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3 hours100 marks

Level 1 - Math II (Physics) Paper 1

A Single Mathematics B-style paper covering 3D vectors, ODE models, Fourier series, multivariable calculus, vector calculus, multiple integrals, and probability.

Instructions

  • Attempt all questions.
  • Show sufficient working to justify each answer.
  • Answers should be exact unless a numerical approximation is requested.
  • Begin each main question on a new page when working on paper.

Section A: Vectors and Kinematics

1. Lines, Angles, and Volume

14 marks
Four points in \(\mathbb R^3\) are given by \(A=(1,0,2)\), \(B=(3,1,0)\), \(C=(0,2,1)\), and \(D=(2,0,3)\).
(a)
4 marks
Find vector equations for the line \(\ell_1\) through \(A\) and \(B\), and the line \(\ell_2\) through \(C\) and \(D\).
(b)
4 marks
Determine the shortest distance between \(\ell_1\) and \(\ell_2\).
(c)
3 marks
Find the cosine of the angle between \(\overrightarrow{BA}\) and \(\overrightarrow{AC}\).
(d)
3 marks
Compute the volume of the tetrahedron with vertices \(A\), \(B\), \(C\), and \(D\).

Section B: Ordinary Differential Equations

2. Linear ODEs and Coupled Populations

14 marks
Answer both independent modelling questions.
(a)
6 marks
Solve \(\frac{dy}{dx}+2xy=4x\), subject to \(y(0)=3\).
(b)
5 marks
Two populations satisfy \(S'=3S-2W\) and \(W'=S-W\). Deduce a second-order differential equation for \(S(t)\).
(c)
3 marks
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.

Section C: Fourier Analysis

3. Fourier Series and Parseval

14 marks
Let \(h\) be the \(2\pi\)-periodic function with \(h(x)=1\) for \(0<x<\pi\), \(h(x)=-1\) for \(-\pi<x<0\), and \(h(0)=h(\pi)=0\).
(a)
3 marks
Describe the graph of \(h\) on \((-2\pi,2\pi)\).
(b)
6 marks
Find the Fourier series of \(h\), giving the first six non-zero terms explicitly.
(c)
5 marks
Use Parseval's theorem to evaluate \(\sum_{k=0}^{\infty}\frac{1}{(2k+1)^2}\).

Section D: Multivariable Calculus

4. Chain Rule and Directional Derivatives

14 marks
Let \(F(u,v)=f(x,y)\), where \(x=u+v\) and \(y=uv\). Assume the mixed partial derivatives of \(f\) are equal.
(a)
5 marks
Find \(F_u\) and \(F_v\) in terms of partial derivatives of \(f\).
(b)
5 marks
Show that \(F_{uv}=f_{xx}+(u+v)f_{xy}+uvf_{yy}+f_y\).
(c)
4 marks
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.

Section E: Critical Points and Vector Calculus

5. Exact Differentials, Divergence, and Curl

14 marks
Answer both parts.
(a)
6 marks
Determine all constants \(a\) and \(b\) for which \(df=(2x+ay)\,dx+(bx+6y)\,dy\) is exact. For those values, find a potential function.
(b)
8 marks
For \(\mathbf V=(x^2y,xz,yz^2)\), compute \(\nabla\cdot\mathbf V\) and \(\nabla\times\mathbf V\).

Section F: Multiple Integrals and Vector Identities

6. Double Integrals and Volumes

15 marks
Evaluate the following integrals and state the geometry used.
(a)
7 marks
Evaluate \(\iint_R (x+y)\,dA\), where \(R\) is the triangle bounded by \(x=0\), \(y=0\), and \(x+y=2\).
(b)
8 marks
Find the volume enclosed by the ellipsoid \(\frac{x^2}{4}+\frac{y^2}{9}+z^2=1\).

Section G: Probability and Statistics

7. Conditional Probability and Approximations

15 marks
A field laboratory monitors three independent warning systems during a week of observations.
(a)
5 marks
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 \(p_A+p_Bp_C-p_Ap_Bp_C\).
(b)
4 marks
Find \(P(A\mid\hbox{alert})\) in terms of \(p_A\), \(p_B\), and \(p_C\).
(c)
3 marks
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.
(d)
3 marks
Over \(40\) independent weeks the weekly count is approximated by \(\operatorname{Poisson}(4)\). Approximate the probability that the sample mean is less than \(3.5\).