科目：进阶数学 Further Mathematics (9231)
年份：2012 冬季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. A particle $P$ is moving in a circle of radius 1.5 m. At time $t$ its velocity is $(k - t^2) \text{ ms}^{-1}$, where $k$ is a positive constant. When $t = 3$, the magnitudes of the radial and transverse components of the acceleration of $P$ are equal. Find the possible values of $k$.
[4]
2. A small bead of mass $m$ is threaded on a thin smooth wire which forms a circle of radius $a$. The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass $4m$ which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed $\sqrt{(kga)}$. Show that, when the angle between the two parts of the 5 Four identical uniform rods, each of mass $m$ and length $2a$, are rigidly joined to form a square frame $ABCD$. Show that the moment of inertia of the frame about an axis through $A$ perpendicular to the plane of the frame is $\frac{40}{3}ma^2$.
The frame is suspended from $A$ and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through $A$. When the frame is at rest with $C$ vertically below $A$, it is given an angular velocity $\sqrt{\left(\frac{6g}{5a}\right)}$. Find the angular velocity of the frame when $AC$ makes an angle $\theta$ with the downward vertical through $A$.
When $AC$ is horizontal, the speed of $C$ is $k\sqrt{ga}$. Find the value of $k$ correct to 3 significant figures.
6 The random variable $X$ has probability density funct9 Experiments are conducted to test the breaking strength of each of two types of rope, $P$ and $Q$. A random sample of 50 ropes of type $P$ and a random sample of 70 ropes of type $Q$ are selected. The breaking strengths, $p$ and $q$, measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p^2 = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q^2 = 3310.0$$ Test, at the 10% significance level, whether the mean breaking strengths of type $P$ and type $Q$ ropes are the same. [10] 10 Delegates who travelled to a conference were asked to report the distance, $y$ km, that they had travelled and the time taken, $x$ minutes. The values reported by a random sample of 8 delegates are given in the following table. | Delegate | $A$ | $B$ | $C$ | $D$ | $E$ | $
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 $8mg$ and natural length $a$. The other end of the string is attached to a fixed point $O$. The particle is pulled vertically downwards a distance $\frac{1}{4}a$ from its equilibrium position and released from rest. Show that the string first becomes slack after a time $\frac{2\pi}{3}\sqrt{\left(\frac{a}{8g}\right)}$.
Find, in terms of $a$, the total distance travelled by $P$ from its release until it subsequently comes to instantaneous rest for the first time.
OR
The continuous random variable $X$ takes values in the interval $0 \leqslant x \leqslant 5$ only. For $0 \leqslant x \leqslant 5$ the graph of its probabiBLANK PAGE
BLANK PAGE
BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.