科目：进阶数学 Further Mathematics (9231)
年份：2014 夏季
试卷：Paper 22
总页数：7
章节：9231 Further Mathematics
1. A small smooth sphere $P$ of mass $2m$ is at rest on a smooth horizontal surface. A horizontal impulse of magnitude $8mu$ is given to $P$. Subsequently $P$ collides directly with a fixed smooth vertical barrier at right angles to $P$'s direction of motion. Given that the coefficient of restitution between $P$ and the barrier is 0.75, find the speed of $P$ after the collision. [2]
2. The point $O$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $OA = 0.5\text{ m}$ and $OB = 0.75\text{ m}$, with $A$ between $O$ and $B$. A particle $P$ of mass $m$ oscillates on $l$ in simple harmonic motion with centre $O$. The ratio of the kinetic energy of $P$ when it is at $A$ to its kinetic energy when it is at $B$ is 12:11. Find the amplitude of the motion. [3]
Given that the g5 A uniform rectangular thin sheet of glass $ABCD$, in which $AB = 8a$ and $BC = 6a$, has mass $\frac{3}{2}M$. Each of the edges $AB, BC, CD$ and $DA$ has a thin strip of metal attached to it, as a border to the glass. The strips along $AB$ and $CD$ each have mass $M$, and the strips along $BC$ and $DA$ each have mass $\frac{1}{2}M$. Show that the moment of inertia of the whole object (glass and metal strips) about an axis through $A$ perpendicular to the plane of the object is $128Ma^2$. [8]
The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with $C$ vertically below $A$. It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion. [5]
9 A random sample of 200 observations of the continuous random variable $X$ was taken and the values are summarised in the following table.
| Interval | $1 \leq x < 2$ | $2 \leq x < 3$ | $3 \leq x < 4$ | $4 \leq x < 5$ | $5 \leq x < 6$ | $6 \leq x < 7$ | $7 \leq x < 8$ |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Observed frequency | 63 | 45 | 32 | 25 | 22 | 7 | 6 |
It is required to test the goodness of fit of the distribution with probability density function $f$ given by
$$f(x) = \begin{cases}
\frac{1}{x \ln 8} & 1 \leq x < 8, \\
0 & \text{otherwise.}
\end{cases}$$
The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
| Interval | $1 \leq x < 2$ | $2 \leq x < 3$ | $3 \leq x < 4$ | $4 \leq x < 5$ | $11 Answer only one of the following two alternatives.
EITHER
The points $C$ and $D$ are at a distance $(2\sqrt{3})a$ apart on a horizontal surface. A rough peg $A$ is fixed at a vertical distance $6a$ above $C$ and a smooth peg $B$ is fixed at a vertical distance $4a$ above $D$. A uniform rectangular frame $PQRS$, with $PQ = 3a$ and $QR = 6a$, is made of rigid thin wire and has weight $W$. It rests in equilibrium in a vertical plane with $PS$ on $A$ and $SR$ on $B$, and with angle $SAC = 30^\circ$ (see diagram).
(i) Show that $AB = 4a$ and that angle $SAB = 30^\circ$.
(ii) Show that the normal reaction at $A$ is $\frac{1}{2}W$.
(iii) Find the frictional force at $A$.
OR
The time taken for a randomly chosen student at College $P$ to complete a particular puzzle has a noBLANK PAGE
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