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2 hours60 marks

Level 1 - Math I (Physics) Paper 1

Real-number methods, trigonometric modelling, calculus, integration, and complex numbers for physics support.

Instructions

  • Attempt all questions.
  • All main questions carry the same number of marks.
  • Show enough working to make the algebra, definitions, and modelling steps clear.
  • Begin each main question on a new page when working on paper.

Section A: Real Numbers and Algebra

1. Calibration algebra and finite sums

15 marks
A dimensionless calibration setting \(x\) is adjusted during a bench test. The same test also uses a small error parameter \(\epsilon\) and a sequence of linearly increasing readings.
(a)
5 marks
Solve the inequality \(0<\frac{2x-1}{x+2}<1\).
(b)
5 marks
Expand \((1+2\epsilon)^5\) up to and including the term in \(\epsilon^3\). Use this to estimate \((1.02)^5\).
(c)
5 marks
Find \(\sum_{r=1}^{n}(4r-1)\). Hence find \(n\) if this sum is \(171\).

Section B: Trigonometry and Limits

2. Oscillatory signal model

15 marks
An oscillatory signal is modelled by \(s(t)=4\sin t+3\cos t\), where \(t\) is measured in radians. A separate small-angle calculation is used to compare two nearby signals.
(a)
5 marks
Write \(s(t)=4\sin t+3\cos t\) in the form \(R\sin(t+\alpha)\), where \(R>0\) and \(0<\alpha<\frac{\pi}{2}\).
(b)
5 marks
Find the maximum value of \(s(t)\), and find the first \(t\) in \([0,2\pi)\) at which it occurs.
(c)
5 marks
Evaluate \(\lim_{u\to0}\frac{\sin(5u)-\sin(2u)}{u}\).

Section C: Differentiation and Integration

3. Pulse rates and accumulated work

15 marks
A displacement model for a short pulse is \(x(t)=t^2e^{-t}\), where \(t\ge0\). A related accumulated work calculation uses a rational force model.
(a)
5 marks
Differentiate \(x(t)=t^2e^{-t}\), and find the stationary time for \(t>0\).
(b)
5 marks
Evaluate \(\int_0^2 t e^{-t}\,dt\).
(c)
5 marks
Resolve \(\frac{3x+5}{x^2+3x+2}\) into partial fractions, and evaluate \(\int_0^1\frac{3x+5}{x^2+3x+2}\,dx\).

Section D: Complex Numbers

4. Phasors and roots

15 marks
A phasor has complex amplitude \(z=\sqrt{3}+i\). Complex roots are used to describe equally spaced phase choices.
(a)
5 marks
Write \(z=\sqrt{3}+i\) in exponential form, and hence find \(z^6\).
(b)
5 marks
Find the three cube roots of \(z\), giving your answers in exponential form.
(c)
5 marks
Let \(A=2e^{i\pi/3}+3e^{-i\pi/6}\). Find the real and imaginary parts of \(A\).