Paper 1
a) Find vector equations for the line \(\ell_1\) through \(A\) and \(B\), and the line \(\ell_2\) through \(C\) and \(D\).
[4 marks]b) Determine the shortest distance between \(\ell_1\) and \(\ell_2\).
[4 marks]c) Find the cosine of the angle between \(\overrightarrow{BA}\) and \(\overrightarrow{AC}\).
[3 marks]d) Compute the volume of the tetrahedron with vertices \(A\), \(B\), \(C\), and \(D\).
[3 marks]a) Solve \(\frac{dy}{dx}+2xy=4x\), subject to \(y(0)=3\).
[6 marks]b) Two populations satisfy \(S'=3S-2W\) and \(W'=S-W\). Deduce a second-order differential equation for \(S(t)\).
[5 marks]c) Solve the second-order equation from part (b) and describe the long-term behaviour of a non-zero solution with a positive coefficient of the growing mode.
[3 marks]a) Describe the graph of \(h\) on \((-2\pi,2\pi)\).
[3 marks]b) Find the Fourier series of \(h\), giving the first six non-zero terms explicitly.
[6 marks]c) Use Parseval's theorem to evaluate \(\sum_{k=0}^{\infty}\frac{1}{(2k+1)^2}\).
[5 marks]a) Find \(F_u\) and \(F_v\) in terms of partial derivatives of \(f\).
[5 marks]b) Show that \(F_{uv}=f_{xx}+(u+v)f_{xy}+uvf_{yy}+f_y\).
[5 marks]c) The temperature in a tank is \(T(x,y,z)=20-x^2-2y^2-z^2\). At \((1,-1,2)\), find the unit direction of greatest increase and the corresponding rate of increase.
[4 marks]a) Determine all constants \(a\) and \(b\) for which \(df=(2x+ay)\,dx+(bx+6y)\,dy\) is exact. For those values, find a potential function.
[6 marks]b) For \(\mathbf V=(x^2y,xz,yz^2)\), compute \(\nabla\cdot\mathbf V\) and \(\nabla\times\mathbf V\).
[8 marks]a) Evaluate \(\iint_R (x+y)\,dA\), where \(R\) is the triangle bounded by \(x=0\), \(y=0\), and \(x+y=2\).
[7 marks]b) Find the volume enclosed by the ellipsoid \(\frac{x^2}{4}+\frac{y^2}{9}+z^2=1\).
[8 marks]a) Let \(A\), \(B\), and \(C\) be independent events with probabilities \(p_A\), \(p_B\), and \(p_C\). An alert is triggered if \(A\) occurs or if both \(B\) and \(C\) occur. Show that the alert probability is \(p_A+p_Bp_C-p_Ap_Bp_C\).
[5 marks]b) Find \(P(A\mid\hbox{alert})\) in terms of \(p_A\), \(p_B\), and \(p_C\).
[4 marks]c) A rare marker appears independently in each of \(800\) samples with probability \(0.005\). Use a Poisson approximation to estimate the probability of at most two appearances.
[3 marks]d) Over \(40\) independent weeks the weekly count is approximated by \(\operatorname{Poisson}(4)\). Approximate the probability that the sample mean is less than \(3.5\).
[3 marks]