科目：进阶数学 Further Mathematics (9231)
年份：2013 冬季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. A particle $P$ is moving in a fixed circle of radius 0.5 m. At time $t$ s its speed is $(2 - 2t + t^2) \text{ m s}^{-1}$. Find the magnitude of each of the radial and transverse components of the acceleration of $P$ when $t = 3$.
2. Three uniform small smooth spheres $A, B$ and $C$, of equal radii and of masses $4m, \lambda m$ and $m$ respectively, are at rest in a straight line on a smooth horizontal plane, with $B$ between $A$ and $C$. Sphere $A$ is projected directly towards $B$ with speed $u$. The coefficient of restitution between $A$ and $B$, and between $B$ and $C$, is $\frac{1}{2}$. Show that the speed of $B$ after it is struck by $A$ is $\frac{6u}{\lambda + 4}$.
Given that the speed of $C$ after it is struck by $B$ is $u$, find the value of $\lambda$.
3. A particle $P$Two parallel vertical smooth walls $EF$ and $CD$ meet a horizontal plane at $E$ and $C$ respectively. A uniform smooth rod $AB$, of weight $2W$ and length $3a$, is freely hinged to the horizontal plane at the point $A$, between $E$ and $C$. The end $B$ rests against $CD$. A uniform smooth circular disc of weight $W$ is in contact with the wall $EF$ at the point $P$ and with the rod at the point $Q$. It is given that angle $BAC$ is $60^\circ$ and that $AQ = a$ (see diagram). The rod and the disc are in equilibrium in the same vertical plane, which is perpendicular to both walls. Show that
(i) the magnitude of the reaction at $P$ is $\sqrt{3}W$.
(ii) the magnitude of the reaction at $B$ is $\frac{7\sqrt{3}}{9}W$.
Find, in the form $kW$, the magnitude of the reaction on $AB$ at $A$, 8 A factory produces china mugs. Random samples of size 6 are selected at regular intervals, and the mugs are inspected for defects. During one week, 100 samples are selected and the numbers of defective mugs found are summarised in the following table. | Number of defective mugs | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Number of samples | 11 | 43 | 35 | 8 | 2 | 1 | 0 | Fit a binomial distribution to the data and carry out a goodness of fit test at the 5% significance level. [10] 9 A random sample of 9 observations of a normally distributed random variable $X$ gave the following summarised data. $$\Sigma x = 94.5 \quad \Sigma x^2 = 993.6$$ Test, at the 5% significance level, whether the population mean of $X$ is 10.2. [7]
11 Answer only one of the following two alternatives.
EITHER
A rigid body consists of a thin uniform rod $AB$, of mass $4m$ and length $6a$, joined at $B$ to a point on the circumference of a uniform circular disc, with centre $O$, mass $8m$ and radius $2a$. The point $C$ on the circumference of the disc is such that $BC$ is a diameter and $ABC$ is a straight line (see diagram). The body rotates about a smooth fixed horizontal axis through $C$, perpendicular to the plane of the disc. The angle between $CA$ and the downward vertical at time $t$ is denoted by $\theta$.
(i) Given that the body is performing small oscillations about the downward vertical, show that the period of these oscillations is approximately $16\pi \sqrt{\left(\frac{a}{11g}\right)}$.
(ii) Given instead that t6 BLANK PAGE
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