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Q1
Answer
Find the general solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2xy = 4x$.
    $y = 2 + Ce^{-x^2}$
    Q2
    Answer
    Solve $(1 + x^2)\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2xy = 4x^2$, given that $y = 1$ when $x = 0$.
      $y = \dfrac{4x^3}{3(1+x^2)} + \dfrac{1}{1+x^2}$
      Q3
      Answer
      A heated object cools in a room of temperature $20°$C. Its temperature $T$ (in $°$C) at time $t$ minutes satisfies $\dfrac{\mathrm{d}T}{\mathrm{d}t} + 0.05T = 1$, with $T = 80$ when $t = 0$.
      1. Solve the differential equation.
        1. Find the temperature of the object after 10 minutes.
        2. Find the time at which the object reaches $30°$C.
        1. $T = 20 + 60e^{-0.05t}$
        2. $T(10) = 20 + 60e^{-0.5} \approx 56.4°$C
        3. $t = -20\ln\left(\dfrac{1}{6}\right) = 20\ln 6 \approx 35.8$ minutes
        Q4
        Answer
        A tank initially contains 100 litres of pure water. Brine containing 2 g/litre of salt flows in at 3 litres/min and the well-mixed solution flows out at 3 litres/min. The mass of salt in the tank at time $t$ minutes is $m$ grams.
        1. Show that $\dfrac{\mathrm{d}m}{\mathrm{d}t} + \dfrac{3m}{100} = 6$.
          1. Find the integrating factor and hence solve the differential equation, given that $m = 0$ when $t = 0$.
            1. Find the mass of salt in the tank after 20 minutes.
            2. Find the limiting mass of salt as $t \to \infty$ and explain this result in context.
            3. Suggest one limitation of the model.
            1. In $= 3 \times 2$ g/min; out $= 3 \times \dfrac{m}{100}$ g/min
            2. $m = 200\left(1 - e^{-\frac{3t}{100}}\right)$
            3. $m(20) = 200\left(1 - e^{-0.6}\right) \approx 90.2$ g
            4. As $t \to \infty$, $m \to 200$ g or 2 g/litre - the same concentration as the inflow
            5. Will not be well-mixed instantly
            Q5
            Answer
            A transformation $T$ is a reflection in the $x$-axis followed by a rotation of $90°$ anticlockwise. Find the single matrix representing $T$.
              $\begin{pmatrix}0&1\\1&0\end{pmatrix}$
              Q6
              Answer
              Find the line of invariant points under the transformation given by $M = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$.
                $x+y=0$
                Q7
                Answer
                A unit square is transformed by $M = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$. Find the area of the image and state whether orientation is preserved.
                  Area $6$. $\det(M) > 0$ orientation is preserved.
                  Q8
                  Answer
                  Find all invariant lines under the transformation $M = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.
                    $y=x$ and $y=-x$
                    Q9
                    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. $(-2,10,15)$
                          Q10
                          Answer
                          Three planes have equations $$\begin{align*}2x + y - z &= 5 \\ x - 2y + 3z &= 1 \\ 3x + 4y + mz &= n\end{align*}$$ where $m$ and $n$ are constants.
                          1. Find the value of $m$ such that the planes do not meet at a unique point.
                            1. For this value of $m$, find $n$ such that the planes are consistent.
                              1. Use a matrix method to find the point of intersection when $m = 0$ and $n = 9$.
                              1. $m = -\dfrac{13}{5}$
                              2. $n = \dfrac{29}{5}$
                              3. $(3, -2, 2)$