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1. Integral of an exponential over a triangle ●○○○○ 10 ptsLet $\gamma$ be the triangle in $\mathbb C$ whose vertices are the three cube roots of $1+i$, oriented clockwise. Find the value of $$\int_\gamma e^{2z+i}\, dz.$$
2. Most economical car speed ●○○○○ 10 ptsA car's fuel use is given by $f(v)=\frac{3}{80}v^{2}-6v+245$, where $v$ is the travelling speed. Find the speed $v$ that minimises the fuel use.
3. Number of solutions of a trig-root equation ●●○○○ 20 ptsFind the number of real solutions $x$ of the equation $\sin\left(\frac{\pi\sqrt x}{4}\right)+\cos\left(\frac{\pi\sqrt{2-x}}{4}\right)=\sqrt 2$.
4. Minimum of f(m) + f'(n) on an interval ●●○○○ 20 ptsThe function $f(x) = -x^3 + ax^2 - 4$ has a local extremum at $x = 2$. For $m, n \in [-1,1]$, find the minimum value of $f(m) + f'(n)$, where $f'$ is the derivative of $f$.
5. Angle between y = x² and y = 1/x ●●○○○ 20 ptsFind the acute angle of intersection, in degrees, between the curves $y = x^2$ and $y = \dfrac{1}{x}$, measured between their tangents at the point of intersection. Round to two decimal places.
6. Corner squares for maximum box volume ●●○○○ 20 ptsAn open-top box is made from a square sheet of metal with sides of 20 cm by cutting equal squares from the corners and folding up the sides. Find the side length (in cm) of the square cut from each corner so that the volume of the box is maximised. Round to two decimal places.
7. Minimum surface area of a square prism ●●○○○ 20 ptsA closed rectangular prism has a square base with side length $L$ and height $H$. Its volume is $10000$. Find the minimum possible surface area. Round to three decimal places.
8. Open-top box with minimum surface area ●●○○○ 20 ptsA container company makes an open-top rectangular box with a square base. The box must have a volume of $108\ \text{cm}^3$. Let the side length of the square base be $x$ cm and the height be $h$ cm. (1) Find the value of $x$ that gives the minimum outer surface area. (2) Find the minimum surface area, in $\text{cm}^2$.
9. Horizontal tangent of a rational function ●●○○○ 20 ptsThe graph of $f(x)=\frac{ax+b}{(x-1)(x-4)}$ has a horizontal tangent line at $(2,-1)$. Determine the values of $a$ and $b$.
10. Rational function with horizontal tangent ●●○○○ 20 ptsThe graph of $g(x) = \frac{ax+b}{(x-1)(x-4)}$ passes through the point $(2,-1)$ and has a horizontal tangent there. Find the values of $a$ and $b$.

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