Questions - A-Level Maths

OCR M2 2009 June Q5

11 marks Standard +0.3

Fig. 1 Fig. 1 shows a uniform lamina $B C D$ in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from $B$ is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina $A B D E$ has dimensions $A B = 12 \mathrm {~cm}$ and $A E = 6 \mathrm {~cm}$. A single plane object is formed by attaching the rectangular lamina to the lamina $B C D$ along $B D$ (see Fig. 2). The mass of $A B D E$ is 3 kg and the mass of $B C D$ is 2 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at $C$ and rests in equilibrium.
Calculate the angle that $A C$ makes with the vertical.

OCR M2 2015 June Q7

11 marks Standard +0.8

Show that $\mu = \frac { 2 } { 3 }$. A small object of weight $a W \mathrm {~N}$ is placed on the ladder at its mid-point and the object $S$ of weight $2 W \mathrm {~N}$ is placed on the ladder at its lowest point $A$.
Given that the system is in equilibrium, find the set of possible values of $a$.

OCR MEI M2 2010 June Q2

18 marks Standard +0.3

Calculate the coordinates of the centre of mass of the stand. A small object of mass 5 kg is fixed to the rod AB at a distance of 40 cm from A .
Show that the coordinates of the centre of mass of the stand with the object are ( 22,68 ). The stand is tilted about the edge PQ until it is on the point of toppling. The angle through which the stand is tilted is called 'the angle of tilt'. This procedure is repeated about the edges QR and RS.
Making your method clear, determine which edge requires the smallest angle of tilt for the stand to topple. The small object is removed. A light string is attached to the stand at A and pulled at an angle of $50 ^ { \circ }$ to the downward vertical in the plane $\mathrm { O } x y$ in an attempt to tip the stand about the edge RS.
Assuming that the stand does not slide, find the tension in the string when the stand is about to turn about the edge RS.

OCR MEI M2 2016 June Q3

18 marks Standard +0.3

Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
Calculate the power of the tension in the string when the block has a speed of $7 \mathrm {~ms} ^ { - 1 }$. Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is $M \mathrm {~kg}$. The sides OP and PQ have thin rigid uniform reinforcements attached with masses $0.6 M \mathrm {~kg}$ and $0.4 M \mathrm {~kg}$, respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.
Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces $Y _ { \mathrm { Q } } \mathrm { N }$ and $Y _ { \mathrm { R } } \mathrm { N }$ act at Q and R respectively.
Calculate the value of $Y _ { \mathrm { Q } }$ and show that $Y _ { \mathrm { R } } = 120$. The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods $\mathrm { AB } , \mathrm { BC } , \mathrm { CA }$ and CD freely pin-jointed together at $\mathrm { A } , \mathrm { B }$ and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces $X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }$ acting at A and $X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }$ acting at D . You are given that $\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }$ and $\cos \beta = \frac { 3 } { 5 }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
\end{figure}
Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D , including those internal to the rods, when the inn sign is hanging from the framework.
Show that $X _ { \mathrm { D } } = 261$.
Calculate the forces internal to the rods $\mathrm { AB } , \mathrm { BC }$ and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of $Y _ { \mathrm { D } }$ and $Y _ { \mathrm { A } }$. [Your working in this part should correspond to your diagram in part (iii).]

Edexcel M3 Q2

7 marks Standard +0.8

Show that $\lambda = 4 m g \sin \alpha$. The particle is now moved and held at rest at $A$ with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.
Find the greatest distance from $A$ that the particle reaches down the plane.

OCR M3 2009 January Q4

10 marks Standard +0.8

Show that $v ^ { 2 } = 9 + 9.8 \sin \theta$.
Find, in terms of $\theta$, the radial and tangential components of the acceleration of $P$.
Show that the tension in the string is $( 3.6 + 5.88 \sin \theta ) \mathrm { N }$ and hence find the value of $\theta$ at the instant when the string becomes slack, giving your answer correct to 1 decimal place.

OCR M3 2010 January Q6

13 marks Challenging +1.2

By considering the total energy of the system, obtain an expression for $v ^ { 2 }$ in terms of $\theta$.
Show that the magnitude of the force exerted on $P$ by the cylinder is $( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }$.
Given that $P$ leaves the surface of the cylinder when $\theta = \alpha$, show that $1.53 < \alpha < 1.54$.

OCR M3 2007 June Q6

13 marks Challenging +1.8

Show that, when $P$ is in equilibrium, $O P = 7.25 \mathrm {~m}$.
Verify that $P$ and $Q$ together just reach the safety net.
At the lowest point of their motion $P$ releases $Q$. Prove that $P$ subsequently just reaches $O$.
State two additional modelling assumptions made when answering this question.

OCR M3 2010 June Q3

8 marks Standard +0.3

Given that the coefficient of restitution between the sphere and the wall is $\frac { 1 } { 2 }$, state the values of $u$ and $v$. Shortly after hitting the wall the sphere $A$ comes into contact with another uniform smooth sphere $B$, which has the same mass and radius as $A$. The sphere $B$ is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_524_371_1503_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
Find, for each sphere, its speed and its direction of motion immediately after the contact. $4 ~ O$ is a fixed point on a horizontal plane. A particle $P$ of mass 0.25 kg is released from rest at $O$ and moves in a straight line on the plane. At time $t \mathrm {~s}$ after release the only horizontal force acting on $P$ has magnitude $$\frac { 1 } { 2400 } \left( 144 - t ^ { 2 } \right) \mathrm { N } \quad \text { for } 0 \leqslant t \leqslant 12$$ and $$\frac { 1 } { 2400 } \left( t ^ { 2 } - 144 \right) \mathrm { N } \text { for } t \geqslant 12 .$$ The force acts in the direction of $P$ 's motion. $P$ 's velocity at time $t \mathrm {~s}$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find an expression for $v$ in terms of $t$, valid for $t \geqslant 12$, and hence show that $v$ is three times greater when $t = 24$ than it is when $t = 12$.
Sketch the $( t , v )$ graph for $0 \leqslant t \leqslant 24$.

OCR M3 2016 June Q3

10 marks Challenging +1.2

Find the speed of $A$ after the collision. Find also the component of the velocity of $B$ along the line of centres after the collision. $B$ subsequently hits the wall.
Explain why $A$ and $B$ will have a second collision if the coefficient of restitution between $B$ and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{4} \includegraphics[alt={},max width=\textwidth]{c0f31235-80aa-4838-844f-b706de55e7cd-3_193_1451_705_306}
\end{figure} $A$ and $C$ are two fixed points, 1.5 m apart, on a smooth horizontal plane. A light elastic string of natural length 0.4 m and modulus of elasticity 20 N has one end fixed to point $A$ and the other end fixed to a particle $B$. Another light elastic string of natural length 0.6 m and modulus of elasticity 15 N has one end fixed to point $C$ and the other end fixed to the particle $B$. The particle is released from rest when $A B C$ forms a straight line and $A B = 0.4 \mathrm {~m}$ (see diagram). Find the greatest kinetic energy of particle $B$ in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-3_586_533_1409_758} One end of a light inextensible string of length $a$ is attached to a fixed point $O$. A particle $P$ of mass $m$ is attached to the other end of the string and hangs at rest. $P$ is then projected horizontally from this position with speed $2 \sqrt { a g }$. When the string makes an angle $\theta$ with the upward vertical $P$ has speed $v$ (see diagram). The tension in the string is $T$.
Find an expression for $T$ in terms of $m , g$ and $\theta$, and hence find the height of $P$ above its initial level when the string becomes slack. $P$ is now projected horizontally from the same initial position with speed $U$.
Find the set of values of $U$ for which the string does not remain taut in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-4_566_1013_255_525} Two uniform rods $A B$ and $A C$ are freely jointed at $A$. Rod $A B$ is of length $2 l$ and weight $W$; $\operatorname { rod } A C$ is of length $4 l$ and weight $2 W$. The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that $A B$ makes an angle of $\theta$ with the horizontal, where $\sin \theta = \frac { 4 } { 5 }$, and $A C$ makes an angle of $\varphi$ with the horizontal, where $\sin \varphi = \frac { 3 } { 5 }$ (see diagram). The force of the step acting on $A B$ at $B$ has vertical component $R$ and horizontal component $F$.
By taking moments about $A$ for the rod $A B$, find an equation relating $W , R$ and $F$.
Show that $R = \frac { 73 } { 50 } W$, and find the vertical component of the force acting on $A C$ at $C$.
The coefficient of friction at $B$ is equal to that at $C$. Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction.

Edexcel M4 2005 January Q5

10 marks Challenging +1.2

Show that, when $\angle B F O = \theta$, the potential energy of the system is $$\frac { 1 } { 10 } m g a ( 8 \cos \theta - 5 ) ^ { 2 } - 2 m g a \cos ^ { 2 } \theta + \text { constant } .$$
Hence find the values of $\theta$ for which the system is in equilibrium.
Determine the nature of the equilibrium at each of these positions.

OCR M4 2008 June Q6

15 marks Challenging +1.3

Show that the moment of inertia of the lamina about the axis through $X$ is $\frac { 4 } { 3 } m a ^ { 2 }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $\omega ^ { 2 } = \frac { 6 g } { 5 a }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $R = 0$, and given also that $\sin \theta = \frac { 4 } { 5 }$ find $S$ in terms of $m$ and $g$.

OCR M4 2010 June Q6

12 marks Challenging +1.2

Show that $\theta = 0$ is a position of stable equilibrium.
Show that the kinetic energy of the system is $4 m a ^ { 2 } \dot { \theta } ^ { 2 }$.
By differentiating the energy equation, then making suitable approximations for $\sin \theta$ and $\cos \theta$, find the approximate period of small oscillations about the equilibrium position $\theta = 0$. \section*{[Question 7 is printed overleaf.]}

OCR M4 2013 June Q5

14 marks Standard +0.8

Find the magnitude and bearing of the velocity of $U$ relative to $P$.
Find the shortest distance between $P$ and $U$ in the subsequent motion.
(ii) Plane $Q$ is flying with constant velocity $160 \mathrm {~ms} ^ { - 1 }$ in the direction which brings it as close as possible to $U$.
Find the bearing of the direction in which $Q$ is flying.
Find the shortest distance between $Q$ and $U$ in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-3_771_769_262_646} A square frame $A B C D$ consists of four uniform rods $A B , B C , C D , D A$, rigidly joined at $A , B , C , D$. Each rod has mass 0.6 kg and length 1.5 m . The frame rotates freely in a vertical plane about a fixed horizontal axis passing through the mid-point $O$ of $A D$. At time $t$ seconds the angle between $A D$ and the horizontal, measured anticlockwise, is $\theta$ radians (see diagram).
1. Show that the moment of inertia of the frame about the axis through $O$ is $3.15 \mathrm {~kg} \mathrm {~m} ^ { 2 }$.
2. Show that $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 5.6 \sin \theta$.
3. Deduce that the frame can make small oscillations which are approximately simple harmonic, and find the period of these oscillations. The frame is at rest with $A D$ horizontal. A couple of constant moment 25 Nm about the axis is then applied to the frame.
4. Find the angular speed of the frame when it has rotated through 1.2 radians.

OCR M4 2015 June Q5

15 marks Challenging +1.8

Taking $H$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$V = m g \left( 2 \lambda r \cos \theta - 2 r \cos ^ { 2 } \theta - \lambda a \right)$$
Find the set of possible values of $\lambda$ so that there is more than one position of equilibrium.
For the case $\lambda = \frac { 3 } { 2 }$, determine whether each equilibrium position is stable or unstable.

Edexcel M5 2002 June Q6

17 marks Challenging +1.8

Show that the moment of inertia of the rod about the edge of the table is $\frac { 7 } { 3 } m a ^ { 2 }$. The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle $\theta$, its angular speed is $\dot { \theta }$. Assuming that the rod has not started to slip,
show that $\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }$,
find the angular acceleration of the rod,
find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is $\mu$.
Show that the rod starts to slip when $\tan \theta = \frac { 4 } { 13 } \mu$ (6)

Edexcel M5 2005 June Q4

11 marks Challenging +1.8

Show that the moment of inertia of the body about $L$ is $\frac { 77 m a ^ { 2 } } { 4 }$. When $P R$ is vertical, the body has angular speed $\omega$ and the centre of the disc strikes a stationary particle of mass $\frac { 1 } { 2 } \mathrm {~m}$. Given that the particle adheres to the centre of the disc,
find, in terms of $\omega$, the angular speed of the body immediately after the impact.

AQA FP1 2013 January Q7

7 marks Standard +0.8

Show that there is a linear relationship between $Y$ and $X$.
The graph of $Y$ against $X$ is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of $n$ and the value of $a$.

AQA FP1 2015 June Q6

5 marks Standard +0.3

Sketch the curve $C _ { 1 }$, stating the values of its intercepts with the coordinate axes.
The curve $C _ { 1 }$ is translated by the vector $\left[ \begin{array} { l } k \\ 0 \end{array} \right]$, where $k < 0$, to give a curve $C _ { 2 }$. Given that $C _ { 2 }$ passes through the origin $( 0,0 )$, find the equations of the asymptotes of $C _ { 2 }$.
[0pt] [3 marks]

OCR FP1 2011 January Q7

9 marks Moderate -0.8

Write down the matrix, $\mathbf { A }$, that represents a shear with $x$-axis invariant in which the image of the point $( 1,1 )$ is $( 4,1 )$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.
The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)$.
(a) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
(b) Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$. 8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.
Show that $p = \frac { 5 } { 6 }$.
Find the value of $q$. 9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)$.
Find, in terms of $a$, the determinant of $\mathbf { M }$.
Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where $k$ is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.
Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
Show that $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.

OCR FP1 2016 June Q8

10 marks Standard +0.3

Show that $\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }$.
Hence find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$, giving your answer as a single fraction.
Find $\sum _ { r = n } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$, giving your answer as a single fraction.

OCR FP2 2010 January Q5