科目：进阶数学 Further Mathematics (9231)
年份：2015 夏季
试卷：Paper 23
总页数：7
章节：9231 Further Mathematics
1 Two uniform small smooth spheres, $A$ and $B$, of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere $A$ receives an impulse of magnitude 8 N s in the direction $AB$. The coefficient of restitution between the spheres is $e$. Find, in terms of $e$, the speeds of $A$ and $B$ after $A$ collides with $B$.
Given that the spheres move in opposite directions after the collision, show that $e > \frac{2}{3}$.
2 A uniform sphere $P$ of mass $m$ is at rest on a smooth horizontal table. The sphere is projected along the table with speed $u$ and strikes a smooth vertical barrier $A$ at an acute angle $\alpha$. It then strikes another smooth vertical barrier $B$ which is at right angles to $A$ (see diagram). The coefficient of The diagram shows a central cross-section $CDEF$ of a uniform solid cube of weight $W$ and with edges of length $2a$. The cube rests on a rough horizontal plane. A thin uniform rod $AB$, of weight $W$ and length $6a$, is hinged to the plane at $A$. The rod rests in smooth contact with the cube at $C$, with angle $CAD$ equal to $30^\circ$. The rod is in the same vertical plane as $CDEF$. The coefficient of friction between the plane and the cube is $\mu$. Given that the system is in equilibrium, show that $\mu \geqslant \frac{1}{25}\sqrt{3}$.
Find the magnitude of the force acting on the rod at $A$.
The end $B$ of a uniform rod $AB$, of mass $3M$ and length $4a$, is rigidly attached to a point on the circumference of a uniform disc. The disc has centre $O$, mass $2M$ and radius $a$, a6 The reliability of the broadband connection received from two suppliers, $A$ and $B$, is classified as good, fair or poor by a random sample of householders. The information collected is summarised in the following table. | | | Reliability | | | :--- | :--- | :--- | :--- | | | Good | Fair | Poor | | Supplier $A$ | 65 | 63 | 33 | | Supplier $B$ | 51 | 44 | 44 | Test, at the 5% significance level, whether reliability is independent of supplier. [8] 7 For a random sample of 10 observations of pairs of values $(x, y)$, the equation of the regression line of $y$ on $x$ is $y = 3.25x - 4.27$. The sum of the ten $x$ values is 15.6 and the product moment correlation coefficient for the sample is 0.56. Find the equation of the regression line of $x$ on $y$. [7] Test, at the 5% sig
10 Answer only one of the following two alternatives.
EITHER
One end of a light inextensible string of length $\frac{3}{2}a$ is attached to a fixed point $O$ on a horizontal surface. The other end of the string is attached to a particle $P$ of mass $m$. The string passes over a small fixed smooth peg $A$ which is at a distance $a$ vertically above $O$. The system is in equilibrium with $P$ hanging vertically below $A$ and the string taut. The particle is projected horizontally with speed $u$ (see diagram). When $P$ is at the same horizontal level as $A$, the tension in the string is $T$. Show that $T = \frac{2m}{a}(u^2 - ag)$.
[3]
The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is 5:1.
(i) Show that $u^BLANK PAGE
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