科目：进阶数学 Further Mathematics (9231)
年份：2016 夏季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
1. A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N. The bullet enters the barrier with speed 320 m/s$^{-1}$ and emerges with speed 20 m/s$^{-1}$. You may assume that the motion takes place in a horizontal straight line. Find
(i) the magnitude of the impulse that acts on the bullet, [2]
(ii) the thickness of the barrier, [2]
(iii) the time taken for the bullet to pass through the barrier. [1]
2. A small smooth sphere $A$ of mass $m$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with an identical sphere $B$ which is initially at rest on the surface. The coefficient of restitution between the spheres is $e$. Sphere $B$ subsequently collides with a fixedA thin uniform rod $AB$ has mass $\frac{3}{4}m$ and length $3a$. The end $A$ of the rod is rigidly attached to a point on the circumference of a uniform disc with centre $C$, mass $m$ and radius $a$. The end $B$ of the rod is rigidly attached to a point on the circumference of a uniform disc with centre $D$, mass $4m$ and radius $2a$. The discs and the rod are in the same plane and $CABD$ is a straight line. The mid-point of $CD$ is $O$. The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis $l$, through $O$ in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about $l$ is $50ma^2$.
The object hangs in equilibrium with $D$ vertically below $C$. It is displaced through a sma8 The random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
2e^{-2x} & x \geqslant 0, \\
0 & \text{otherwise.}
\end{cases}$$
(i) Find the distribution function of $X$. [2]
(ii) Find the median value of $X$. [3]
The random variable $Y$ is defined by $Y = e^X$.
(iii) Find the probability density function of $Y$. [4]
9 Applicants for a national teacher training course are required to pass a mathematics test. Each year, the applicants are tested in groups of 6 and the number of successful applicants in each group is recorded. The overall proportion of successful applicants has remained constant over the years and is equal to 60% of the applicants. The results from 150 randomly chosen groups are shown in the following table.
| Number o11 Answer only one of the following two alternatives.
EITHER
The end $A$ of a uniform rod $AB$, of length $2a$ and weight $W$, is freely hinged to a vertical wall. The end $B$ of the rod is attached to a light elastic string of natural length $\frac{3}{2}a$ and modulus of elasticity $3W$. The other end of the string is attached to the point $C$ on the wall, where $C$ is vertically above $A$ and $AC = 2a$. A particle of weight $2W$ is attached to the rod at the point $D$, where $DB = \frac{1}{2}a$. The angle $ABC$ is equal to $\theta$ (see diagram). Show that $\cos \theta = \frac{3}{4}$ and find the tension in the string in terms of $W$.
Find the magnitude of the reaction force at the hinge.
OR
Petra is studying a particular species of bird. She takes a random sample of 12 biBLANK PAGE
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