3 hours105 marks

Level 1 - Math I (Physics) Paper 1

Differentiation, limits, integration, complex numbers, series, Taylor approximation, matrices, linear systems, and eigenvalue methods for Level 1 Math I.

Instructions

Attempt all questions.
All main questions carry the same number of marks.
Show sufficient working to justify each answer.
Begin each main question on a new page.

Section A: Differentiation and Limits

1. Rates, limits, and first principles

15 marks

This question concerns differentiation rules, limits, and the derivative definition.

(a)

4 marks

Differentiate \(f(x)=(x^2+1)\sin(3x)\), and evaluate \(f'\left(\frac{\pi}{6}\right)\).

(b)

3 marks

Evaluate \(\displaystyle \lim_{x\to1}\frac{x^2+5x-6}{x-1}\).

(c)

3 marks

Evaluate \(\displaystyle \lim_{t\to\infty}\frac{\sqrt{9t^2+2t}}{4t-1}\).

(d)

5 marks

Using first principles, find the derivative of \(g(x)=\frac{2x-1}{x+3}\), where \(x\ne-3\).

Section B: Integration

2. Integrals and a complex sine identity

15 marks

This question uses trigonometric identities, rational integration, and complex exponential definitions of trigonometric functions.

(a)

5 marks

Evaluate \(\displaystyle \int \sin^3 x\cos^2 x\,dx\).

(b)

5 marks

Evaluate \(\displaystyle \int \frac{3x+1}{x^2+x-2}\,dx\).

(c)

5 marks

Using \(\sin z=\frac{e^{iz}-e^{-iz}}{2i}\), show that \(\sin(a+ib)=\sin a\cosh b+i\cos a\sinh b\). Hence find \(\sin(ib)\).

Section C: Complex Numbers

3. Complex form, powers, and roots

15 marks

Use principal arguments in \((-\pi,\pi]\) unless a general argument is requested.

(a)

3 marks

Find the modulus, principal argument, and rectangular form of \(z=e^{1-i\frac{2\pi}{3}}\).

(b)

3 marks

State De Moivre's theorem and use it to evaluate \(\left(\cos\frac{\pi}{10}+i\sin\frac{\pi}{10}\right)^5\).

(c)

5 marks

Use De Moivre's theorem to express \(\cos(4\theta)\) as a polynomial in \(\cos\theta\).

(d)

4 marks

Solve \(z^4+4z^2+16=0\), giving all roots in exponential form and rectangular form.

Section D: Series

4. Convergence tests

15 marks

This question concerns convergence tests for numerical and power series.

(a)

4 marks

Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\frac{3^n}{n!}\) converges.

(b)

4 marks

Use the root test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\left(\frac{2n+1}{5n-4}\right)^n\) converges.

(c)

4 marks

Use the alternating series test, and then test absolute convergence, for \(\displaystyle \sum_{n=2}^{\infty}(-1)^n\frac{1}{\sqrt n}\).

(d)

3 marks

Find the interval of convergence of \(\displaystyle \sum_{n=1}^{\infty}\frac{(x-2)^n}{n4^n}\).

Section E: Taylor Series and Matrices

5. Approximation and matrix conditions

15 marks

This question combines Taylor approximation with elementary vector and determinant reasoning.

(a)

5 marks

Find the cubic Taylor polynomial for \(f(x)=\ln(1+x)\) about \(0\), and use it to approximate \(\ln(1.2)\).

(b)

4 marks

Use the Lagrange remainder to bound the error in the approximation from part (a).

(c)

3 marks

Find the value of \(k\) for which \((1,-2,3)\) and \((2,k,6)\) are linearly dependent.

(d)

3 marks

For \(A=\begin{pmatrix}1&a&0\\2&1&1\\0&3&a\end{pmatrix}\), find the real values of \(a\) for which \(A\) is singular.

Section F: Linear Systems

6. A parameter-dependent system

15 marks

Consider the parameter-dependent linear system \[ \begin{aligned} x+y+z&=2,\\ 2x+3y+5z&=7,\\ x+2y+az&=b. \end{aligned} \]

(a)

4 marks

Find the values of \(a\) for which the system has a unique solution.

(b)

5 marks

Classify the non-unique cases as no solution or infinitely many solutions.

(c)

4 marks

For the unique case \(a=5\), find the inverse of the coefficient matrix.

(d)

2 marks

Use your inverse to solve the system when \(a=5\), leaving the answer in terms of \(b\).

Section G: Eigenvalues and Matrix Structure

7. Diagonalisation and matrix statements

15 marks

Let \[ A=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}. \] You are given eigenvectors \[ \mathbf v_1=\begin{pmatrix}1\\1\\1\end{pmatrix},\qquad \mathbf v_2=\begin{pmatrix}1\\-1\\0\end{pmatrix},\qquad \mathbf v_3=\begin{pmatrix}1\\0\\-1\end{pmatrix}. \]

(a)

4 marks

Verify the eigenvalue belonging to each supplied eigenvector.

(b)

4 marks

Construct matrices \(P\) and \(D\) such that \(A=PDP^{-1}\), and justify that this is a valid diagonalisation.

(c)

3 marks

Use the diagonalisation to write a formula for \(A^n\), where \(n\) is a positive integer.

(d)

4 marks

Decide whether each statement is true or false, giving a proof or counterexample. (i) Every matrix with a repeated eigenvalue is diagonalizable. (ii) Every \(n imes n\) matrix with \(n\) linearly independent eigenvectors is diagonalizable.