科目：进阶数学 Further Mathematics (9231)
年份：2013 夏季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
A uniform rod $AB$, of mass $m$ and length $4a$, rests with the end $A$ on rough horizontal ground. The point $C$ on $AB$ is such that $AC = 3a$. A light inextensible string has one end attached to the point $P$ which is at a distance $5a$ vertically above $A$, and the other end attached to $C$. The rod and the string are in the same vertical plane and the system is in equilibrium with angle $ACP$ equal to $90^\circ$ (see diagram). The coefficient of friction between the rod and the ground is $\mu$. Show that the least possible value of $\mu$ is $\frac{24}{43}$.
2 Three uniform small smooth spheres, $A, B$ and $C$, have equal radii. Their masses are $4m, 2m$ and $m$ respectively. They lie in a straight line on a smooth horizontal surface with $B$ between $A$ and $C$. Initially $A$ is moThe end $A$ of a uniform rod $AB$, of mass $4m$ and length $3a$, is rigidly attached to a point on a uniform spherical shell, of mass $\lambda m$ and radius $3a$. The end $B$ of the rod is rigidly attached to a point on a uniform ring. The ring has centre $O$, mass $4m$ and radius $\frac{1}{2}a$. The ring and the rod are in the same vertical plane. The line $OBA$, extended, passes through the centre of the spherical shell. $BC$ is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through $C$ perpendicular to the plane of the ring, is $(30 + 55\lambda)ma^2$.
Given that the system performs small oscillations of period $2\pi \sqrt{\left(\frac{5a}{g}\right)}$ about this axis, find the value of $\lambda$.
For a random sampl7 A random sample of 80 observations of the continuous random variable $X$ was taken and the values are summarised in the following table.
| Interval | $2 \leqslant x < 3$ | $3 \leqslant x < 4$ | $4 \leqslant x < 5$ | $5 \leqslant x < 6$ |
| :--- | :---: | :---: | :---: | :---: |
| Observed frequency | 36 | 29 | 9 | 6 |
It is required to test the goodness of fit of the distribution having probability density function $f$ given by
$$f(x) = \begin{cases}
\frac{3}{x^2} & 2 \leqslant x < 6, \\
0 & \text{otherwise.}
\end{cases}$$
Show that the expected frequency for the interval $2 \leqslant x < 3$ is 40 and calculate the remaining expected frequencies. [4]
Carry out a goodness of fit test, at the 10% significance level. [5]
8 The continuous random variable $X$ has proba10 Answer only one of the following two alternatives.
EITHER
A light elastic string has modulus of elasticity $\frac{3}{2}mg$ and natural length $a$. A particle of mass $m$ is attached to one end of the string. The other end of the string is attached to a fixed point $A$. The particle is released from rest at $A$. Show that when the particle has fallen a distance $ka$ from $A$, where $k > 1$, its kinetic energy is
$$\frac{1}{4}mga(10k - 3 - 3k^2).$$
[3]
Show that the particle first comes to instantaneous rest at the point $B$ which is at a distance $3a$ vertically below $A$. [3]
Show that the time taken by the particle to travel from $A$ to $B$ is
$$\sqrt{\left(\frac{2a}{g}\right)} + \frac{2\pi}{3}\sqrt{\left(\frac{2a}{3g}\right)}.$$
[8]
OR
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