Part IA, 2015, Paper 1

Analysis I

• Paper 1, Section I, $3 \mathbf{F}$

  Analysis I | Part IA, 2015

  Find the following limits: (a) $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ (b) $\lim _{x \rightarrow 0}(1+x)^{1 / x}$ (c) $\lim _{x \rightarrow \infty} \frac{(1+x)^{\frac{x}{1+x}} \cos ^{4} x}{e^{x}}$

  Carefully justify your answers.

  [You may use standard results provided that they are clearly stated.]

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• Paper 1, Section I, E

  Analysis I | Part IA, 2015

  Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

  Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.

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• Paper 1, Section II, $11 D$

  Analysis I | Part IA, 2015

  (i) State and prove the intermediate value theorem.

  (ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

  $$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$$

  or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

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• Paper 1, Section II, 10D

  Analysis I | Part IA, 2015

  (a) For real numbers $a, b$ such that $a<b$, let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is bounded on $[a, b]$, and that $f$ attains its supremum and infimum on $[a, b]$.

  (b) For $x \in \mathbb{R}$, define

  $$g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.$$

  Find the set of points $x \in \mathbb{R}$ at which $g(x)$ is continuous.

  Does $g$ attain its supremum on $[0, \pi] ?$

  Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

  Justify your answers.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2015

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

  Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

  [You may use the mean value theorem provided that it is clearly stated.]

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• Paper 1, Section II, F

  Analysis I | Part IA, 2015

  Let $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

  $$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$$

  Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

  Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

  Justify your answers.

  [You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

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Vectors and Matrices

• Paper 1, Section I, $2 \mathrm{C}$

  Vectors and Matrices | Part IA, 2015

  Precisely one of the four matrices specified below is not orthogonal. Which is it?

  Give a brief justification.

  $$\frac{1}{\sqrt{6}}\left(\begin{array}{rcc} 1 & -\sqrt{3} & \sqrt{2} \\ 1 & \sqrt{3} & \sqrt{2} \\ -2 & 0 & \sqrt{2} \end{array}\right) \quad \frac{1}{3}\left(\begin{array}{ccc} 1 & 2 & -2 \\ 2 & -2 & -1 \\ 2 & 1 & 2 \end{array}\right) \quad \frac{1}{\sqrt{6}}\left(\begin{array}{rrr} 1 & -2 & 1 \\ -\sqrt{6} & 0 & \sqrt{6} \\ 1 & 1 & 1 \end{array}\right) \quad \frac{1}{9}\left(\begin{array}{rrr} 7 & -4 & -4 \\ -4 & 1 & -8 \\ -4 & -8 & 1 \end{array}\right)$$

  Given that the four matrices represent transformations of $\mathbb{R}^{3}$ corresponding (in no particular order) to a rotation, a reflection, a combination of a rotation and a reflection, and none of these, identify each matrix. Explain your reasoning.

  [Hint: For two of the matrices, $A$ and $B$ say, you may find it helpful to calculate $\operatorname{det}(A-I)$ and $\operatorname{det}(B-I)$, where $I$ is the identity matrix.]

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• Paper 1, Section I, B

  Vectors and Matrices | Part IA, 2015

  (a) Describe geometrically the curve

  $$|\alpha z+\beta \bar{z}|=\sqrt{\alpha \beta}(z+\bar{z})+(\alpha-\beta)^{2},$$

  where $z \in \mathbb{C}$ and $\alpha, \beta$ are positive, distinct, real constants.

  (b) Let $\theta$ be a real number not equal to an integer multiple of $2 \pi$. Show that

  $$\sum_{m=1}^{N} \sin (m \theta)=\frac{\sin \theta+\sin (N \theta)-\sin (N \theta+\theta)}{2(1-\cos \theta)}$$

  and derive a similar expression for $\sum_{m=1}^{N} \cos (m \theta)$.

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• Paper 1, Section II, $6 \mathrm{C}$

  Vectors and Matrices | Part IA, 2015

  (i) Consider the map from $\mathbb{R}^{4}$ to $\mathbb{R}^{3}$ represented by the matrix

  $$\left(\begin{array}{rrrr} \alpha & 1 & 1 & -1 \\ 2 & -\alpha & 0 & -2 \\ -\alpha & 2 & 1 & 1 \end{array}\right)$$

  where $\alpha \in \mathbb{R}$. Find the image and kernel of the map for each value of $\alpha$.

  (ii) Show that any linear map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ may be written in the form $f(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}$ for some fixed vector $\mathbf{a} \in \mathbb{R}^{n}$. Show further that $\mathbf{a}$ is uniquely determined by $f$.

  It is given that $n=4$ and that the vectors

  $$\left(\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{r} 2 \\ -1 \\ 0 \\ -2 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ 1 \\ 1 \end{array}\right)$$

  lie in the kernel of $f$. Determine the set of possible values of a.

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• Paper 1, Section II, 5B

  Vectors and Matrices | Part IA, 2015

  (i) State and prove the Cauchy-Schwarz inequality for vectors in $\mathbb{R}^{n}$. Deduce the inequalities

  $$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}| \text { and }|\mathbf{a}+\mathbf{b}+\mathbf{c}| \leqslant|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|$$

  for $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{n}$.

  (ii) Show that every point on the intersection of the planes

  $$\mathbf{x} \cdot \mathbf{a}=A, \quad \mathbf{x} \cdot \mathbf{b}=B$$

  where $\mathbf{a} \neq \mathbf{b}$, satisfies

  $$|\mathbf{x}|^{2} \geqslant \frac{(A-B)^{2}}{|\mathbf{a}-\mathbf{b}|^{2}}$$

  What happens if $\mathbf{a}=\mathbf{b} ?$

  (iii) Using your results from part (i), or otherwise, show that for any $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{y}_{1}, \mathbf{y}_{2} \in \mathbb{R}^{n}$,

  $$\left|\mathbf{x}_{1}-\mathbf{y}_{1}\right|-\left|\mathbf{x}_{1}-\mathbf{y}_{2}\right| \leqslant\left|\mathbf{x}_{2}-\mathbf{y}_{1}\right|+\left|\mathbf{x}_{2}-\mathbf{y}_{2}\right|$$

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2015

  (a) A matrix is called normal if $A^{\dagger} A=A A^{\dagger}$. Let $A$ be a normal $n \times n$ complex matrix.

  (i) Show that for any vector $\mathbf{x} \in \mathbb{C}^{n}$,

  $$|A \mathbf{x}|=\left|A^{\dagger} \mathbf{x}\right|$$

  (ii) Show that $A-\lambda I$ is also normal for any $\lambda \in \mathbb{C}$, where $I$ denotes the identity matrix.

  (iii) Show that if $\mathbf{x}$ is an eigenvector of $A$ with respect to the eigenvalue $\lambda \in \mathbb{C}$, then $\mathbf{x}$ is also an eigenvector of $A^{\dagger}$, and determine the corresponding eigenvalue.

  (iv) Show that if $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are eigenvectors of $A$ with respect to distinct eigenvalues $\lambda$ and $\mu$ respectively, then $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are orthogonal.

  (v) Show that if $A$ has a basis of eigenvectors, then $A$ can be diagonalised using an orthonormal basis. Justify your answer.

  [You may use standard results provided that they are clearly stated.]

  (b) Show that any matrix $A$ satisfying $A^{\dagger}=A$ is normal, and deduce using results from (a) that its eigenvalues are real.

  (c) Show that any matrix $A$ satisfying $A^{\dagger}=-A$ is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.

  (d) Show that any matrix $A$ satisfying $A^{\dagger}=A^{-1}$ is normal, and deduce using results from (a) that its eigenvalues have unit modulus.

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2015

  (i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:

  $$A=\left(\begin{array}{rrr} 1 & 1 & -1 \\ -1 & 3 & -1 \\ -1 & 1 & 1 \end{array}\right), \quad B=\left(\begin{array}{rcr} 1 & 4 & -3 \\ -4 & 10 & -4 \\ -3 & 4 & 1 \end{array}\right)$$

  (ii) Show that, if two real $n \times n$ matrices can both be diagonalised using the same basis transformation, then they commute.

  (iii) Suppose now that two real $n \times n$ matrices $C$ and $D$ commute and that $D$ has $n$ distinct eigenvalues. Show that for any eigenvector $\mathbf{x}$ of $D$ the vector $C \mathbf{x}$ is a scalar multiple of $\mathbf{x}$. Deduce that there exists a common basis transformation that diagonalises both matrices.

  (iv) Show that $A$ and $B$ satisfy the conditions in (iii) and find a matrix $S$ such that both of the matrices $S^{-1} A S$ and $S^{-1} B S$ are diagonal.

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