Problems

Let \(a,b,c,d,p\) and \(q\) be positive integers. Prove that:

1. if \(b > a\) and \(c > 1,\) then \(bc\geqslant2c\geqslant2+c\);
2. if \(a < b\) and \(d < c\), then \(bc-ad\geqslant a+c\);
3. if \({\displaystyle \frac{a}{b} < p < \frac{c}{d}}\), then \(\left(bc-ad\right)p\geqslant a+c\);
4. if \({\displaystyle \frac{a}{b} < \frac{p}{q} < \frac{c}{d}},\) then \({\displaystyle p\geqslant\frac{a+c}{bc-ad}}\) and \({\displaystyle q\geqslant\frac{b+d}{bc-ad}}\).

A damped system with feedback is modelled by the equation \[ \mathrm{f}'(t)+\mathrm{f}(t)-k\mathrm{f}(t-1)=0,\mbox{ }\tag{\(\dagger\)} \] where \(k\) is a given non-zero constant. Show that (non-zero) solutions for \(\mathrm{f}\) of the form \(\mathrm{f}(t)=A\mathrm{e}^{pt},\) where \(A\) and \(p\) are constants, are possible provided \(p\) satisfies \[ p+1=k\mathrm{e}^{-p}.\mbox{ }\tag{*} \] Show also, by means of a sketch, or otherwise, that equation \((*)\) can have \(0,1\) or \(2\) real roots, depending on the value of \(k\), and find the set of values of \(k\) for which such solutions of \((\dagger)\) exist. For what set of values of \(k\) do such solutions tend to zero as \(t\rightarrow+\infty\)?

Show Solution

---

Solution: Suppose \(f(t) = Ae^{pt}\) is a solution, then \begin{align*} && 0 &= Ape^{pt} + Ae^{pt} - Ake^{p(t-1)} \\ \Leftrightarrow && 0 &= p +1 - ke^{-p} \\ \Leftrightarrow && p+1 &= ke^{-p} \end{align*}

+ - ↺

If we sketch \(y = x+1\) and \(y = ke^{-x}\) we can see that when \(k \geq 0\) there will be exactly one solution. If \(k < 0\) there can be no solutions (if \(k\) is large and negative) and two solutions if \(k\) is small and negative. There will be exactly one real root if \(y = x+1\) is tangent to \(y = ke^{-x}\). The gradient of \(y = ke^{-x}\) is \(-ke^{-x}\) so to have a gradient \(1\) we must have \(x+1 = -1 \Rightarrow x = -2\), but also \(x = -\ln(-k) \Rightarrow k = -e^{-2}\). Therefore there will be so solution as long as \(k \geq -e^{-2}\). The solutions will tend to \(0\) as \(t \to + \infty\) as long as the intersection point is less than zero. For \(k \geq 0\) they intersect at \(0\) when \(k =1\), so we would want \(k < 1\). All negative values of k will work since the intersection has to happen for negative \(p\). Therefore the range is \(-e^{-2} \leq k < 1\)

The functions \(\mathrm{x}\) and \(\mathrm{y}\) are related by \[ \mathrm{x}(t)=\int_{0}^{t}\mathrm{y}(u)\,\mathrm{d}u, \] so that \(\mathrm{x}'(t)=\mathrm{y}(t)\). Show that \[ \int_{0}^{1}\mathrm{x}(t)\mathrm{y}(t)\,\mathrm{d}t=\tfrac{1}{2}\left[\mathrm{x}(1)\right]^{2}. \] In addition, it is given that \(\mbox{y}(t)\) satisfies \[ \mathrm{y}''+(\mathrm{y}^{2}-1)\mathrm{y}'+\mathrm{y}=0,\mbox{ }(*) \] with \(\mathrm{y}(0)=\mathrm{y}(1)\) and \(\mathrm{y}'(0)=\mathrm{y}'(1)\). By integrating \((*)\), prove that \(\mathrm{x}(1)=0.\) By multiplying \((*)\) by \(\mathrm{x}(t)\) and integrating by parts, prove the relation \[ \int_{0}^{1}\left[\mathrm{y}(t)\right]^{2}\,\mathrm{d}t=\tfrac{1}{3}\int_{0}^{1}\left[\mathrm{y}(t)\right]^{4}\,\mathrm{d}t. \] Prove also the relation \[ \int_{0}^{1}\left[\mathrm{y}'(t)\right]^{2}\,\mathrm{d}t=\int_{0}^{1}\left[\mathrm{y}(t)\right]^{2}\,\mathrm{d}t. \]

Show by means of a sketch that the parabola \(r(1+\cos\theta)=1\) cuts the interior of the cardioid \(r=4(1+\cos\theta)\) into two parts. Show that the total length of the boundary of the part that includes the point \(r=1,\theta=0\) is \(18\sqrt{3}+\ln(2+\sqrt{3}).\)

Show Solution

---

Solution:

+ - ↺

The curves will intersect when: \begin{align*} && \frac{1}{1+\cos \theta} &= 4 (1 + \cos \theta) \\ \Rightarrow && 1 + \cos \theta &= \pm \frac{1}{2} \\ \Rightarrow && \cos \theta &= -\frac12 \\ \Rightarrow && \theta &= \pm \frac{2\pi}{3}, \end{align*} Therefore we can measure the two sides of the boundaries. For the cardioid it will be: \begin{align*} s &= \int_{-2\pi/3}^{2 \pi /3} \sqrt{r^2 + \left ( \frac{\d r}{\d \theta} \right)^2} \d \theta \\ &= \int_{-2\pi/3}^{2 \pi /3}\sqrt{r^2 + \left ( \frac{\d r}{\d \theta} \right)^2} \d \theta \\ &= \int_{-2\pi/3}^{2 \pi /3}\sqrt{16(1 + \cos \theta)^2 + 16 \sin^2 \theta} \d \theta \\ &= 4\int_{-2\pi/3}^{2 \pi /3}\sqrt{2 + 2 \cos \theta} \d \theta \\ &= 8\int_{-2\pi/3}^{2 \pi /3}\sqrt{\cos^2 \frac{\theta}{2}} \d \theta \\ &= 8\int_{-2\pi/3}^{2 \pi /3}|\cos \frac{\theta}{2}| \d \theta \\ &= 16\int_{\pi}^{2\pi/3}(-\cos \frac{\theta}{2}) \d \theta + 8\int_{-\pi}^{\pi}\cos \frac{\theta}{2} \d \theta \\ &= 16 \cdot \left [ 2\sin \frac{\theta}{2}\right]_{\pi}^{2\pi/3}+ 8 \cdot 4 \\ &= 16 \cdot (\sqrt{3}-2) + 8 \cdot 4 \\ &= 16\sqrt{3} \end{align*} For the parabola we have that \(\sqrt{x^2+y^2} + x = 1 \Rightarrow x^2 + y^2 = 1 - 2x + x^2 \Rightarrow y^2 = 1-2x\). So we can parameterise our parabola as \(y = t, x = \frac{1-t^2}2\). And we are interested in the points \(t = -\sqrt{3}\) and \(t =\sqrt3\) \begin{align*} &&s &= \int_{-\sqrt3}^\sqrt3 \sqrt{\left ( \frac{\d x}{\d t} \right)^2 + \left ( \frac{\d y}{\d t} \right)^2 } \d t \\ &&&= \int_{-\sqrt3}^\sqrt3 \sqrt{t^2+1^2} \d t \\ \sinh u = t, \frac{\d t}{\d u} = \cosh u&&&= \int_{-\sinh^{-1} \sqrt3}^{\sinh^{-1}\sqrt3} \cosh^2 u \d u \\ &&&= \left [\frac12 u + \frac14 \sinh(2u) \right ]_{-\sinh^{-1} \sqrt3}^{\sinh^{-1}\sqrt3} \\ &&&= \sinh^{-1} \sqrt{3} + 2\sqrt{3} \\ &&&= \ln(2 + \sqrt{3}) + 2\sqrt{3} \end{align*} Therefore the total distance is as required.

Two square matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfies \(\mathbf{AB=0}.\) Show that either \(\det\mathbf{A}=0\) or \(\det\mathbf{B}=0\) or \(\det\mathbf{A}=\det\mathbf{B}=0\). If \(\det\mathbf{B}\neq0\), what must \(\mathbf{A}\) be? Give an example to show that the condition \(\det\mathbf{A}=\det\mathbf{B}=0\) is not sufficient for the equation \(\mathbf{AB=0}\) to hold. Find real numbers \(p,q\) and \(r\) such that \[ \mathbf{M}^{3}+2\mathbf{M}^{2}-5\mathbf{M}-6\mathbf{I}=(\mathbf{M}+p\mathbf{I})(\mathbf{M}+q\mathbf{I})(\mathbf{M}+r\mathbf{I}), \] where \(\mathbf{M}\) is any square matrix and \(\mathbf{I}\) is the appropriate identity matrix. Hence, or otherwise, find all matrices \(\mathbf{M}\) of the form $\begin{pmatrix}a & c\\ 0 & b \end{pmatrix}$ which satisfy the equation \[ \mathbf{M}^{3}+2\mathbf{M}^{2}-5\mathbf{M}-6\mathbf{I}=\mathbf{0}. \]

A disc is free to rotate in a horizontal plane about a vertical axis through its centre. The moment of inertia of the disc about this axis is \(mk^{2}.\) Along one diameter is a narrow groove in which a particle of mass \(m\) slides freely. At time \(t=0,\) the disc is rotating with angular speed \(\Omega,\) and the particle is at a distance \(a\) from the axis and is moving towards the axis with speed \(V\), where \(k^{2}V^{2}=\Omega^{2}a^{2}(k^{2}+a^{2}).\) Show that, at a later time \(t,\) while the particle is still moving towards the axis, the angular speed \(\omega\) of the disc and the distance \(r\) of the particle from the axis are related by \[ \omega=\frac{\Omega(k^{2}+a^{2})}{k^{2}+r^{2}}\qquad\mbox{ and }\qquad\frac{\mathrm{d}r}{\mathrm{d}t}=-\frac{\Omega r(k^{2}+a^{2})}{k(k^{2}+r^{2})^{\frac{1}{2}}}. \] Deduce that \[ k\frac{\mathrm{d}r}{\mathrm{d}\theta}=-r(k^{2}+r^{2})^{\frac{1}{2}}, \] where \(\theta\) is the angle through which the disc has turned at time \(t\). By making the substitution \(u=1/r\), or otherwise, show that \(r\sinh(\theta+\alpha)=k,\) where \(\sinh\alpha=k/a.\) Hence, or otherwise, show that the particle never reaches the axis.

A straight staircase consists of \(N\) smooth horizontal stairs each of height \(h\). A particle slides over the top stair at speed \(U\), with velocity perpendicular to the edge of the stair, and then falls down the staircase, bouncing once on every stair. The coefficient of restitution between the particle and each stair is \(e\), where \(e<1\). Show that the horizontal distance \(d_{n}\) travelled between the \(n\)th and \((n+1)\)th bounces is given by \[ d_{n}=U\left(\frac{2h}{g}\right)^{\frac{1}{2}}\left(e\alpha_{n}+\alpha_{n+1}\right), \] where \({\displaystyle \alpha_{n}=\left(\frac{1-e^{2n}}{1-e^{2}}\right)^{\frac{1}{2}}}\). If \(N\) is very large, show that \(U\) must satisfy \[ U=\left(\frac{L^{2}g}{2h}\right)^{\frac{1}{2}}\left(\frac{1-e}{1+e}\right)^{\frac{1}{2}}, \] where \(L\) is the horizontal distance between the edges of successive stairs.

A thin non-uniform rod \(PQ\) of length \(2a\) has its centre of gravity a distance \(a+d\) from \(P\). It hangs (not vertically) in equilibrium suspended from a small smooth peg \(O\) by means of a light inextensible string of length \(2b\) which passes over the peg and is attached at its ends to \(P\) and \(Q\). Express \(OP\) and \(OQ\) in terms of \(a,b\) and \(d\). By considering the angle \(POQ\), or otherwise, show that \(d < a^{2}/b\).

Show Solution

---

Solution:

+ - ↺

Resolving horizontally, it's clear that \(\angle POG = \angle GOQ\), in particular applying the sine rule: \begin{align*} && \sin \angle POG &= \frac{a+d}{2b-x} \sin \angle PGO \\ && \sin \angle GOP &= \frac{a-d}{x} \sin \angle OGQ \\ \Rightarrow && \frac{a+d}{2b-x} &= \frac{a-d}{x} \\ \Rightarrow && x(a+d) &= (2b-x)(a-d) \\ \Rightarrow && 2ax &= 2b(a-d) \\ \Rightarrow && x &= b - \frac{db}{a} \\ \Rightarrow && PO &= b+\frac{db}{a} \\ && OQ &= b - \frac{d}{a} \end{align*} Applying the cosine rule: \begin{align*} && \cos POQ &= \frac{(b + \frac{db}{a})^2 + (b - \frac{db}{a})^2 -4a^2}{2(b^2 - \frac{d^2b^2}{a^2})} \\ &&&= \frac{2b^2 + \frac{2d^2b^2}{a^2}-4a^2}{2(b^2 - \frac{d^2b^2}{a^2})} \\ &&&= \frac{2a^2b^2 + 2d^2b^2-4a^4}{2b^2(a^2 - d^2)} \\ &&&< 1 \\ \Leftrightarrow && 2a^2b^2 + 2d^2b^2-4a^4 &< 2b^2(a^2-d^2) \\ \Leftrightarrow && 2d^2b^2-4a^4 &< -2b^2d^2 \\ \Leftrightarrow && 4d^2b^2&< 4a^4 \\ \Leftrightarrow && d^2&< \frac{a^4}{b^2} \\ \Leftrightarrow && d&< \frac{a^2}{b} \\ \end{align*}

The identical uniform smooth spherical marbles \(A_{1},A_{2},\ldots,A_{n},\) where \(n\geqslant3,\) each of mass \(m,\) lie in that order in a smooth straight trough, with each marble touching the next. The marble \(A_{n+1},\) which is similar to \(A_{n}\) but has mass \(\lambda m,\) is placed in the trough so that it touches \(A_{n}.\) Another marble \(A_{0},\) identical to \(A_{n},\) slides along the trough with speed \(u\) and hits \(A_{1}.\) It is given that kinetic energy is conserved throughout.

1. Show that if \(\lambda<1,\) there is a possible subsequent motion in which only \(A_{n}\) and \(A_{n+1}\) move (and \(A_{0}\) is reduced to rest), but that if \(\lambda>1,\) such a motion is not possible.
2. If \(\lambda>1,\) show that a subsequent motion in which only \(A_{n-1},A_{n}\) and \(A_{n+1}\) move is not possible.
3. If \(\lambda>1,\) find a possible subsequent motion in which only two marbles move.

A target consists of a disc of unit radius and centre \(O\). A certain marksman never misses the target, and the probability of any given shot hitting the target within a distance \(t\) from \(O\) it \(t^{2}\), where \(0\leqslant t\leqslant1\). The marksman fires \(n\) shots independently. The random variable \(Y\) is the radius of the smallest circle, with centre \(O\), which encloses all the shots. Show that the probability density function of \(Y\) is \(2ny^{2n-1}\) and find the expected area of the circle. The shot which is furthest from \(O\) is rejected. Show that the expected area of the smallest circle, with centre \(O\), which encloses the remaining \((n-1)\) shots is \[ \left(\frac{n-1}{n+1}\right)\pi. \]