科目：进阶数学 Further Mathematics (9231)
年份：2012 夏季
试卷：Paper 22
总页数：3
章节：9231 Further Mathematics
1. A circular flywheel of radius 0.3 m, with moment of inertia about its axis 18 kg m$^2$, is rotating freely with angular speed 6 rad s$^{-1}$. A tangential force of constant magnitude 4 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to 2 rad s$^{-1}$.
2. Two particles, of masses $3m$ and $m$, are moving in the same straight line towards each other with speeds $2u$ and $u$ respectively. When they collide, the impulse acting on each particle has magnitude $4mu$. Show that the total loss in kinetic energy is $\frac{4}{3}mu^2$.
3. A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt{\left(\frac{2}{3}ga\right)}$ from the lowest point of the inside of a fixed hollow smo6 The probability that a particular type of light bulb is defective is 0.01. A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the $N$th to be tested. Write down the value of $E(N)$. [1] Find the least value of $n$ such that $P(N \leq n)$ is greater than 0.9. [3] 7 A random sample of 8 swimmers from a swimming club were timed over a distance of 100 metres, once in an outdoor pool and once in an indoor pool. Their times, in seconds, are given in the following table. | Swimmer | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $H$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Outdoor time | 66.2 | 62.4 | 60.8 | 6
11 Answer only one of the following two alternatives.
EITHER
A particle $P$ of mass $m$ is attached to one end of a light elastic string of modulus of elasticity $4mg$ and natural length $l$. The other end of the string is attached to a fixed point $O$. The particle rests in equilibrium at the point $E$, vertically below $O$. The particle is pulled down a vertical distance $\frac{1}{4}l$ from $E$ and released from rest. Show that the motion of $P$ is simple harmonic with period $\pi \sqrt{\left(\frac{l}{g}\right)}$. [4]
At an instant when $P$ is moving vertically downwards through $E$, the string is cut. When $P$ has descended a further distance $\frac{7}{10}l$ under gravity, it strikes a fixed smooth plane which is inclined at $30^\circ$ to the horizontal. The coefficient of rest