科目：进阶数学 Further Mathematics (9231)
年份：2016 冬季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
The point $C$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $AC = 4\text{ m}$ and $CB = 2\text{ m}$, with $C$ between $A$ and $B$. The point $M$ is the mid-point of $AB$ (see diagram). A particle $P$ of mass $m$ oscillates between $A$ and $B$ in simple harmonic motion. When $P$ is at $C$, its speed is $4\text{ m s}^{-1}$. Find
(i) the magnitude of the maximum acceleration of $P$,
(ii) the number of complete oscillations made by $P$ in one minute,
(iii) the time that $P$ takes to travel directly from $A$ to $C$.
Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of $60^\circ$. A small smooth sphere $P$ is moving on the horizontal surface with speed $u$ when it collides with the first vertical wall at the point The end $P$ of a uniform rod $PQ$, of weight $kW$ and length $8a$, is rigidly attached to a point on the surface of a uniform sphere with centre $C$, weight $W$ and radius $a$. The end $Q$ is rigidly attached to a point on the surface of an identical sphere with centre $D$. The points $C, P, Q$ and $D$ are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point $A$, and the other sphere in contact with a smooth vertical wall, at the point $B$. The angle between $CD$ and the horizontal is $\theta$. The point $B$ is at a height of $7a$ above the base of the wall (see diagram). The points $A, B, C, D, P$ and $Q$ are all in the same vertical plane.
(i) Show that $\sin \theta = \frac{3}{5}$.
The ob5 The distance, $X$ km, completed by a new car before any mechanical fault occurs has distribution function $F$ given by
$$F(x) = \begin{cases}
1 - e^{-ax} & x \geqslant 0, \\
0 & \text{otherwise},
\end{cases}$$
where $a$ is a positive constant. The mean value of $X$ is 10 000. Find
(i) the value of $a$, [1]
(ii) the probability that a new car completes less than 15 000 km before any mechanical fault occurs. [2]
The probability that a new car completes at least $d$ km before any mechanical fault occurs is 0.75.
(iii) Find the value of $d$. [3]
6 A random sample of 8 observations of a normal random variable $X$ has mean $\bar{x}$, where
$$\bar{x} = 6.246 \quad \text{and} \quad \Sigma(x - \bar{x})^2 = 0.784.$$
Test, at the 5% significance level, whether the pop9 The number of visitors arriving at an art exhibition is recorded for each 10-minute period of time during the ten hours that it is open on a particular day. The results are as follows. | Number of visitors in a 10-minute period | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ≥9 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Number of 10-minute periods | 2 | 2 | 12 | 8 | 11 | 13 | 4 | 7 | 1 | 0 | (i) Calculate the mean and variance for this sample and explain whether your answers support a suggestion that a Poisson distribution might be a suitable model for the number of visitors in a 10-minute period. (ii) Use an appropriate Poisson distribution to find the two expected frequencies missing from the following table. | Number of visitors in a 10-minu
10 Answer only one of the following two alternatives.
EITHER
A thin uniform rod $AB$ has mass $2m$ and length $3a$. Two identical uniform discs each have mass $\frac{1}{2}m$ and radius $a$. The centre of one of the discs is rigidly attached to the end $A$ of the rod and the centre of the other disc is rigidly attached to the end $B$ of the rod. The plane of each disc is perpendicular to the rod $AB$. A second thin uniform rod $OC$ has mass $m$ and length $2a$. The end $C$ of this rod is rigidly attached to the mid-point of $AB$, with $OC$ perpendicular to $AB$ (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis $l$, through $O$, which is perpendicular to both rods.
(i) Show that the moment of inertia of one of the discs abouBLANK PAGE
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