科目：进阶数学 Further Mathematics (9231)
年份：2024 冬季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
1. A particle $P$ is projected with speed $u \text{ m s}^{-1}$ at an angle $\tan^{-1} 2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance $56 \text{ m}$ horizontally from $O$, it is at a vertical height $H \text{ m}$ above the plane. When $P$ has travelled a distance $84 \text{ m}$ horizontally from $O$, it is at a vertical height $\frac{1}{2} H \text{ m}$ above the plane.
Find, in either order, the value of $u$ and the value of $H$. [5]2 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 particle $P$ is held at the point $A$ with the string taut. It is given that $OA$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta = \frac{1}{4}$. The particle $P$ is projected perpendicular to $OA$ in an upwards direction with speed $\sqrt{5}ag$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $AOB$ is a right angle, the tension in the string is $T$.
Find $T$ in terms of $m$ and $g$. [5]3 A particle $P$ of mass $m\text{kg}$ is attached to one end of a light elastic string of natural length $2\text{m}$ and modulus of elasticity $2mg\text{N}$. The other end of the string is attached to a fixed point $O$. The particle $P$ hangs in equilibrium vertically below $O$. The particle $P$ is pulled down vertically a distance $d\text{m}$ below its equilibrium position and released from rest.
(a) Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$. [6](b) Hence find the speed of $P$ when it is 2 m below $O$. [2]
An object is formed by removing a cylinder of radius $\frac{2}{3}a$ and height $kh$ ($k < 1$) from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points $A$ and $B$ are the opposite ends of a diameter of the upper face of the object (see diagram).
(a) Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $AB$. [4]When the object is suspended from $A$, the angle between $AB$ and the vertical is $\theta$, where $\tan \theta = \frac{3}{2}$.
(b) Given that $h = \frac{8}{3}a$, find the possible values of $k$. [3]5 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x$ m from a fixed point $O$ on the line and velocity $v$ m/s$^{-1}$ at time $t$ s. The only horizontal force acting on $P$ is a variable force $F_N$ which can be expressed as a function of $t$. It is given that
$$\frac{v}{x} = \frac{3-t}{1+t}$$
and when $t = 0, x = 5$.
(a) Find an expression for $x$ in terms of $t$. [4](b) Find the magnitude of $F$ when $t = 3$.
A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m. The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $P$.
The particle $P$ moves in a horizontal circle of radius 0.8 m with angular speed $\omega \text{ rad s}^{-1}$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \text{ rad s}^{-1}$. The centres of the circles are vertically below $O$, and $O, P$ and $Q$ are always in the same vertical plane. The strings $OP$ and $PQ$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
(a) Find the tension in the string $O(b) Find the value of $\omega$. (c) Find the value of $\beta$.
7 A particle $P$ is projected with speed $u$ at an angle $\tan^{-1}\left(\frac{4}{3}\right)$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ is moving horizontally, it strikes a smooth inclined plane at the point $A$. This plane is inclined to the horizontal at an angle $\alpha$, and the line of greatest slope through $A$ lies in the vertical plane through $O$ and $A$.
As a result of the impact, $P$ moves vertically upwards. The coefficient of restitution between $P$ and the inclined plane is $e$.
(a) Show that $e \tan^2\alpha = 1$. [4]In its subsequent motion, the greatest height reached by $P$ above $A$ is $\frac{3}{16}$ of the vertical height of $A$ above the horizontal plane.
(b) Find the value of $e$. [6]Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
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