科目：进阶数学 Further Mathematics (9231)
年份：2015 冬季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
A uniform ladder $AB$, of length $3a$ and weight $W$, rests with the end $A$ in contact with smooth horizontal ground and the end $B$ against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point $C$, where $AC = a$. The other end of the rope is fixed to the point $D$ at the base of the wall and the rope $DC$ is in the same vertical plane as the ladder $AB$. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle $\theta$ with the horizontal and the rope making an angle $\alpha$ with the horizontal (see diagram). It is given that $\tan \theta = 2\tan \alpha$. Find, in terms of $W$ and $\alpha$, the tension in the rope and the magnitudes of the forces acting on the ladder at $A$ and at $B$.
4 A particle $P$ of mass $m$ 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 $O$. When $P$ is hanging at rest vertically below $O$, it is projected horizontally. In the subsequent motion $P$ completes a vertical circle. The speed of $P$ when it is at its highest point is $u$. Show that the least possible value of $u$ is $\sqrt{(ag)}$.
It is now given that $u = \sqrt{(ag)}$. When $P$ passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass $\frac{1}{4} m$. Find the speed of the combined particle immediately after the collision.
In the subsequent motion, when $OP$ makes an angle $\theta$ with the upward vertical the tension in the string is $T$. Find an 8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. | Number of goals | 0 | 1 | 2 | 3 | 4 | 5 | 6 or more | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Frequency | 12 | 16 | 31 | 25 | 13 | 3 | 0 | Fit a Poisson distribution to the data, and test its goodness of fit at the 5% significance level. [10] 9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, $x$, and in French, $y$, are summarised as follows. $$\Sigma x = 472 \quad \Sigma x^2 = 29950 \quad \Sigma y = 400 \quad \Sigma y^2 = 21226 \quad \Sigma xy = 24879$$ Another student scored 72 marks in the Mathematics examination but wa
10 Answer only one of the following two alternatives.
EITHER
An object is formed by attaching a thin uniform rod $PQ$ to a uniform rectangular lamina $ABCD$. The lamina has mass $m$, and $AB = DC = 6a$, $BC = AD = 3a$. The rod has mass $M$ and length $3a$. The end $P$ of the rod is attached to the mid-point of $AB$. The rod is perpendicular to $AB$ and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis $l_1$, through $Q$ and perpendicular to the plane of the lamina, is $3(8m + M)a^2$.
[4]
Show that the moment of inertia of the object about a smooth horizontal axis $l_2$, through the mid-point of $PQ$ and perpendicular to the plane of the lamina, is $\frac{4}{3}(17m + M)a^2$.
[2]
Find expressions for the pBLANK PAGE
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