2 hours60 marks

Level 1 - Math I (Physics) Paper 1

Functions, limits, differentiation, integration, and complex-number identities for physics mathematics.

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 Inverses

1. Logarithmic response functions

15 marks

A logarithmic response model is \(R(q)=2+\ln(q+1)\), where \(q>-1\). A read-back formula is \(A(y)=e^{y-2}-1\).

(a)

5 marks

Find the range of \(R\), and find an expression for \(R^{-1}(y)\).

(b)

5 marks

Simplify \(A(R(q))\), and state what this means for the read-back formula.

(c)

5 marks

Find the setting \(q\) for which \(R(q)=\ln6\).

Section B: Limits and Derivatives

2. Continuity, first principles, and small angles

15 marks

A removable discontinuity is used to test a numerical routine. A separate polynomial model is differentiated from first principles.

(a)

5 marks

Let \(f(x)=\frac{x^2-4}{x-2}\) for \(x\ne2\), and let \(f(2)=k\). Find \(k\) so that \(f\) is continuous at \(x=2\).

(b)

5 marks

Using first principles, differentiate \(g(x)=x^2-4x\).

(c)

5 marks

Evaluate \(\lim_{x\to0}\frac{\cos(3x)-1}{x^2}\).

Section C: Integration

3. Definite integrals from rate models

15 marks

Accumulated quantities are computed from three rate models over short intervals: a polynomial rate, a trigonometric rate, and an exponential rate.

(a)

5 marks

Evaluate \(\int_0^1(6u^2-4u+5)\,du\).

(b)

5 marks

Use a substitution to evaluate \(\int_0^{\pi/4}\frac{\sin\theta}{\cos^3\theta}\,d\theta\).

(c)

5 marks

Use integration by parts to evaluate \(\int_0^1 u e^{2u}\,du\).

Section D: Complex Numbers and De Moivre's Theorem

4. Complex powers and roots

15 marks

Let \(z=\cos\theta+i\sin\theta\). De Moivre's theorem connects powers of \(z\) with trigonometric identities and roots of complex numbers.

(a)

5 marks

Use De Moivre's theorem to show that \(z^3+z^{-3}=2\cos3\theta\).

(b)

5 marks

Deduce that \(\cos3\theta=4\cos^3\theta-3\cos\theta\).

(c)

5 marks

Find all fourth roots of \(-16\), giving your answers in exponential form.