Paper 1
a) Find the range of \(h\), and find an expression for \(h^{-1}(y)\).
[5 marks]b) Find the voltage as a function of \(\theta\), written as \(V(h(\theta))\). Hence find the value of \(\theta\) for which the voltage is \(4.70\,\mathrm{V}\).
[5 marks]c) During a sweep, \(\theta(t)=-\frac{\pi}{6}+\frac{\pi t}{12}\) for \(0\le t\le8\). Find the time at which the voltage is \(4.70\,\mathrm{V}\). A power model is \(P(V)=(V-3.20)^2+2\); find \(P(V(h(\theta)))\) in simplest form.
[5 marks]a) Using first principles, show that if \(f(t)=t^2+3t\), then \(f'(t)=2t+3\).
[5 marks]b) Differentiate \(x(t)=\frac{t^2}{t+1}\) and find the cart velocity at \(t=2\).
[5 marks]c) Evaluate \(\lim_{u\to0}\frac{\sqrt{1+4u}-1}{u}\) and state what type of rate this limit represents.
[5 marks]a) A variable force is \(F(x)=2x+3\) for \(0\le x\le4\). Find the work done and state the definite-integral interpretation.
[4 marks]b) Use substitution to evaluate \(\int_0^1 4t(1+t^2)^3\,dt\).
[5 marks]c) A control input is \(I(t)=t e^{-t}\). Use integration by parts to find \(\int_0^2 t e^{-t}\,dt\).
[6 marks]a) Write \(z=\sqrt{3}+i\) in polar form \(r(\cos\theta+i\sin\theta)\).
[4 marks]b) Let \(w=3\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)\). Find \(zw\) in polar form and in the form \(a+bi\).
[4 marks]c) A signal is represented by \(\operatorname{Re}\left(6e^{i(3t+\frac{\pi}{3})}\right)\). Write it as a real trigonometric function and find its value at \(t=\frac{\pi}{18}\).
[4 marks]d) Write \(5-5i\) in the form \(Re^{i\phi}\), and interpret \(R\) and \(\phi\) for an oscillatory signal.
[3 marks]