Math problem sheet from Markdown to PDF

---
title: Problem sheet 4, linear algebra and calculus
author: Department of Mathematics
---

# Problem sheet 4

**Due Friday 24 April 2026, 16:00. Hand in problems 1 to 4. Problem 5 is for discussion in the tutorial.**

Show your working. Where a result from the lectures is used, name it. A correct answer without a derivation earns no marks[^marks].

## Problem 1: the matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$

1. Find the eigenvalues $\lambda_1, \lambda_2$ of $A$ by solving $\det(A - \lambda I) = 0$.
2. Find an eigenvector for each eigenvalue and check that they are orthogonal.
3. Write $A = Q \Lambda Q^{\top}$ with $Q$ orthogonal, and use it to compute $A^{10}$ without multiplying matrices ten times.

## Problem 2: a definite integral

Evaluate $\displaystyle I = \int_0^{1} \frac{\ln(1 + x)}{1 + x^2}\,\mathrm{d}x$.

> [!TIP]
> Substitute $x = \tan\theta$ first. You should reach an integral of the form $\int_0^{\pi/4} \ln(1 + \tan\theta)\,\mathrm{d}\theta$, then use the symmetry $\theta \mapsto \frac{\pi}{4} - \theta$.

## Problem 3: a series

Decide whether each series converges, and name the test you used.

| Series | Converges? | Test |
|:-------|:----------:|:-----|
| $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 + n}$ | | |
| $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ | | |
| $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}$ | | |
| $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n \ln n}$ | | |

## Problem 4: a differential equation

A tank holds $V = 500$ liters of brine. Fresh water flows in at $q = 5$ liters a minute and the mixture leaves at the same rate. With $m(t)$ the mass of salt in kilograms,

$ \frac{\mathrm{d}m}{\mathrm{d}t} = -\frac{q}{V}\, m, \qquad m(0) = m_0 . $

1. Solve for $m(t)$.
2. After how long is $m(t) = m_0 / 2$? Give the answer in minutes to one decimal place.

## Problem 5, for the tutorial: Euler's identity

> Prove that $e^{i\pi} + 1 = 0$ from the Taylor series of $e^x$, $\sin x$ and $\cos x$.

## Worked example from the lecture

The eigenvalues of $B = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$:

$ \begin{aligned} \det(B - \lambda I) &= \begin{vmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{vmatrix} \\ &= (4 - \lambda)(3 - \lambda) - 2 \\ &= \lambda^2 - 7\lambda + 10 \\ &= (\lambda - 2)(\lambda - 5) \end{aligned} $

$ \boxed{\lambda_1 = 2, \quad \lambda_2 = 5} $

Check: $\lambda_1 + \lambda_2 = 7 = \operatorname{tr} B$ and $\lambda_1 \lambda_2 = 10 = \det B$. ✓

- [ ] Problem 1 handed in
- [ ] Problem 2 handed in
- [ ] Problems 3 and 4 handed in

[^marks]: Marking scheme: 10 marks per problem, of which at most 2 for the final numerical answer. Partial credit is given for a correct method with an arithmetic slip.