科目：进阶数学 Further Mathematics (9231)
年份：2021 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
A uniform lamina $ABCD$ consists of two isosceles triangles $ABD$ and $BCD$. The diagonals of $ABCD$ meet at the point $O$. The length of $AO$ is $3a$, the length of $OC$ is $6a$ and the length of $BD$ is $16a$ (see diagram). Find the distance of the centre of mass of the lamina from $DB$. [3]
2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point $O$ on a smooth plane. The plane is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac{3}{5}$. A particle $P$ of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of $O$. The particle is projected with speed $\sqrt{2} \text{ m/s}^{-1}$ directly down the plane from the position where $OP$ is equal to the natural length of the string.
Find the maximum extension of the string during the subsequent motion. [5]Particles $A$ and $B$, of masses $3m$ and $m$ respectively, are connected by a light inextensible string of length $a$ that passes through a fixed smooth ring $R$. Particle $B$ hangs in equilibrium vertically below the ring. Particle $A$ moves in horizontal circles on a smooth horizontal surface with speed $\frac{2}{5}\sqrt{ga}$. The angle between $AR$ and $BR$ is $\theta$ (see diagram). The normal reaction between $A$ and the surface is $\frac{12}{5}mg$.
(a) Find $\cos \theta$. [3](b) Find, in terms of $a$, the distance of $B$ below the ring. [3]
A particle 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 particle is initially held with the string taut at the point $A$, where $OA$ makes an angle $\theta$ with the downward vertical through $O$. The particle is then projected with speed $u$ perpendicular to $OA$ and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point $B$ where angle $AOB$ is a right angle. The speed of the particle when it is at $B$ is $\frac{1}{2}u$ (see diagram).
Find the tension in the string at $A$, giving your answer in terms of $m$ and $g$. [8]7 © UCLES 2021 9231/33/M/J/21 [Turn over]
5 A particle $P$ of mass $m \text{ kg}$ is projected vertically upwards from a point $O$, with speed $20 \text{ ms}^{-1}$, and moves under gravity. There is a resistive force of magnitude $2mv \text{ N}$, where $v \text{ ms}^{-1}$ is the speed of $P$ at time $t \text{ s}$ after projection.
(a) Find an expression for $v$ in terms of $t$, while $P$ is moving upwards. [6]The displacement of $P$ from $O$ is $x\text{ m}$ at time $t\text{ s}$.
(b) Find an expression for $x$ in terms of $t$, while $P$ is moving upwards. [2]
(c) Find, correct to 3 significant figures, the greatest height above $O$ reached by $P$. [2]Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $km$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides with sphere $B$ which is at rest. Immediately before the collision, $A$’s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{1}{3}$.
(a) Show that the speed of $B$ after the collision is $\frac{4u\cos \theta}{3(1+k)}$. [3]70% of the total kinetic energy of the spheres is lost as a result of the collision.
(b) Given that $\tan \theta = \frac{1}{3}$, find the value of $k$. [6]7 A particle $P$ is projected with speed $u$ at an angle $\theta$ 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 $x$ and $y$ respectively. (a) Use the equation of the trajectory given in the List of formulae (MF19), together with the condition $y = 0$, to establish an expression for the range $R$ in terms of $u, \theta$ and $g$. [2] (b) Deduce an expression for the maximum height $H$, in terms of $u, \theta$ and $g$. [2]
It is given that $R = \frac{4H}{\sqrt{3}}$.
(c) Show that $\theta = 60^\circ$. [1]
It is given also that $u = \sqrt{40} \text{ m s}^{-1}$.
(d) Find, by differentiating the equation of the trajectory or otherwise, the set of values of $x$ for which the direction of motion makes an angle of less than $45^\circ$ with the horizontal. [4]Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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