科目：进阶数学 Further Mathematics (9231)
年份：2008 冬季
试卷：Paper 2
总页数：7
章节：9231 Further Mathematics
A uniform wire, of length $24a$ and mass $m$, is bent into the form of a triangle $ABC$ with angle $ABC = 90^\circ$, $AB = 6a$ and $BC = 8a$ (see diagram). Find the moment of inertia of the wire about an axis through $A$ perpendicular to the plane of the wire.
2. A small bead $B$ of mass $m$ is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius $a$ and centre $O$. The highest point of the circle is $A$. The bead is slightly displaced from rest at $A$. When angle $AOB = \theta$, where $\theta < \cos^{-1}\left(\frac{2}{3}\right)$, the force exerted on the bead by the wire has magnitude $R_1$. When angle $AOB = \pi + \theta$, the force exerted on the bead by the wire has magnitude $R_2$. Show that $R_2 - R_1 = 4mg$.
3. A uniform disc, of mass $m$ and 4 Two smooth spheres $A$ and $B$, of equal radii, have masses 0.1 kg and $m$ kg respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of $A$ is 5 m s$^{-1}$, and the speed of $B$ is 2 m s$^{-1}$.
(i) Assume that in the collision $A$ does not change direction. The speeds of $A$ and $B$ after the collision are $\nu_A$ m s$^{-1}$ and $\nu_B$ m s$^{-1}$ respectively. Express $m$ in terms of $\nu_A$ and $\nu_B$, and hence show that $m < 0.25$.
(ii) Assume instead that $m = 0.2$ and that the coefficient of restitution between the spheres is $\frac{1}{2}$. Find the magnitude of the impulse acting on $A$ in the collision.
5 A particle of mass $m$ moves in a straight line $AB$ of length 9 A sample of 100 observations of the continuous random variable $T$ was obtained and the values are summarised in the following table.
| Interval | $1 \leq t < 1.5$ | $1.5 \leq t < 2$ | $2 \leq t < 2.5$ | $2.5 \leq t < 3$ |
| :--- | :---: | :---: | :---: | :---: |
| Frequency | 64 | 17 | 16 | 3 |
It is required to test the goodness of fit of the distribution with probability density function given by
$$f(t) = \begin{cases}
\frac{9}{4t^3} & 1 \leq t < 3, \\
0 & \text{otherwise.}
\end{cases}$$
The relevant expected values are as follows.
| Interval | $1 \leq t < 1.5$ | $1.5 \leq t < 2$ | $2 \leq t < 2.5$ | $2.5 \leq t < 3$ |
| :--- | :---: | :---: | :---: | :---: |
| Expected frequency | 62.5 | 21.875 | 10.125 | 5.5 |
Show how the expected value 10.125 is obtained.11 Answer only one of the following two alternatives.
EITHER
The diagram shows a central cross-section CDEF of a uniform solid cube of weight $kW$ with edges of length $4a$. The cube rests on a rough horizontal floor. One of the vertical faces of the cube is parallel to a smooth vertical wall and at a distance $5a$ from it. A uniform ladder, of length $10a$ and weight $W$, is represented by $AB$. The ladder rests in equilibrium with $A$ in contact with the rough top surface of the cube and $B$ in contact with the wall. The distance $AC$ is $a$ and the vertical plane containing $AB$ is perpendicular to the wall. The coefficients of friction between the ladder and the cube, and between the cube and the floor, are both equal to $\mu$. A small dog of weight $\frac{1}{4}W$ climbs the laddBLANK PAGE
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