Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 34
9231 Mathematics - Further November 2025 Question Paper 34
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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 $a$.
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 $t$ s are denoted by $x$ m and $y$ m respectively.
Use the equation of the trajectory given in the list of formulae (MF19) to show that $y = 2x - \dfrac{25x^2}{u^2}$.
In the subsequent motion, $P$ passes through the point with coordinates $(8, 12)$. The particle then hits a fixed vertical barrier 7 m high that is at a horizontal distance of $D$ m from the point of projection. Find the set of possible values of $D$.
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 $AB$.
Show that $\bar x = \dfrac{h^2 - 6ah + 11a^2}{3(3a - h)}$ and find a corresponding expression for $\bar y$.
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.
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 $\cos\theta = \tfrac{1}{18}$ (see diagram).
Show that $v \geqslant \tfrac13\sqrt{26ag}$.
It is given that $v = \tfrac13\sqrt{26ag}$. Find, in terms of $m$ and $g$, an expression for the greatest possible value of the normal reaction between the shell and the particle.
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 $0.025mv^2$ 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 $x$ m at time $t$ s after projection.
Find an expression for $v$ in terms of $x$, while $P$ is moving upwards.
Find an expression for $v$ in terms of $t$, while $P$ is moving upwards.
$A$ 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 $B$.
Show that, when the system is in equilibrium, $BP = \dfrac{57a - 2ak}{4}$.
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 $BP = 8a$. Find the value of $k$.
$X$ 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 $\sin\alpha = \tfrac45$. 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 $kT$. Find, in terms of $k$, the coefficient of restitution between $Q$ and Wall $Y$.