Paper 1
a) Solve the inequality \(0<\frac{2x-1}{x+2}<1\).
[5 marks]b) Expand \((1+2\epsilon)^5\) up to and including the term in \(\epsilon^3\). Use this to estimate \((1.02)^5\).
[5 marks]c) Find \(\sum_{r=1}^{n}(4r-1)\). Hence find \(n\) if this sum is \(171\).
[5 marks]a) Write \(s(t)=4\sin t+3\cos t\) in the form \(R\sin(t+\alpha)\), where \(R>0\) and \(0<\alpha<\frac{\pi}{2}\).
[5 marks]b) Find the maximum value of \(s(t)\), and find the first \(t\) in \([0,2\pi)\) at which it occurs.
[5 marks]c) Evaluate \(\lim_{u\to0}\frac{\sin(5u)-\sin(2u)}{u}\).
[5 marks]a) Differentiate \(x(t)=t^2e^{-t}\), and find the stationary time for \(t>0\).
[5 marks]b) Evaluate \(\int_0^2 t e^{-t}\,dt\).
[5 marks]c) Resolve \(\frac{3x+5}{x^2+3x+2}\) into partial fractions, and evaluate \(\int_0^1\frac{3x+5}{x^2+3x+2}\,dx\).
[5 marks]a) Write \(z=\sqrt{3}+i\) in exponential form, and hence find \(z^6\).
[5 marks]b) Find the three cube roots of \(z\), giving your answers in exponential form.
[5 marks]c) Let \(A=2e^{i\pi/3}+3e^{-i\pi/6}\). Find the real and imaginary parts of \(A\).
[5 marks]