科目：进阶数学 Further Mathematics (9231)
年份：2013 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1. A bullet of mass $m$ kg is fired into a fixed vertical barrier. It enters the barrier horizontally with speed $280 \text{ m/s}^{-1}$ and emerges horizontally after 0.01 s with speed $30 \text{ m/s}^{-1}$. There is a constant horizontal resisting force of magnitude $1500 \text{ N}$. Find $m$.
2. A particle $P$ travels on a smooth surface whose vertical cross-section is in the form of two arcs of circles. The first arc $AB$ is a quarter of a circle of radius $\frac{5}{4}a$ and centre $O$. The second arc $BC$ is a quarter of a circle of radius $a$ and centre $Q$. The two arcs are smoothly joined at $B$. The point $Q$ is vertically below $O$ and the two arcs are in the same vertical plane. The particle $P$ is projected vertically downwards from $A$ with speed $u$. When $P$ is on the arc ABCD is a uniform rectangular lamina of mass $m$ in which $AB = 4a$ and $BC = 2a$. The lines AC and BD intersect at O. The mid-points of OA, OB, OC, OD are E, F, G, H respectively. The rectangle EFGH, in which $EF = 2a$ and $FG = a$, is removed from ABCD (see diagram). The resulting lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis through A and perpendicular to the plane of ABCD. Show that the moment of inertia of this lamina about the axis is $\frac{85}{16}ma^2$.
The lamina hangs in equilibrium under gravity with $C$ vertically below A. The point $C$ is now given a speed $u$. Given that the lamina performs complete revolutions, show that
$$u^2 > \frac{192\sqrt{5}}{17}ag.$$
6 Six pairs of values of variables $x$ and $y$ are measured. Draw a sketch 8 The number, $x$, of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country $A$. The number, $y$, of the same type of shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country $B$. The results are summarised as follows. $$\Sigma x = 1752 \quad \Sigma x^2 = 5500 \quad \Sigma y = 1220 \quad \Sigma y^2 = 33500$$ Find a 95% confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country $A$ and in country $B$. 9 A researcher records a random sample of $n$ pairs of values of $(x, y)$, giving the following summarised data. $$\Sigma x = 24 \quad \Sigma x^2 = 160 \quad \Sigma y = 34 \quad \Sigma y^2 = 324 \quad \Sig
11 Answer only one of the following two alternatives.
EITHER
A uniform rod $AB$ rests in limiting equilibrium in a vertical plane with the end $A$ on rough horizontal ground and the end $B$ against a rough vertical wall that is perpendicular to the plane of the rod. The angle between the rod and the ground is $\theta$. The coefficient of friction between the rod and the wall is $\mu$, and the coefficient of friction between the rod and the ground is $2\mu$. Show that $\tan \theta = \frac{1 - 2\mu^2}{4\mu}$.
Given that $\theta \leq 45^\circ$, find the set of possible values of $\mu$.
OR
A researcher is investigating the relationship between the political allegiance of university students and their childhood environment. He chooses a random sample of 100 students and finds thaBLANK PAGE
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