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2 hours60 marks

Level 1 - Math I (Physics) Paper 1

Real-number reasoning, trigonometry, limits, calculus, and complex methods 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. Real-number and algebraic structure

15 marks
This question uses exact arithmetic, inequalities, binomial coefficients, and summation notation.
(a)
3 marks
Prove that \(\sqrt{3}+\sqrt{7}\) is irrational.
(b)
4 marks
Solve the inequality \(\frac{2x+1}{x-3}\ge1\).
(c)
4 marks
Find the coefficient of \(x^5\) in the expansion of \((2-x)^7\).
(d)
4 marks
Evaluate \(\sum_{r=1}^{10}(2r^2-r)\).

Section B: Trigonometry and Limits

2. Trigonometric identities and limiting values

15 marks
Use radians unless degrees are explicitly stated. Exact trigonometric values are preferred.
(a)
4 marks
Show that \(\frac{\sin(2x)}{1+\cos(2x)}=\tan x\) wherever both sides are defined.
(b)
3 marks
Find the exact value of \(\sin\left(\arctan\left(\frac{5}{12}\right)\right)\).
(c)
4 marks
Solve \(\arctan x+\arctan(2x)=\frac{\pi}{4}\).
(d)
4 marks
Evaluate \(\lim_{h\to0}\frac{1-\cos(4h)}{h^2}\).

Section C: Differentiation and Integration

3. Reconstructing volume from a rate

15 marks
A storage vessel contains \(30\) litres of liquid at \(t=0\). For \(0\le t\le5\), the signed rate of change of volume is \(R(t)=2t^2-8t+6\), measured in litres per minute.
(a)
3 marks
Determine when the volume is increasing and when it is decreasing.
(b)
4 marks
Find an expression for the volume \(V(t)\).
(c)
4 marks
Find the minimum and maximum volumes during \(0\le t\le5\).
(d)
4 marks
Find the average volume over the interval \(0\le t\le5\).

Section D: Complex Numbers

4. Complex arithmetic and De Moivre's theorem

15 marks
Write complex answers in exact form. Use \(i^2=-1\).
(a)
3 marks
Evaluate \(\frac{(2-i)(3+4i)}{1+i}\) in the form \(a+bi\).
(b)
4 marks
Use De Moivre's theorem to evaluate \((1+i)^6\).
(c)
4 marks
Find all complex solutions of \(z^3=27\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)\).
(d)
4 marks
Describe the moduli and arguments of the roots from part (c), and state the geometric shape they form in the complex plane.