科目：进阶数学 Further Mathematics (9231)
年份：2012 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1 Two smooth spheres $A$ and $B$, of equal radii and of masses $3m$ and $6m$ respectively, are at rest on a smooth horizontal surface. Sphere $A$ is projected directly towards $B$ with speed $u$. The coefficient of restitution between $A$ and $B$ is $e$. Show that the kinetic energy lost in the collision between $A$ and $B$ is $mu^2(1 - e^2)$. 2 Two light elastic strings, each of natural length $a$ and modulus of elasticity $2mg$, are attached to a particle $P$ of mass $m$. The strings join the particle to the points $A$ and $B$ which are fixed and at a distance $4a$ apart on a smooth horizontal surface. The particle is at rest at the mid-point $O$ of $AB$. The particle is now displaced a small distance in a direction perpendicular to $AB$, on the surface, and released from rest. At tim
A uniform rod $AB$, of mass $m$ and length $6a$, is rigidly attached at $B$ to a point on the circumference of a uniform circular lamina of mass $m$, radius $2a$ and centre $O$. The lamina and the rod are in the same vertical plane, and $ABO$ is a straight line (see diagram). Show that the moment of inertia of the system about an axis $l$ through $A$ perpendicular to the plane of the lamina is $78ma^2$.
A particle of mass $2m$ is now attached at $B$ and the system is free to rotate in a vertical plane about the fixed axis $l$ which is horizontal. Initially $AB$ is horizontal, with $O$ moving downwards and the system having angular velocity $\frac{3}{5}\sqrt{\left(\frac{g}{a}\right)}$. At time $t$, $AB$ makes an angle $\theta$ with the downward vertical through $A$.
(i) Find, in terms8 Residents of three towns $A, B$ and $C$ were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table.
| | Good | Satisfactory | Poor |
| :--- | :---: | :---: | :---: |
| Town $A$ | 24 | 34 | 14 |
| Town $B$ | 58 | 60 | 26 |
| Town $C$ | 20 | 34 | 30 |
Test, at the 2.5% significance level, whether grade of reliability is independent of town. [7]
Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question. [2]
9 The continuous random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
\frac{1}{2a} & -a \leq x \leq a11 Answer only one of the following two alternatives.
EITHER
The diagram shows a uniform rod $AB$, of length $4a$ and weight $W$, resting in equilibrium with its end $A$ on rough horizontal ground. The rod rests at $C$ on the surface of a smooth cylinder whose axis is horizontal. The cylinder rests on the ground and is fixed to it. The rod is in a vertical plane perpendicular to the axis of the cylinder and is inclined at an angle $\theta$ to the horizontal, where $\cos \theta = \frac{2}{3}$. A particle of weight $kW$ is attached to the rod at $B$. Given that $AC = 3a$, show that the least possible value of the coefficient of friction $\mu$ between the rod and the ground is $\frac{8(2k + 1)}{13k + 19}$.
Given that $\mu = \frac{9}{10}$, find the set of values of $k$ for which equilBLANK PAGE
BLANK PAGE
BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.