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2 hours60 marks

Level 1 - Math I (Physics) Paper 1

Functions, trigonometry, limits, differentiation, integration, and complex numbers for physics modelling.

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: Functions and Trigonometry

1. Sensor calibration functions

15 marks
A sensor arm rotates through an angle \(\theta\), where \(-\frac{\pi}{2}\le \theta\le \frac{\pi}{2}\). Its height above a reference rail is modelled by \(h(\theta)=1.20+0.50\sin\theta\), measured in metres. A calibration circuit converts height to voltage by \(V(h)=6h-4\).
(a)
5 marks
Find the range of \(h\), and find an expression for \(h^{-1}(y)\).
(b)
5 marks
Find the voltage as a function of \(\theta\), written as \(V(h(\theta))\). Hence find the value of \(\theta\) for which the voltage is \(4.70\,\mathrm{V}\).
(c)
5 marks
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.

Section B: Limits and Differentiation

2. Rates from limits and derivatives

15 marks
A test cart has displacement \(x(t)=\frac{t^2}{t+1}\), where \(x\) is in metres and \(t\) is in seconds. A separate sensor response uses square-root behaviour for small changes.
(a)
5 marks
Using first principles, show that if \(f(t)=t^2+3t\), then \(f'(t)=2t+3\).
(b)
5 marks
Differentiate \(x(t)=\frac{t^2}{t+1}\) and find the cart velocity at \(t=2\).
(c)
5 marks
Evaluate \(\lim_{u\to0}\frac{\sqrt{1+4u}-1}{u}\) and state what type of rate this limit represents.

Section C: Integration

3. Accumulation by integration

15 marks
In a straight-track experiment, accumulated quantities are found by integrating simple rate or force models.
(a)
4 marks
A variable force is \(F(x)=2x+3\) for \(0\le x\le4\). Find the work done and state the definite-integral interpretation.
(b)
5 marks
Use substitution to evaluate \(\int_0^1 4t(1+t^2)^3\,dt\).
(c)
6 marks
A control input is \(I(t)=t e^{-t}\). Use integration by parts to find \(\int_0^2 t e^{-t}\,dt\).

Section D: Complex Numbers

4. Complex phasor arithmetic

15 marks
Complex numbers are used to represent amplitudes and phases of oscillatory signals. Write \(i^2=-1\).
(a)
4 marks
Write \(z=\sqrt{3}+i\) in polar form \(r(\cos\theta+i\sin\theta)\).
(b)
4 marks
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 \(a+bi\).
(c)
4 marks
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 \(t=\frac{\pi}{18}\).
(d)
3 marks
Write \(5-5i\) in the form \(Re^{i\phi}\), and interpret \(R\) and \(\phi\) for an oscillatory signal.