科目：进阶数学 Further Mathematics (9231)
年份：2014 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1 Two small smooth spheres $A$ and $B$ have equal radii and have masses $m$ and $km$ respectively. They are moving in a straight line in the same direction on a smooth horizontal table. The speed of $A$ is $u$ and the speed of $B$ is $\frac{4}{3}u$. Sphere $A$ collides directly with sphere $B$. The coefficient of restitution between the spheres is $\frac{4}{5}$.
(i) Show that the speed of $A$ after the collision is $\frac{u(2k+5)}{5(k+1)}$. [5]
(ii) Given that the magnitude of the impulse experienced by $B$ during the collision is $\frac{2}{5}mu$, find the value of $k$. [2]
2 A particle $P$ of mass $m$ kg moves on an arc of a circle with centre $O$ and radius $a$ metres. At time $t=0$ the particle is at the point $A$. At time $t$ seconds, angle $POA = \sin^2 2t$. Show that the radA uniform rod $AB$ has mass $m$ and length $2d$. The rod rests in equilibrium on a smooth peg $C$, with the end $A$ resting on a rough horizontal plane. The distance $AC$ is $2a$ and the angle between $AB$ and the horizontal is $\alpha$, where $\cos \alpha = \frac{3}{5}$. A particle of mass $\frac{1}{2}m$ is attached to the rod at $B$ (see diagram). Find the normal reaction at $A$ and deduce that $d < \frac{25}{6}a$.
The coefficient of friction between the rod and the plane is $\mu$. Show that $\mu \geqslant \frac{8d}{25a - 6d}$.
A uniform rectangular lamina $ABCD$, in which $AB = 8a$ and $BC = 6a$, has mass $M$. A uniform circular lamina of radius $\frac{3}{2}a$ has mass $\frac{1}{2}M$. The two laminas are fixed together in the same plane with their centres coinciding at the point $6 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm. To try to reduce absence, the company decides to introduce ‘flexi-time’ and allow employees to work their seven hours each day at any time between 7 am and 9 pm. For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table. | Employee | A | B | C | D | E | F | G | H | I | J | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Before | 42 | 35 | 96 | 74 | 20 | 5 | 78 | 45 | 146 | 0 | | After | 34 | 32 | 100 | 72 | 31 | 2 | 61 | 35 | 140 | 0 | Use a paired sample $t$-test to test, at the 10% significance level, whether the population mean nu
10 The lengths of a random sample of eight fish of a certain species are measured, in cm, as follows.
$$17.3 \quad 15.8 \quad 18.2 \quad 15.6 \quad 16.0 \quad 18.8 \quad 15.3 \quad 15.0$$
Assuming that lengths are normally distributed,
(i) test, at the 10% significance level, whether the population mean length of fish of this species is greater than 15.8 cm.
(ii) calculate a 95% confidence interval for the population mean length of fish of this species.
11 Answer only one of the following two alternatives.
EITHER
A particle $P$ of mass $m$ is suspended from a fixed point by a light elastic string of natural length $l$, and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is $\frac{11}{7} l$. The particle is releaBLANK PAGE
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