科目：进阶数学 Further Mathematics (9231)
年份：2014 冬季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. Two smooth spheres $A$ and $B$, of equal radii and masses $2m$ and $m$ respectively, lie at rest on a smooth horizontal table. The spheres $A$ and $B$ are projected directly towards each other with speeds $4u$ and $3u$ respectively. The coefficient of restitution between the spheres is $e$. Find the set of values of $e$ for which the direction of motion of $A$ is reversed in the collision.
[5]
2. A small smooth ball $P$ is moving on a smooth horizontal plane with speed $4\text{ m s}^{-1}$. It strikes a smooth vertical barrier at an angle $\alpha$ (see diagram). The coefficient of restitution between $P$ and the barrier is 0.4. Given that the speed of $P$ is halved as a result of the collision, find the value of $\alpha$.
[5]
3. A smooth cylinder of radius $a$ is fixed with iA uniform rod $AB$, of length $l$ and mass $m$, rests in equilibrium with its lower end $A$ on a rough horizontal floor and the end $B$ against a smooth vertical wall. The rod is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$, and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring $CD$ which is in compression in a vertical line with its lower end $D$ fixed on the floor. The upper end $C$ is attached to the rod at a distance $\frac{1}{4}l$ from $B$ (see diagram). The coefficient of friction at $A$ between the rod and the floor is $\frac{1}{4}$ and the system is in limiting equilibrium.
(i) Show that the normal reaction of the floor at $A$ has magnitude $\frac{1}{2}mg$ and find the force in the spring.
(ii) Given7 The time, $T$ seconds, between successive cars passing a particular checkpoint on a wide road has probability density function $f$ given by
$$f(t) = \begin{cases}
\frac{1}{100} e^{-0.01t} & t \geqslant 0, \\
0 & \text{otherwise.}
\end{cases}$$
(i) State the expected value of $T$. [1]
(ii) Find the median value of $T$. [3]
Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes. [2]
8 The numbers of a particular type of laptop computer sold by a store on each of 100 consecutive Saturdays are summarised in the following table.
| Number sold | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | $\geqslant 11 Answer only one of the following two alternatives.
EITHER
A uniform plane object consists of three identical circular rings, $X, Y$ and $Z$, enclosed in a larger circular ring $W$. Each of the inner rings has mass $m$ and radius $r$. The outer ring has mass $3m$ and radius $R$. The centres of the inner rings lie at the vertices of an equilateral triangle of side $2r$. The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis $AB$ is the diameter of $W$ that passes through the centre of $X$ and the point of contact of $Y$ and $Z$ (see diagram). It is given that $R = \left(1 + \frac{2}{3}\sqrt{3}\right)r$.
(i) Show that the moment of inertia of the object about $AB$ is $(7 + 2\sqrt{3})mr^2$.
[8]
The line $CD$ is the diameter ofBLANK PAGE
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