theta - \begin{pmatrix}n\\3\end{pmatrix}\sin3\theta + ...$$ and by considering $\mathrm{C} + i\mathrm{S}$, find an expression for $\mathrm{C}$.
      1. Factor out $\mathrm{e}^{i\frac{\theta}{2}}$
      2. $\left(-2i\right)^n\sin^n\dfrac{\theta}{2}\cos\dfrac{n\theta}{2}$
      Q2
      Answer
      Find the general solution of the differential equation $$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2\dfrac{\mathrm{d}y}{\mathrm{d}x} + y = \mathrm{e}^x$$
        $y = (A+Bx)\mathrm{e}^x + \dfrac{1}{2}x^2\mathrm{e}^x$
        Q3
        Answer
        Three planes have equations $$\begin{align*}x + 2y - z &= 3 \\ 2x - y + kz &= 1 \\ x + 5y - 3z &= c\end{align*}$$ where $k$ and $c$ are constants.
        1. Find the value of $k$ such that the planes do not meet at a single point.
          1. For this value of $k$, find $c$ such that the planes form a sheaf.
            1. Use a matrix method to find the point of intersection of the planes when $k = 1$ and $c = 3$.
              1. $\dfrac{4}{3}$
              2. $6$
              3. $(-2,10,15)$
              Q4
              Answer
              Two planes have equations $$\Pi_1: 2x - y + z = 5$$ $$\Pi_2: x + y - 3z = 1$$
              1. Find the acute angle between the planes.
                1. Show that the point $A(3, 1 0)$ lies on $\Pi_1$.
                  1. Find the shortest distance between $A$ and $\Pi_2$.
                    1. $75.7^{\circ}$
                    2. Substitute
                    3. $\dfrac{3}{\sqrt{11}}$