科目：进阶数学 Further Mathematics (9231)
年份：2022 冬季
试卷：Paper 32
总页数：14
章节：9231 Further Mathematics
1. 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 $O$. The string is held taut with $OP$ making an angle $\alpha$ with the downward vertical, where $\cos \alpha = \frac{2}{3}$. The particle $P$ is projected perpendicular to $OP$ in an upwards direction with speed $\sqrt{3ag}$. It then starts to move along a circular path in a vertical plane.
Find the cosine of the angle between the string and the upward vertical when the string first becomes slack. [4]A uniform lamina is in the form of a triangle $ABC$ in which angle $B$ is a right angle, $AB = 9a$ and $BC = 6a$. The point $D$ is on $BC$ such that $BD = x$ (see diagram). The region $ABD$ is removed from the lamina. The resulting shape $ADC$ is placed with the edge $DC$ on a horizontal surface and the plane $ADC$ is vertical. Find the set of values of $x$, in terms of $a$, for which the shape is in equilibrium. [6]
3 One end of a light elastic string, of natural length $a$ and modulus of elasticity $\frac{16}{3}Mg$, is attached to a fixed point $O$. A particle $P$ of mass $4M$ is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass $2M$ is attached to $P$ and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance $\frac{1}{4}a$ is $v$.
Find an expression for $v$ in terms of $g$ and $a$. [6]5 © UCLES 2022 9231/32/O/N/22 [Turn over]
4 A particle $P$ of mass 5 kg moves along a horizontal straight line. At time $t$ s, the velocity of $P$ is $\nu$ m/s$^{-1}$ and its displacement from a fixed point $O$ on the line is $x$ m. The forces acting on $P$ are a force of magnitude $\frac{500}{\nu}$ N in the direction $OP$ and a resistive force of magnitude $\frac{1}{2}\nu^2$ N. When $t = 0$, $x = 0$ and $\nu = 5$.
(a) Find an expression for $\nu$ in terms of $x$. [6](b) State the value that the speed approaches for large values of $x$. [1]
5 A particle $P$ is projected with speed $u \text{ m s}^{-1}$ at an angle of $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t$ are denoted by $x\text{ m}$ and $y\text{ m}$ respectively.
(a) Show that the equation of the trajectory is given by
$$y = x \tan \theta - \frac{gx^2}{2u^2}(1 + \tan^2 \theta).$$
[4]In the subsequent motion $P$ passes through the point with coordinates (30, 20).
(b) Given that one possible value of $\tan \theta$ is $\frac{4}{3}$, find the other possible value of $\tan \theta$. [5]A light inextensible string is threaded through a fixed smooth ring $R$ which is at a height $h$ above a smooth horizontal surface. One end of the string is attached to a particle $A$ of mass $m$. The other end of the string is attached to a particle $B$ of mass $\frac{6}{7} m$. The particle $A$ moves in a horizontal circle on the surface. The particle $B$ hangs in equilibrium below the ring and above the surface (see diagram).
When $A$ has constant angular speed $\omega$, the angle between $AR$ and $BR$ is $\theta$ and the normal reaction between $A$ and the surface is $N$.
When $A$ has constant angular speed $\frac{3}{2}\omega$, the angle between $AR$ and $BR$ is $\alpha$ and the normal reaction between $A$ and the surface is $\frac{1}{2} N$.
(a) Show that $\cos \theta = \frac{4(b) Find $N$ in terms of $m$ and $g$ and find the value of $\cos \alpha$. [4]
Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac{1}{2} m$ respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere $A$ is travelling with speed $u$ and its direction of motion makes an angle $\alpha$ with the line of centres. Sphere $B$ is travelling with speed $2u$ and its direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{5}{8}$ and $\alpha + \beta = 90^\circ$.
(a) Find the component of the velocity of $B$ parallel to the line of centres after the collision, giving your answer in terms of $u$ and $\alpha$. [4]The direction of motion of $B$ after the collision is parallel to the direction of motion of $A$ before the collision. (b) Find the value of $\tan \alpha$. [5]
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