科目：进阶数学 Further Mathematics (9231)
年份：2011 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. Three small spheres, $A, B$ and $C$, of masses $m, km$ and $6m$ respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between $A$ and $B$ is $\frac{1}{2}$, and the coefficient of restitution between $B$ and $C$ is $e$. Sphere $A$ is projected towards $B$ with speed $u$ and is brought to rest by the subsequent collision. Show that $k = 2$.
Given that there are no further collisions after $B$ has collided with $C$, show that $e \leq \frac{1}{3}$.
2. A uniform circular disc with centre $A$ has mass $M$ and radius $3a$. A second uniform circular disc with centre $B$ has mass $\frac{1}{6}M$ and radius $a$. The two discs are rigidly joined together so that they lie in the same 4 A particle $P$ of mass $m$ is suspended from a fixed point $O$ by a light inextensible string of length $a$. When hanging at rest under gravity, $P$ is given a horizontal velocity of magnitude $\sqrt{3ag}$ and subsequently moves freely in a vertical circle. Show that the tension $T$ in the string when OP makes an angle $\theta$ with the downward vertical is given by
$$T = mg(1 + 3\cos \theta).$$
[4]
When the string is horizontal, it comes into contact with a small smooth peg $Q$ which is at the same horizontal level as $O$ and at a distance $x$ from $O$, where $x < a$. Given that $P$ completes a vertical circle about $Q$, find the least possible value of $x$. [8]
5 The continuous random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
0.09 The marks achieved by a random sample of 15 college students in a Physics examination $(x)$ and in a General Studies examination $(y)$ are summarised as follows.
$$\Sigma x = 752 \quad \Sigma x^2 = 38814 \quad \Sigma y = 773 \quad \Sigma y^2 = 45351 \quad \Sigma xy = 40236$$
(i) Find the mean values, $\bar{x}$ and $\bar{y}$. [1]
(ii) Another college student achieved a mark of 56 in the General Studies examination, but was unable to take the Physics examination. Use the equation of a suitable regression line to estimate the mark that the student would have obtained in the Physics examination. [4]
(iii) Find the product moment correlation coefficient for the given data. [2]
(iv) Stating your hypotheses, test at the 5% level of significance whether there is a non-zero product10 Answer only one of the following two alternatives.
EITHER
One end of a light elastic string is attached to a fixed point $O$. A particle of mass $m$ is attached to the other end of the string and hangs freely under gravity. In the equilibrium position, the extension of the string is $d$. Show that the period of small vertical oscillations about the equilibrium position is $2\pi \sqrt{\frac{d}{g}}$.
The particle is now pulled down and released from rest at a distance $2d$ below the equilibrium position. Given that the particle does not reach $O$ in the subsequent motion, show that the time taken until the particle first comes to instantaneous rest is $\sqrt{3 + \frac{2}{3}\pi} \sqrt{\frac{d}{g}}$.
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