科目：进阶数学 Further Mathematics (9231)
年份：2008 夏季
试卷：Paper 2
总页数：7
章节：9231 Further Mathematics
1. A bullet of mass 20 grams is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed $240 \text{ m s}^{-1}$ and emerges $0.004 \text{ s}^{-1}$ later with speed $10 \text{ m s}^{-1}$. Assuming that the motion occurs in a horizontal straight line, find
(i) the magnitude of the impulse that acts on the bullet.
(ii) the magnitude of the resisting force, assuming that it is constant.
2. A particle $P$ moves on a straight line $AOA'$ in simple harmonic motion with period $T$. The centre of the motion is $O$ and $P$ is instantaneously at rest at $A$ and $A'$. The mid-points of $OA$ and $OA'$ are $B$ and $B'$ respectively.
(i) Find, in terms of $T$, the time for $P$ to move directly from $B$ to $B'$.
(ii) Find the ratio of the sTwo identical smooth balls, $P$ and $Q$, are projected simultaneously towards each other from two points on horizontal ground. $P$ is projected with speed $u$ at an angle $\tan^{-1}\left(\frac{4}{3}\right)$ to the horizontal and $Q$ is projected with speed $ku$ at an angle $\tan^{-1}\left(\frac{4}{3}\right)$ to the horizontal (see diagram). The balls collide when they are moving horizontally. It may be assumed that there is no air resistance. Find the value of $k$.
The coefficient of restitution between the balls is $e$. Find, in terms of $e$, $u$ and $g$, the distance between the points where the balls first hit the ground.
A uniform circular disc has radius 0.5 m, centre $O$ and mass per unit area 6.20 kg m$^{-2}$. A rectangle $ABCD$, where $AC$ and $BD$ intersect at $O$, is remove7 The time for a certain kind of plaster to dry has a normal distribution with mean $\mu$ minutes. In order to find a confidence interval for $\mu$, the times for a random sample of 10 specimens to dry were measured. The results, $x$ minutes, are summarised by
$$\Sigma x = 285.4, \quad \Sigma(x - \bar{x})^2 = 448.65.$$
Calculate a 95% confidence interval for $\mu$. [5]
Estimate the width of a 95% confidence interval for $\mu$ based on a random sample of 20 specimens. [3]
8 The number of serious faults on a randomly chosen 1 km stretch of motorway has a Poisson distribution with mean 2.1. The random variable $X$ km is the distance between two successive serious faults on the motorway.
(i) Explain, using a Poisson distribution, why, for $x \geqslant 0$,
$$P(X > x) = e^{-2.111 Answer only one of the following two alternatives.
EITHER
A particle $P$ of mass $m$ moves on an arc of a circle with centre $O$ and radius $a$. At time $t = 0$ the particle is at the point $A$ and at time $t$ the angle $POA = \theta$, where $\theta = \sin^2 kt$ and $k$ is a positive constant. Find constants $b$ and $c$ such that $\ddot{\theta} = b + c\theta$, and deduce that $P$ moves in simple harmonic motion along the arc. State the centre and amplitude of the motion. [9]
Show that the magnitude of the resultant force acting on $P$ at time $t$ is $mak^2(1 + \cos^2 2kt)$. [5]
OR
Each of 100 coins is tossed repeatedly until a head is obtained, and the number, $x$, of tosses needed is recorded. The results are summarised in the following table.
| $x$ | 1 | 2 | 3 | 4 | BLANK PAGE
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