Problems

1995 Paper 3 Q2

D: 1700.0 B: 1586.3

integration reduction formula definite integral trigonometric substitution recurrence relation Wallis integral beta function

If \[ \mathrm{I}_{n}=\int_{0}^{a}x^{n+\frac{1}{2}}(a-x)^{\frac{1}{2}}\,\mathrm{d}x, \] show that $\mathrm{I}_{0}=\pi a^{2}/8.$ Show that $(2n+4)\mathrm{I}_{n}=(2n+1)a\mathrm{I}_{n-1}$ and hence evaluate $\mathrm{I}_{n}$.

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1995 Paper 3 Q3

D: 1700.0 B: 1500.0

differential equation second order auxiliary equation damped oscillation SHM overdamped critically damped underdamped logarithmic decrement exponential decay geometric series

What is the general solution of the differential equation \[ \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+2k\frac{\mathrm{d}x}{\mathrm{d}t}+x=0 \] for each of the cases: (i) $k>1;$ (ii) $k=1$; (iii) $0 < x < 1$? In case (iii) the equation represents damped simple harmonic motion with damping factor $k$. Let $x(0)=0$ and let $x_{1},x_{2},\ldots,x_{n},\ldots$ be the sequence of successive maxima and minima, so that if $x_{n}$ is a maximum then $x_{n+1}$ is the next minimum. Show that $\left|x_{n+1}/x_{n}\right|$ takes a value $\alpha$ which is independent of $n$, and that \[ k^{2}=\frac{(\ln\alpha)^{2}}{\pi^{2}+(\ln\alpha)^{2}}. \]

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1995 Paper 3 Q4

D: 1700.0 B: 1499.3

trigonometry summation geometric series complex numbers inequality De Moivre trigonometric sums

Let \[ \mathrm{C}_{n}(\theta)=\sum_{k=0}^{n}\cos k\theta \] and let \[ \mathrm{S}_{n}(\theta)=\sum_{k=0}^{n}\sin k\theta, \] where $n$ is a positive integer and $0<\theta<2\pi.$ Show that \[ \mathrm{C}_{n}(\theta)=\frac{\cos(\tfrac{1}{2}n\theta)\sin\left(\frac{1}{2}(n+1)\theta\right)}{\sin(\frac{1}{2}\theta)}, \] and obtain the corresponding expression for $\mathrm{S}_{n}(\theta)$. Hence, or otherwise, show that for $0<\theta<2\pi,$ \[ \left|\mathrm{C}_{n}(\theta)-\frac{1}{2}\right|\leqslant\frac{1}{2\sin(\frac{1}{2}\theta)}. \]

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1995 Paper 3 Q5

D: 1700.0 B: 1500.0

differential equation second order Maclaurin series Leibniz rule proof by induction inverse trigonometric functions repeated differentiation general term

Show that $y=\sin^{2}(m\sin^{-1}x)$ satisfies the differential equation \[ (1-x^{2})y^{(2)}=xy^{(1)}+2m^{2}(1-2y), \] and deduce that, for all $n\geqslant1,$ \[ (1-x^{2})y^{(n+2)}=(2n+1)xy^{(n+1)}+(n^{2}-4m^{2})y^{(n)}, \] where $y^{(n)}$ denotes the $n$th derivative of $y$. Derive the Maclaurin series for $y$, making it clear what the general term is.

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1995 Paper 3 Q6

D: 1700.0 B: 1501.9

complex numbers modulus argument Argand diagram locus circle reciprocal of complex number curve sketching

The variable non-zero complex number $z$ is such that \[ \left|z-\mathrm{i}\right|=1. \] Find the modulus of $z$ when its argument is $\theta.$ Find also the modulus and argument of $1/z$ in terms of $\theta$ and show in an Argand diagram the loci of points which represent $z$ and $1/z$. Find the locus $C$ in the Argand diagram such that $w\in C$ if, and only if, the real part of $(1/w)$ is $-1$.

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1995 Paper 3 Q7

D: 1654.7 B: 1516.0

groups group axioms closure identity element inverse isomorphism complex numbers modular arithmetic matrix rotation matrix unit modulus

Consider the following sets with the usual definition of multiplication appropriate to each. In each case you may assume that the multiplication is associative. In each case state, giving adequate reasons, whether or not the set is a group.

the complex numbers of unit modulus;
the integers modulo 4;
the matrices \[ \mathrm{M}(\theta)=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}, \] where $0\leqslant\theta<2\pi$;
the integers $1,3,5,7$ modulo 8;
the $2\times2$ matrices all of whose entries are integers;
the integers $1,2,3,4$ modulo 5.

In the case of each pair of groups above state, with reasons, whether or not they are isomorphic.

Solution:

$\{ z \in \mathbb{C} : |z| = 1\}$ is a group.
1. (Closure) $|z_1z_2| = |z_1||z_2| = 1$. Set is closed under multiplication
2. (Associativity) Multiplication of complex numbers is associative
3. (Identity) $|1| = 1$
4. (Inverses) $| \frac{1}{z} | = \frac{1}{|z|} = \frac{1}{1} = 1$, the set contains inverses
the integers $\pmod{4}$ are not a group under multiplication, $2$ has no inverse, since $0 \times k \equiv 0 \pmod{4}$
The set of rotation matrices is a group:
1. (Closure) \begin{align*} \begin{pmatrix}\cos\theta_1 & -\sin\theta_1\\ \sin\theta_1 & \cos\theta_1 \end{pmatrix} \begin{pmatrix}\cos\theta_2 & -\sin\theta_2\\ \sin\theta_2 & \cos\theta_2 \end{pmatrix} &= {\scriptscriptstyle\begin{pmatrix}\cos\theta_1 \cos \theta_2 - \sin \theta_1\sin \theta_2 & -\sin\theta_1\ \cos \theta_1 - \sin \theta_2\cos\theta_1\\ \sin\theta_1\ \cos \theta_1 + \sin \theta_2\cos\theta_1 & \cos\theta_1 \cos \theta_2 - \sin \theta_1\sin \theta_2 \end{pmatrix}} \\ &= \begin{pmatrix}\cos(\theta_1+\theta_2) & -\sin(\theta_1+\theta_2)\\ \sin(\theta_1+\theta_2) & \cos(\theta_1+\theta_2) \end{pmatrix} \end{align*} Since $\cos, \sin$ are periodic with period $2\pi$, we can find $\theta_3 = \theta_1 + \theta_2 + 2k\pi$ such that $0 \leq \theta_3 < 2 \pi$, so our set is closed
2. (Associativity) Matrix multiplication is associative
3. (Identity) Consider $\theta = 0$
4. (Inverses) Consider $2\pi - \theta$
$\{1, 3, 5, 7\} \pmod{8}$ is a group:
1 3 5 7
1 1 3 5 7
3 3 1 7 5
5 5 7 1 3
7 7 5 3 1
1. (Closure) See Cayley table
2. (Associativity) Integer multiplication is associative
3. (Identity) $1$
4. (Inverses) $x \mapsto x$ (See Cayley table)
$2\times2$ matrices are not a group, consider $0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$, then $\mathbf{0}\mathbf{M} = \mathbf{0}$ for all other matrices.
1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
1. (Closure) See Cayley table
2. (Associativity) Integer multiplication is associative
3. (Identity) $1$
4. (Inverses) $1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2, 4 \mapsto 4$ (See Cayley table)

	(i)	(iii)	(iv)	(vi)
(i)	$\checkmark$	$\checkmark$ consider \(z \mapsto \begin{pmatrix} \cos \arg (z)	- \sin \arg(z)
\sin \arg(z)	\cos \arg(z) \end{pmatrix}\)	not finite	not finite
(iii)		$\checkmark$	not finite	not finite
(iv)			$\checkmark$	no element order $4$
(vi)				$\checkmark$

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1995 Paper 3 Q8

D: 1700.0 B: 1500.0

vectors plane geometry scalar product cross product perpendicular distance sphere geometric proof normal to plane 3D geometry

A plane $\pi$ in 3-dimensional space is given by the vector equation $\mathbf{r}\cdot\mathbf{n}=p,$ where $\mathbf{n}$ is a unit vector and $p$ is a non-negative real number. If $\mathbf{x}$ is the position vector of a general point $X$, find the equation of the normal to $\pi$ through $X$ and the perpendicular distance of $X$ from $\pi$. The unit circles $C_{i},$ $i=1,2,$ with centres $\mathbf{r}_{i},$ lie in the planes $\pi_{i}$ given by $\mathbf{r}\cdot\mathbf{n}_{i}=p_{i},$ where the $\mathbf{n}_{i}$ are unit vectors, and $p_{i}$ are non-negative real numbers. Prove that there is a sphere whose surface contains both circles only if there is a real number $\lambda$ such that \[ \mathbf{r}_{1}+\lambda\mathbf{n}_{1}=\mathbf{r}_{2}\pm\lambda\mathbf{n}_{2}. \] Hence, or otherwise, deduce the necessary conditions that \[ (\mathbf{r}_{1}-\mathbf{r}_{2})\cdot(\mathbf{n}_{1}\times\mathbf{n}_{2})=0 \] and that \[ (p_{1}-\mathbf{n}_{1}\cdot\mathbf{r}_{2})^{2}=(p_{2}-\mathbf{n}_{2}\cdot\mathbf{r}_{1})^{2}. \] Interpret each of these two conditions geometrically.

Solution: The equation of the normal to $\pi$ through $X$ is $\mathbf{x} + \lambda \mathbf{n}$. The distance is $|\mathbf{x}\cdot \mathbf{n}-p|$ We know that the centre of the sphere must lie above the centre of the circle following the normal, ie $\mathbf{c} = \mathbf{r}_1+\lambda_1 \mathbf{n}_1 = \mathbf{r}_2+\lambda_2 \mathbf{n}_2$

We can also see that $R^2 = 1 + \lambda_1^2 = 1 + \lambda_2^2 \Rightarrow \lambda_1 = \pm \lambda_2 $, from which we obtain the desired result. Therefore the condition is \begin{align*} && \mathbf{r}_1+\lambda \mathbf{n}_1 &= \mathbf{r}_2 \pm \lambda \mathbf{n}_2 \tag{1}\\ && \mathbf{r}_1 - \mathbf{r}_2 &= \lambda(\pm \mathbf{n}_1 - \mathbf{n}_2) \\ \Rightarrow && (\mathbf{r}_{1}-\mathbf{r}_{2})\cdot(\mathbf{n}_{1}\times\mathbf{n}_{2}) &= (\lambda(\pm \mathbf{n}_1 - \mathbf{n}_2))\cdot(\mathbf{n}_{1}\times\mathbf{n}_{2}) \\ &&&= \lambda \left (\pm \mathbf{n}_1 \cdot ( \mathbf{n}_{1}\times\mathbf{n}_{2}) - \mathbf{n}_2 \cdot (\mathbf{n}_{1}\times\mathbf{n}_{2})\right) \\ &&&= 0 \\ \\ \mathbf{n}_1 \cdot (1)&& \mathbf{r}_1 \cdot \mathbf{n}_1+\lambda \mathbf{n}_1 \cdot \mathbf{n}_1 &= \mathbf{r}_2 \cdot \mathbf{n}_1 \pm \lambda \mathbf{n}_2 \cdot \mathbf{n}_1 \\ && p_1 + \lambda &= \mathbf{r}_2 \cdot \mathbf{n}_1 \pm \lambda \mathbf{n}_2 \cdot \mathbf{n}_1 \\ \\ \mathbf{n}_2 \cdot (1)&& \mathbf{r}_1 \cdot \mathbf{n}_2+\lambda \mathbf{n}_1 \cdot \mathbf{n}_2 &= \mathbf{r}_2 \cdot \mathbf{n}_2 \pm \lambda \mathbf{n}_2 \cdot \mathbf{n}_2 \\ && \mathbf{r}_1 \cdot \mathbf{n}_2+\lambda \mathbf{n}_1 \cdot \mathbf{n}_2 &= p_2 \pm \lambda \\ && \pm \lambda -\lambda \mathbf{n}_1\cdot\mathbf{n}_2 &= \mathbf{r}_1 \cdot \mathbf{n}_2 - p_2 \\ &&&= \pm (\mathbf{r}_2\cdot \mathbf{n}_1 - p_1) \\ \Rightarrow && (p_{1}-\mathbf{n}_{1}\cdot\mathbf{r}_{2})^{2}&=(p_{2}-\mathbf{n}_{2}\cdot\mathbf{r}_{1})^{2} \end{align*} The first condition means the line between the centres lies in the plane spanned by the normal of the two planes $\pi_1$ and $\pi_2$. The second condition means that the distance of the center to the other plane is the same for both centres/planes.

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1995 Paper 3 Q9

D: 1700.0 B: 1470.3

moments of inertia circular disc non-uniform density rotational dynamics stable equilibrium energy conservation optimisation compound pendulum

A thin circular disc of mass $m$, radius $r$ and with its centre of mass at its centre $C$ can rotate freely in a vertical plane about a fixed horizontal axis through a point $O$ of its circumference. A particle $P$, also of mass $m,$ is attached to the circumference of the disc so that the angle $OCP$ is $2\alpha,$ where $\alpha\leqslant\pi/2$.

In the position of stable equilibrium $OC$ makes an angle $\beta$ with the vertical. Prove that \[ \tan\beta=\frac{\sin2\alpha}{2-\cos2\alpha}. \]
The density of the disc at a point distant $x$ from $C$ is $\rho x/r.$ Show that its moment of inertia about the horizontal axis through $O$ is $8mr^{2}/5$.
The mid-point of $CP$ is $Q$. The disc is held at rest with $OQ$ horizontal and $C$ lower than $P$ and it is then released. Show that the speed $v$ with which $C$ is moving when $P$ passes vertically below $O$ is given by \[ v^{2}=\frac{15gr\sin\alpha}{2(2+5\sin^{2}\alpha)}. \] Find the maximum value of $v^{2}$ as $\alpha$ is varied.

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1995 Paper 3 Q10

D: 1700.0 B: 1500.0

projectiles projectile on slope inclined plane optimisation range maximisation trigonometric identity kinematics

A cannon is situated at the bottom of a plane inclined at angle $\beta$ to the horizontal. A (small) cannon ball is fired from the cannon at an initial speed $u.$ Ignoring air resistance, find the angle of firing which will maximise the distance up the plane travelled by the cannon ball and show that in this case the ball will land at a distance \[ \frac{u^{2}}{g(1+\sin\beta)} \] from the cannon.

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1995 Paper 3 Q11

D: 1700.0 B: 1500.0

vectors relative velocity wind navigation optimisation trigonometry ratio

A ship is sailing due west at $V$ knots while a plane, with an airspeed of $kV$ knots, where $k>\sqrt{2},$ patrols so that it is always to the north west of the ship. If the wind in the area is blowing from north to south at $V$ knots and the pilot is instructed to return to the ship every thirty minutes, how long will her outward flight last? Assume that the maximum distance of the plane from the ship during the above patrol was $d_{w}$ miles. If the air now becomes dead calm, and the pilot's orders are maintained, show that the ratio $d_{w}/d_{c}$ of $d_{w}$ to the new maximum distance, $d_{c}$ miles, of the plane from the ship is \[ \frac{k^{2}-2}{2k(k^{2}-1)}\sqrt{4k^{2}-2}. \]

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1995 Paper 3 Q12

D: 1700.0 B: 1484.0

bivariate data joint probability density function normal distribution radial symmetry ratio distribution Cauchy distribution polar coordinates geometric proof

The random variables $X$ and $Y$ are independently normally distributed with means 0 and variances 1. Show that the joint probability density function for $(X,Y)$ is \[ \mathrm{f}(x,y)=\frac{1}{2\pi}\mathrm{e}^{-\frac{1}{2}(x^{2}+y^{2})}\qquad-\infty < x < \infty,-\infty < y < \infty. \] If $(x,y)$ are the coordinates, referred to rectangular axes, of a point in the plane, explain what is meant by saying that this density is radially symmetrical. The random variables $U$ and $V$ have a joint probability density function which is radially symmetrical (in the above sense). By considering the straight line with equation $U=kV,$ or otherwise, show that \[ \mathrm{P}\left(\frac{U}{V} < k\right)=2\mathrm{P}(U < kV,V > 0). \] Hence, or otherwise, show that the probability density function of $U/V$ is \[ \mathrm{g}(k)=\frac{1}{\pi(1+k^{2})}\qquad-\infty < k < \infty. \]

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1995 Paper 3 Q13

D: 1700.0 B: 1500.0

probability conditional probability binary digits message transmission error probability Bayes' theorem approximation optimisation

A message of $10^{k}$ binary digits is sent along a fibre optic cable with high probabilities $p_{0}$ and $p_{1}$ that the digits 0 and 1, respectively, are received correctly. If the probability of a digit in the original message being a 1 is $\alpha,$ find the probability that the entire message is received correctly. Find the probability $\beta$ that a randomly chosen digit in the message is received as a 1 and show that $\beta=\alpha$ if, and only if \[ \alpha=\frac{q_{0}}{q_{1}+q_{0}}, \] where $q_{0}=1-p_{0}$ and $q_{1}=1-p_{1}.$ If this condition is satisfied and the received message consists entirely of zeros, what is the probability that it is correct? If now $q_{0}=q_{1}=q$ and $\alpha=\frac{1}{2},$ find the approximate value of $q$ which will ensure that a message of one million binary digits has a fifty-fifty chance of being received entirely correctly. The probability of error $q$ is proportional to the square of the length of the cable. Initially the length is such that the probability of a message of one million binary bits, among which 0 and 1 are equally likely, being received correctly is $\frac{1}{2}.$ What would this probability become if a booster station were installed at its mid-point, assuming that the booster station re-transmits the received version of the message, and assuming that terms of order $q^{2}$ may be ignored?

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1995 Paper 3 Q14

D: 1700.0 B: 1516.0

moment generating functions probability density function gamma distribution convolution sum of independent random variables MGF product property

A candidate finishes examination questions in time $T$, where $T$ has probability density function \[ \mathrm{f}(t)=t\mathrm{e}^{-t}\qquad t\geqslant0, \] the probabilities for the various questions being independent. Find the moment generating function of $T$ and hence find the moment generating function for the total time $U$ taken to finish two such questions. Show that the probability density function for $U$ is \[ \mathrm{g}(u)=\frac{1}{6}u^{3}\mathrm{e}^{-u}\qquad u\geqslant0. \] Find the probability density function for the total time taken to answer $n$ such questions.

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1994 Paper 1 Q1

D: 1500.0 B: 1500.0

coordinate geometry 3D geometry volume surface area trigonometry optimisation roof ridge practical modelling

My house has an attic consisting of a horizontal rectangular base of length $2q$ and breadth $2p$ (where $p < q$) and four plane roof sections each at angle $\theta$ to the horizontal. Show that the length of the roof ridge is independent of $\theta$ and find the volume of the attic and the surface area of the roof.

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1994 Paper 1 Q2

D: 1484.0 B: 1468.1

differentiation exponential functions power functions logarithmic differentiation chain rule iterated exponentials

Given that $a$ is constant, differentiate the following expressions with respect to $x$:

$x^{a}$;
$a^{x}$;
$x^{x}$;
$x^{(x^{x})}$;
$(x^{x})^{x}.$

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