q^p\,$. \item Show by means of a sketch that the straight line through the points $(p, \ln p)$ and $(q, \ln q)$, where $\e\le pp^q$. \end{questionparts} \end{question} \vspace{-0.8cm} %%%%%%%%%%Q2 \begin{question} For $n\ge 0$, let \[ I_n = \int_0^1 x^n(1-x)^n\d x\,. \] \begin{questionparts} \item For $n\ge 1$, show by means of a substitution that \[ \int_0^1 x^{n-1}(1-x)^n\d x = \int_0^1 x^n(1-x)^{n-1}\d x\, \] and deduce that \[ 2 \int_0^1 x^{n-1}(1-x)^n\d x = I_{n-1}\,. \] Show also, for $n\ge1$, that \[ I_n = \frac n {n+1} \int_0^1 x^{n-1} (1-x)^{n+1} \d x \] and hence that $I_n = \dfrac{n}{2(2n+1)} I_{n-1}\,.$ \item When $n$ is a positive integer, show that \[ I_n = \frac{(n!)^2}{(2n+1)!}\,. \] \item Use the substitution $x= \sin^2 \theta$ to show that $I_{\frac12}= \frac \pi 8$, and evaluate $I_{\frac32}$. \end{questionparts} \end{question} %%%%%%%%% Q3 \begin{question} \begin{questionparts} \item Given that the cubic equation $x^3+3ax^2 + 3bx +c=0$ has three distinct real roots and $c<0$, show with the help of sketches that either exactly one of the roots is positive or all three of the roots are positive. \item Given that the equation $x^3 +3ax^2+3bx+c=0$ has three distinct real positive roots show that \begin{equation*} a^2>b>0, \ \ \ \ a<0, \ \ \ \ c<0\,. \tag{$*$} \end{equation*} \noindent[{\bf Hint}: Consider the turning points.] \item Given that the equation $x^3 +3ax^2+3bx+c=0$ has three distinct real roots and that \begin{equation*} ab<0, \ \ \ \ c>0\,, \end{equation*} determine, with the help of sketches, the signs of the roots. \item Show by means of an explicit example (giving values for $a$, $b$ and $c$) that it is possible for the conditions ($*$) to be satisfied even though the corresponding cubic equation has only one real root. \end{questionparts} \end{question} %%%%%% Q4 \begin{question} The line passing through the point $(a,0)$ with gradient $b$ intersects the circle of unit radius centred at the origin at $P$ and $Q$, and $M$ is the midpoint of the chord $PQ$. Find the coordinates of $M$ in terms of $a$ and $b$. \begin{questionparts} \item Suppose $b$ is fixed and positive. As $a$ varies, $M$ traces out a curve (the {\em locus} of $M$). Show that $x=- by$ on this curve. Given that $a$ varies with $-1\le a \le 1$, show that the locus is a line segment of length~$2b/(1+b^2)^\frac12$. Give a sketch showing the locus and the unit circle. \item Find the locus of $M$ in the following cases, giving in each case its cartesian equation, describing it geometrically and sketching it in relation to the unit circle: \begin{itemize} \item [({\bf a})] $a$ is fixed with $0 \dfrac{27} 4\,$. Find also all the roots of the equation $\f(x) = \dfrac{343}{36}\,$ and state the ranges of values of $x$ for which $\f(x) > \dfrac{343}{36}$. \end{questionparts} \end{question} %%%%%%%%% Q6 \begin{question} In this question, the following theorem may be used.\newline {\sl Let $u_1$, $u_2$, $\ldots$ be a sequence of (real) numbers. If the sequence is bounded above (that is, $u_n\le b$ for all $n$, where $b$ is some fixed number) and increasing (that is, $u_n\ge u_{n-1}$ for all $n$), then the sequence tends to a limit (that is, converges).} The sequence $u_1$, $u_2$, $\ldots$ is defined by $u_1=1$ and \[ u_{n+1} = 1+\frac 1{u_n} \ \ \ \ \ \ \ \ \ \ (n\ge1)\,. \tag{$*$} \] \begin{questionparts} \item Show that, for $n\ge3$, \[ u_{n+2}-u_n = \frac{u_{n} - u_{n-2}}{(1+u_n)(1+u_{n-2})} . \] \item Prove, by induction or otherwise, that $1\le u_n \le 2$ for all $n$. \item Show that the sequence $u_1$, $u_3$, $u_5$, $\ldots$ tends to a limit, and that the sequence $u_2$, $u_4$, $u_6$, $\ldots$ tends to a limit. Find these limits and deduce that the sequence $u_1$, $u_2$, $u_3$, $\ldots\,$ tends to a limit. Would this conclusion change if the sequence were defined by $(*)$ and $u_1=3$? \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} \begin{questionparts} \item Write down a solution of the equation \[ x^2-2y^2 =1\,, \tag{$*$} \] for which $x$ and $y$ are non-negative integers. Show that, if $x=p$, $y=q$ is a solution of ($*$), then so also is $x=3p+4q$, $y=2p+3q$. Hence find two solutions of $(*)$ for which $x$ is a positive odd integer and $y$ is a positive even integer. \item Show that, if $x$ is an odd integer and $y$ is an even integer, $(*)$ can be written in the form \[ n^2 = \tfrac12 m(m+1)\,, \] where $m$ and $n$ are integers. \item The positive integers $a$, $b$ and $c$ satisfy \[ b^3=c^4-a^2\,, \] where $b$ is a prime number. Express $a$ and $c^2$ in terms of $b$ in the two cases that arise. Find a solution of $a^2+b^3=c^4$, where $a$, $b$ and $c$ are positive integers but $b$ is not prime. \end{questionparts} \end{question} \vspace{-0.9cm} %%%%%%%%% Q8 \begin{question} The function $\f$ satisfies $\f(x)>0$ for $x\ge0$ and is strictly decreasing (which means that $\f(b)<\f(a)$ for $b>a$). \begin{questionparts} \item For $t\ge0$, let $A_0(t)$ be the area of the largest rectangle with sides parallel to the coordinate axes that can fit in the region bounded by the curve $y=\f(x)$, the $y$-axis and the line $y=\f(t)$. Show that $A_0(t)$ can be written in the form \[ A_0(t) =x_0\left( \f(x_0) -\f(t)\right), \] where $x_0$ satisfies $x_0 \f'(x_0) +\f(x_0) = \f(t)\,$. \item The function g is defined, for $t> 0$, by \[ \g(t) =\frac 1t \int_0^t \f(x) \d x\,. \] Show that $t \g'(t) = \f(t) -\g(t)\,$. Making use of a sketch show that, for $t>0$, \[ \int_0^t \left( \f(x) - \f(t)\right) \d x > A_0(t) \] and deduce that $-t^2 \g'(t)> A_0(t)$. \item In the case $\f(x)= \dfrac 1 {1+x}\,$, use the above to establish the inequality \[ \ln \sqrt{1+t} > 1 - \frac 1 {\sqrt{1+t}} \,, \] for $t>0$. \end{questionparts} \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} The diagram shows three identical discs in equilibrium in a vertical plane. Two discs rest, not in contact with each other, on a horizontal surface and the third disc rests on the other two. The angle at the upper vertex of the triangle joining the centres of the discs is $2\theta$. \begin{center} \psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3pt 0,linewidth=0.3pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-7,-0.42)(7,6.86) \psline(-7,0)(7,0) \pscircle(-3,2){1.4} \pscircle(3,2){1.4} \pscircle(0,4.64){1.4} \psline(0,4.64)(-3,2) \psline(0,4.64)(3,2) \psline(0,4.64)(0,0.9) \parametricplot{-1.5707963267948966}{-0.722030440522891}{1*cos(t)+0|1*sin(t)+4.64} \rput[tl](0.16,4.25){$\theta$} \end{pspicture*} \end{center} \noindent The weight of each disc is $W$. The coefficient of friction between a disc and the horizontal surface is $\mu$ and the coefficient of friction between the discs is also $\mu$. \begin{questionparts} \item Show that the normal reaction between the horizontal surface and a disc in contact with the surface is $\frac32 W\,$. \item Find the normal reaction between two discs in contact and show that the magnitude of the frictional force between two discs in contact is $\dfrac{W\sin\theta}{2(1+\cos\theta)}\,$. \item Show that if $\mu <2- \surd3\,$ there is no value of $\theta$ for which equilibrium is possible. \end{questionparts} \end{question} %%%%%%%%%% Q10 \begin{question} A particle is projected at an angle of elevation $\alpha$ (where $\alpha>0$) from a point $A$ on horizontal ground. At a general point in its trajectory the angle of elevation of the particle from $A$ is $\theta$ and its direction of motion is at an angle $\phi$ above the horizontal (with $\phi\ge0$ for the first half of the trajectory and $\phi\le0$ for the second half). Let $B$ denote the point on the trajectory at which $\theta = \frac12 \alpha$ and let $C$ denote the point on the trajectory at which $\phi = -\frac12\alpha$. \begin{questionparts} \item Show that, at a general point on the trajectory, $2\tan\theta = \tan \alpha + \tan\phi\,$. \item Show that, if $B$ and $C$ are the same point, then $ \alpha = 60^\circ\,$. \item Given that $\alpha < 60^\circ\,$, determine whether the particle reaches the point $B$ first or the point $C$ first. \end{questionparts} \end{question} %%%%%%%%%% Q11 \begin{question} Three identical particles lie, not touching one another, in a straight line on a smooth horizontal surface. One particle is projected with speed $u$ directly towards the other two which are at rest. The coefficient of restitution in all collisions is $e$, where $01$, $\P(S=2n)>\P(A=2n)$ and determine the corresponding relationship between $\P(S=2n+1)$ and $\P(A=2n+1)$. [You are advised {\em not} to use $p+q=1$ in this part.] \end{questionparts} \end{question} \end{document}