CAIE Further Paper 3 (Further Paper 3) 2024 November

Question 1

1 A particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 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 m horizontally from $O$, it is at a vertical height $H \mathrm {~m}$ above the plane. When $P$ has travelled a distance 84 m horizontally from $O$, it is at a vertical height $\frac { 1 } { 2 } H \mathrm {~m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$.
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Question 2

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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 $O A$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta = \frac { 3 } { 4 }$. The particle $P$ is projected perpendicular to $O A$ in an upwards direction with speed $\sqrt { 5 a g }$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $A O B$ is a right angle, the tension in the string is $T$. Find $T$ in terms of $m$ and $g$.

Question 3

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3 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity $2 m g \mathrm {~N}$. 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 $d$ m below its equilibrium position and released from rest.

Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$.
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Hence find the speed of $P$ when it is 2 m below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-06_785_729_255_708} An object is formed by removing a cylinder of radius $\frac { 2 } { 3 } a$ and height $k h ( k < 1 )$ 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).
Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $A B$.

When the object is suspended from $A$, the angle between $A B$ and the vertical is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$.
Given that $h = \frac { 8 } { 3 } a$, find the possible values of $k$.

Question 5

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5 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ s. The only horizontal force acting on $P$ is a variable force $F \mathrm {~N}$ which can be expressed as a function of $t$. It is given that $$\frac { v } { x } = \frac { 3 - t } { 1 + t }$$ and when $t = 0 , x = 5$.

Find an expression for $x$ in terms of $t$.
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Find the magnitude of $F$ when $t = 3$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_559_1257_255_445} 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 $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$.
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Find the value of $\omega$.
Find the value of $\beta$.

Question 7

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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 $e$.

Show that $e \tan ^ { 2 } \alpha = 1$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-12_2716_36_106_2014} 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.
Find the value of $e$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
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