科目：进阶数学 Further Mathematics (9231)
年份：2011 夏季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
1. A particle oscillates in simple harmonic motion with centre $O$. When its distance from $O$ is 3 m its speed is 16 m s$^{-1}$, and when its distance from $O$ is 4 m its speed is 12 m s$^{-1}$. Find the period and amplitude of the motion. [5]
2. A particle of mass $m$ is attached to the mid-point of a light elastic string. The string is stretched between two points $A$ and $B$ on a smooth horizontal surface, where $AB = 2a$. The string has modulus of elasticity $\lambda$ and natural length $2l$, where $l < a$. The particle is in motion on the surface along a line passing through the mid-point of $AB$ and perpendicular to $AB$. When the displacement of the particle from $AB$ is $x$, the tension in the string is $T$. Given that $x$ is small enough for $x^2$ to be neglected, show that
5 A particle $P$ of mass $m$ is placed at the point $Q$ on the outer surface of a fixed smooth sphere with centre $O$ and radius $a$. The acute angle between $OQ$ and the upward vertical is $\alpha$, where $\cos \alpha = \frac{9}{10}$. The particle is released from rest and begins to move in a vertical circle on the surface of the sphere. Show that $P$ loses contact with the sphere when OP makes an angle $\theta$ with the upward vertical, where $\cos \theta = \frac{3}{5}$, and find the speed of $P$ at this instant.
Show that, in the subsequent motion, when $P$ is at a distance $\frac{7}{5}a$ from the vertical diameter through $O$, its distance below the horizontal through $O$ is $\frac{11}{30}a$.
6 A random sample of residents in a town took part in a survey. They were asked whether 10 The mid-day temperature, $x$ °C, and the amount of sunshine, $y$ hours, were recorded at a winter holiday resort on each of 12 days, chosen at random during the winter season. The results are summarised as follows. $$\Sigma x = 18.7 \quad \Sigma x^2 = 106.43 \quad \Sigma y = 34.7 \quad \Sigma y^2 = 133.43 \quad \Sigma xy = 92.01$$ (i) Find the product moment correlation coefficient for the data. [3] (ii) Stating your hypotheses, test at the 1% significance level whether there is a non-zero correlation between mid-day temperature and amount of sunshine. [4] (iii) Use the equation of a suitable regression line to estimate the number of hours of sunshine on a day when the mid-day temperature is 2 °C. [3]
11 Answer only one of the following two alternatives.
EITHER
A rigid body is made from uniform wire of negligible thickness and is in the form of a square $ABCD$ of mass $M$ enclosed within a circular ring of radius $a$ and mass $2M$. The centres of the square and the circle coincide at $O$ and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through $O$, perpendicular to the plane of the body, is $\frac{8}{3}Ma^2$.
Hence find the moment of inertia of the body about an axis $l$, through $A$, in the plane of the body and tangential to the circle.
A particle $P$ of mass $M$ is now attached to the body at $C$. The system is able to rotate freely about the fixed axis $l$, which is horizontal. The system is rBLANK PAGE
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