科目：进阶数学 Further Mathematics (9231)
年份：2016 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. A particle $P$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle moves in complete vertical circles with centre $O$. The tension in the string when $P$ is at its lowest point is twice the tension in the string when $P$ is at its highest point. Find, in terms of $a$ and $g$, the greatest speed of $P$ during the motion. [6] 2. A uniform rod $AB$ of length $2a$ and weight $W$ rests with its lower end $A$ in contact with a rough horizontal surface. The rod rests on a smooth peg at the point $P$. A particle of weight $2W$ is suspended from the end $B$ of the rod. The system is in limiting equilibrium with the angle between the rod and the horizontal surface equal to $\theta$, where $\tan \theta =
6 Pia is practising for a high jump competition. In order to qualify for the competition, she needs to succeed in jumping a certain height. At each attempt the probability that she succeeds is $\frac{1}{4}$. The number of attempts that Pia makes, up to and including her first success, is denoted by the random variable $N$. State the mean and variance of $N$.
Find the probability that Pia
(i) succeeds in jumping the qualifying height in at most 4 attempts,
(ii) takes more than 6 attempts to succeed in jumping the qualifying height.
7 The lifetime, in months, of a particular type of light bulb is a random variable $T$ and its probability density function, $f$, is given by
$$f(t) = \begin{cases} \lambda e^{-2t} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$
where $\10 A Physics teacher takes a random sample of size 5 from his students. Their marks in a written test $(x)$ and a practical test $(y)$ are given in the following table. | Student | $A$ | $B$ | $C$ | $D$ | $E$ | | :---: | :---: | :---: | :---: | :---: | :---: | | $x$ | 11 | 14 | 16 | 12 | 15 | | $y$ | 9 | 12 | 14 | 13 | 15 | (i) Find the product moment correlation coefficient. [3] (ii) Test, at the 5% significance level, whether there is evidence of positive correlation. [4] The teacher takes a second random sample, this time of size 7, and the marks for this sample are summarised as follows. $$\Sigma x = 96 \quad \Sigma x^2 = 1366 \quad \Sigma y = 86 \quad \Sigma y^2 = 1096 \quad \Sigma xy = 1220$$ (iii) Use the combined sample of size 12 to find the equation of the re
11 Answer only one of the following two alternatives.
EITHER
The fixed points $A$ and $B$ are such that $AB = 3.2\text{ m}$ and $A$ is vertically above $B$. One end of a light elastic string, of natural length 0.8 m and modulus of elasticity 8 N, is attached to a particle $P$ of mass 0.2 kg. The other end of the string is attached to $A$. One end of a second light elastic string, of natural length 1.2 m and modulus of elasticity 4 N, is attached to $P$ and the other end is attached to $B$.
(i) Find the length of $AP$ when the particle is in equilibrium. [4]
The particle is now pulled vertically down from its equilibrium position and released from rest at the point where $AP = 1.65\text{ m}$.
(ii) Show that $P$ performs simple harmonic motion and find the period of the motionBLANK 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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the we examination series. Cambridge International Examinations is part of