科目：进阶数学 Further Mathematics (9231)
年份：2022 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
1 A uniform lamina $OABC$ is a trapezium whose vertices can be represented by coordinates in the $x-y$ plane. The coordinates of the vertices are $O(0,0), A(15,0), B(9,4)$ and $C(3,4)$. Find the $x$-coordinate of the centre of mass of the lamina. [4]
2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\frac{4}{3}mg$. The other end of the string is attached to a fixed point $O$ on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between $P$ and the surface is $\frac{1}{3}$. The particle is projected along the surface in the direction $OP$ with a speed of $\frac{1}{2}\sqrt{ga}$.
Find the greatest extension of the string during the subsequent motion. [5]3 A particle $P$ is projected with speed $25 \text{ m/s}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. After 2 s the speed of $P$ is $15 \text{ m/s}^{-1}$.
(a) Find the value of $\sin \theta$. [5](b) Find the range of the flight. [3]
4 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. A particle of mass $m$ is attached to the other end of the string and is held with the string taut at the point $A$. At $A$ the string makes an angle $\theta$ with the upward vertical through $O$. The particle is projected perpendicular to the string in a downward direction from $A$ with a speed $u$. It moves along a circular path in the vertical plane.
When the string makes an angle $\alpha$ with the downward vertical through $O$, the speed of the particle is $2u$ and the magnitude of the tension in the string is 10 times its magnitude at $A$.
It is given that $u = \sqrt{\frac{2}{5}}ga$.
(a) Find, in terms of $m$ and $g$, the magnitude of the tension in the string at $A$. [6](b) Find the value of $\cos \alpha$. [2]
5 A particle $P$ of mass 4 kg is moving in a horizontal straight line. At time $ts$ the velocity of $P$ is $\nu m s^{-1}$ and the displacement of $P$ from a fixed point $O$ on the line is $x m$. The only force acting on $P$ is a resistive force of magnitude $(4 e^{-x} + 12) e^{-x} N$. When $t = 0, x = 0$ and $\nu = 4$.
(a) Show by integration that $\nu = \frac{1 + 3 e^{x}}{e^{x}}$. [4](b) Find an expression for $x$ in terms of $t$. [4]
AB and BC are two fixed smooth vertical barriers on a smooth horizontal surface, with angle $ABC = 60^\circ$. A particle of mass $m$ is moving with speed $u$ on the surface. The particle strikes AB at an angle $\theta$ with AB. It then strikes BC and rebounds at an angle $\beta$ with BC (see diagram). The coefficient of restitution between the particle and each barrier is $e$ and $\tan \theta = 2$. The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision. (a) Find the value of $e$. [4]
(b) Find the size of angle $\beta$. [4]
A uniform cylinder with a rough surface and of radius $a$ is fixed with its axis horizontal. Two identical uniform rods $AB$ and $BC$, each of weight $W$ and length $2a$, are rigidly joined at $B$ with $AB$ perpendicular to $BC$. The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with $AB$ at an angle $\theta$ to the horizontal. $D$ and $E$ are the midpoints of $AB$ and $BC$ respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is $\mu$. The normal reaction between $AB$ and the cylinder is $R$ and the normal reaction between $BC$ and the cylinder is $N$. (a) Find the ratio $R : N$ in terms of $\m
(b) Given that $\mu = \frac{1}{3}$, find the value of $\tan \theta$. [3]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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