科目：进阶数学 Further Mathematics (9231)
年份：2020 夏季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
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1 A particle $P$ is projected with speed $u$ at an angle of $30^\circ$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time $T$ after projection.
Find, in terms of $u$, the speed of $P$ at time $\frac{2}{3} T$ after projection. [5]A light inextensible string of length $a$ is threaded through a fixed smooth ring $R$. One end of the string is attached to a particle $A$ of mass $3m$. The other end of the string is attached to a particle $B$ of mass $m$. The particle $A$ hangs in equilibrium at a distance $x$ vertically below the ring. The angle between $AR$ and $BR$ is $\theta$ (see diagram). The particle $B$ moves in a horizontal circle with constant angular speed $2\sqrt{\frac{g}{a}}$.
Show that $\cos \theta = \frac{1}{3}$ and find $x$ in terms of $a$. [5]5 © UCLES 2020 9231/31/M/J/20 [Turn over]
3 One end of a light elastic spring, of natural length $a$ and modulus of elasticity $5mg$, is attached to a fixed point $A$. The other end of the spring is attached to a particle $P$ of mass $m$. The spring hangs with $P$ vertically below $A$. The particle $P$ is released from rest in the position where the extension of the spring is $\frac{1}{2}a$.
(a) Show that the initial acceleration of $P$ is $\frac{3}{2}g$ upwards. [3](b) Find the speed of $P$ when the spring first returns to its natural length. [4]
A uniform square lamina $ABCD$ has sides of length 10 cm. The point $E$ is on $BC$ with $EC = 7.5\text{cm}$, and the point $F$ is on $DC$ with $CF = x\text{cm}$. The triangle $EFC$ is removed from $ABCD$ (see diagram). The centre of mass of the resulting shape $ABEFD$ is a distance $\bar{x}$ cm from $CB$ and a distance $\bar{y}$ cm from $CD$.
(a) Show that $\bar{x} = \frac{400 - x^2}{80 - 3x}$ and find a corresponding expression for $\bar{y}$. [4]The shape $ABEFD$ is in equilibrium in a vertical plane with the edge $DF$ resting on a smooth horizontal surface.
(b) Find the greatest possible value of $x$, giving your answer in the form $a+b\sqrt{2}$, where $a$ and $b$ are constants to be determined. [3]5 A particle $P$ is moving along a straight line with acceleration $3ku - kv$ where $v$ is its velocity at time $t$, $u$ is its initial velocity and $k$ is a constant. The velocity and acceleration of $P$ are both in the direction of increasing displacement from the initial position. (a) Find the time taken for $P$ to achieve a velocity of $2u$. [3]
(b) Find an expression for the displacement of $P$ from its initial position when its velocity is $2u$. [5]
6 A particle $P$ of mass $m$ is moving with speed $u$ on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of $P$ makes an angle $\alpha$ with the barrier. The coefficient of restitution between $P$ and the barrier is $e$. As a result of the impact, the direction of motion of $P$ is turned through $90^\circ$.
(a) Show that $\tan^2 \alpha = \frac{1}{e}$. [3]The particle $P$ loses two-thirds of its kinetic energy in the impact. (b) Find the value of $\alpha$ and the value of $e$. [5]
7 A hollow cylinder of radius $a$ is fixed with its axis horizontal. A particle $P$, of mass $m$, moves in part of a vertical circle of radius $a$ and centre $O$ on the smooth inner surface of the cylinder. The speed of $P$ when it is at the lowest point $A$ of its motion is $\sqrt{\frac{7}{2}}ga$.
The particle $P$ loses contact with the surface of the cylinder when $OP$ makes an angle $\theta$ with the upward vertical through $O$.
(a) Show that $\theta = 60^\circ$. [5](b) Show that in its subsequent motion $P$ strikes the cylinder at the point $A$. [5]