科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 34
总页数：14
章节：9231 Further Mathematics
A light inextensible string of length 12a is threaded through a fixed smooth ring $R$. 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 0.5m. Particle $A$ hangs in equilibrium vertically below the ring. Particle $B$ moves with constant angular speed $\omega$ in a horizontal circle with particle $A$ at its centre. The angle between $AR$ and $BR$ is $\theta$ (see diagram). Express $\omega$ in terms of $g$ and $a$. [5]
2 A particle $P$ is projected with speed $u \text{ m s}^{-1}$ at an angle $\theta$, where $\tan \theta = 2$, 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 time $ts$ are denoted by $x\text{ m}$ and $y\text{ m}$ respectively.
(a) Use the equation of the trajectory given in the list of formulae (MF 19) to show that
$$y = 2x - \frac{25x^2}{u^2}.$$
[1]
(b) In the subsequent motion, $P$ passes through the point with coordinates $(8,12)$. The particle then hits a fixed vertical barrier $7\text{ m}$ high that is at a horizontal distance of $D\text{ m}$ from the point of projection.
Find the set of possible values of $D$.
[5]The lamina $BFDE$ is obtained by removing triangles $AED$ and $BCF$ from a uniform square lamina $ABCD$ of side $2a$. The length of side $AE$ is $a$ and the length of side $FC$ is $h$ (see diagram). The centre of mass of $BFDE$ is at a distance $\bar{x}$ from $AD$, and at a distance $\bar{y}$ from $AB$.
(a) Show that $\bar{x} = \frac{h^2 - 6ah + 11a^2}{3(3a - h)}$ and find a corresponding expression for $\bar{y}$. [5](b) The lamina $BFDE$ is placed vertically on its edge $EB$ on a smooth horizontal surface. Find, in terms of $a$, the set of possible values of $h$ for which the lamina remains in equilibrium. [2]
A fixed smooth spherical shell has centre $O$ and radius $a$. A particle of mass $m$ moves in complete vertical circles on the smooth inner surface of the shell, where the plane of the circular motion is vertical and passes through $O$. The particle has speed $v$ when it is at point $A$, where $OA$ makes an angle $\theta$ with the upward vertical through $O$, and $\cos \theta = \frac{1}{18}$ (see diagram).
(a) Show that $v \geqslant \frac{1}{3}\sqrt{26ag}$. [5]It is given that $v = \frac{1}{3}\sqrt{26ag}$.
(b) Find, in terms of $m$ and $g$, an expression for the greatest possible value of the normal reaction between the shell and the particle. [3]5 A particle $P$ of mass $m$ kg is projected vertically upwards from a point $O$ with an initial speed of $20 \text{ m/s}^{-1}$ and moves under gravity. There is a resistive force of magnitude $0.025mv^2 \text{ N}$, where $v \text{ m/s}^{-1}$ is the speed of $P$ at time $t_s$ after projection. The displacement of $P$ from $O$ is $x \text{ m}$ at time $t_s$ after projection.
(a) Find an expression for $v$ in terms of $x$, while $P$ is moving upwards. [5](b) Find an expression for $v$ in terms of $t$, while $P$ is moving upwards. [4]
6 $A$ and $B$ are two fixed points at a distance $22a$ apart, with $B$ vertically below $A$. A light elastic string of natural length $4a$ and modulus of elasticity $5mg$ has one end attached to $A$ and the other end attached to a particle $P$ of mass $km$. Another light elastic string of natural length $8a$ and modulus of elasticity $6mg$ has one end attached to $B$ and the other end attached to $P$. Particle $P$ is vertically above $B$.
(a) Show that, when the system is in equilibrium, $BP = \frac{57a - 2ak}{4}$. [5]The particle $P$ is pulled vertically upwards so that $BP = 18a$, and is then released from rest. In its subsequent motion, $P$ first comes to instantaneous rest at the point where $BP = 8a$. (b) Find the value of $k$. [4]
X and Y are two fixed smooth vertical walls on a smooth horizontal surface. The walls are parallel and at a distance $d$ apart. The points $P_1, P_2$ and $P_3$ all lie on the surface.
A particle $Q$ is projected horizontally from the point $P_1$ on Wall X with speed $u$, and moves along the surface. The particle $Q$ strikes Wall Y at the point $P_2$. Immediately before the collision, the direction of motion of $Q$ makes an angle $\alpha$ with Wall Y, where $\sin \alpha = \frac{4}{5}$. Immediately after the collision, the direction of motion of $Q$ makes an angle $\theta$ with Wall Y. The particle $Q$ then strikes Wall X at the point $P_3$ (see diagram).
The time that it takes $Q$ to travel the distance $P_1P_2$ is $T$. The time that it takes $Q$ to travel the distance $P_2P_3$ is $kT13 © UCLES 2025 9231/34/O/N/25
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