Fera Academy

Paper 1

Time2 hours

Marks60

SetSet 4

PaperLevel 1 - Math I (Physics) Paper 1

Information

Section A: Functions and Inverses
Section B: Limits and Derivatives
Section C: Integration
Section D: Complex Numbers and De Moivre's Theorem

Candidate name

Candidate number

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.
Answer spaces are provided after each question part.

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

[5 marks]

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

[5 marks]

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

[5 marks]

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) 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 \(x=2\).

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

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

[5 marks]

END