科目：进阶数学 Further Mathematics (9231)
年份：2013 冬季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
Three identical uniform rods, $AB$, $BC$ and $CD$, each of mass $M$ and length $2a$, are rigidly joined to form three sides of a square. A uniform circular disc, of mass $\frac{2}{3}M$ and radius $a$, has the opposite ends of one of its diameters attached to $A$ and $D$ respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis $AD$. [6]
2 The point $O$ is on the fixed line $l$. The point $A$ on $l$ is such that $OA = 3\text{ m}$. A particle $P$ oscillates on $l$ in simple harmonic motion with centre $O$ and period $\pi$ seconds. When $P$ is at $A$ its speed is $12\text{ m s}^{-1}$. Find the speed of $P$ when it is at the point $B$ on $l$, where $OB = 6\text{ m}$ and $B$ is on the same side of $O$ as $A$. [4]
FA uniform circular disc, with centre $O$ and weight $W$, rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at $A$ and with the wall at $B$. A force of magnitude $P$ acts tangentially on the disc at the point $C$ on the edge of the disc, where the radius $OC$ makes an angle $\theta$ with the upward vertical, and tan $\theta = \frac{3}{4}$ (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is $\frac{1}{2}$. Show that the sum of the magnitudes of the frictional forces at $A$ and $B$ is equal to $P$.
Given that the equilibrium is limiting at both $A$ and $B$,
(i) show that $P = \frac{15}{34} W$.
(ii) find 8 The lifetime, in years, of an electrical component is the random variable $T$, with probability density function $f$ given by
$$f(t) = \begin{cases} A e^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$
where $A$ and $\lambda$ are positive constants.
(i) Show that $A = \lambda$. [2]
It is known that out of 100 randomly chosen components, 16 failed within the first year.
(ii) Find an estimate for the value of $\lambda$, and hence find an estimate for the median value of $T$. [6]
9 For a random sample of 10 observations of pairs of values $(x, y)$, the equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are
$$y = 4.21x - 0.862 \quad \text{and} \quad x = 0.043y + 6.36,$$
respectively.
(i) Find the value of the product moment correl11 Answer only one of the following two alternatives.
EITHER
A smooth sphere, with centre $O$ and radius $a$, is fixed on a smooth horizontal plane $\Pi$. A particle $P$ of mass $m$ is projected horizontally from the highest point of the sphere with speed $\sqrt{\left(\frac{1}{2}ga\right)}$. While $P$ remains in contact with the sphere, the angle between $OP$ and the upward vertical is denoted by $\theta$. Show that $P$ loses contact with the sphere when $\cos \theta = \frac{4}{5}$. [6]
Subsequently the particle collides with the plane $\Pi$. The coefficient of restitution between $P$ and $\Pi$ is $\frac{5}{9}$. Find the vertical height of $P$ above $\Pi$ when the vertical component of the velocity of $P$ first becomes zero. [6]
OR
A factory produces bottles of spring water.6 BLANK PAGE
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