Questions - A-Level Maths

OCR MEI M3 2016 June Q1

19 marks Standard +0.3

1

In an investigation, small spheres are dropped into a long column of a viscous liquid and their terminal speeds measured. It is thought that the terminal speed $V$ of a sphere depends on a product of powers of its radius $r$, its weight $m g$ and the viscosity $\eta$ of the liquid, and is given by $$V = k r ^ { \alpha } ( m g ) ^ { \beta } \eta ^ { \gamma } ,$$ where $k$ is a dimensionless constant.
1. Given that the dimensions of viscosity are $\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }$ find $\alpha , \beta$ and $\gamma$. A sphere of mass 0.03 grams and radius 0.2 cm has a terminal speed of $6 \mathrm {~ms} ^ { - 1 }$ when falling through a liquid with viscosity $\eta$. A second sphere of radius 0.25 cm falling through the same liquid has a terminal speed of $8 \mathrm {~ms} ^ { - 1 }$.
2. Find the mass of the second sphere.
A manufacturer is testing different types of light elastic ropes to be used in bungee jumping. You may assume that air resistance is negligible. A bungee jumper of mass 80 kg is connected to a fixed point A by one of these elastic ropes. The natural length of this rope is 25 m and its modulus of elasticity is 1600 N . At one instant, the jumper is 30 m directly below A and he is moving vertically upwards at $15 \mathrm {~ms} ^ { - 1 }$. He comes to instantaneous rest at a point B , with the rope slack.
1. Find the distance AB . The same bungee jumper now tests a second rope, also of natural length 25 m . He falls from rest at A . It is found that he first comes instantaneously to rest at a distance 54 m directly below A .
2. Find the modulus of elasticity of this second rope. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_559_705_262_680} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
  \end{figure} The region R shown in Fig. 2.1 is bounded by the curve $y = k ^ { 2 } - x ^ { 2 }$, for $0 \leqslant x \leqslant k$, and the coordinate axes. The $x$-coordinate of the centre of mass of a uniform lamina occupying the region R is 0.75 .
3. Show that $k = 2$. A uniform solid S is formed by rotating the region R through $2 \pi$ radians about the $x$-axis.
4. Show that the centre of mass of S is at $( 0.625,0 )$. Fig. 2.2 shows a solid T made by attaching the solid S to the base of a uniform solid circular cone C . The cone $C$ is made of the same material as $S$ and has height 8 cm and base radius 4 cm . \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_455_794_1521_639} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
  \end{figure}
5. Show that the centre of mass of T is at a distance of 6.75 cm from the vertex of the cone. [You may quote the standard results that the volume of a cone is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ and its centre of mass is $\frac { 3 } { 4 } h$ from its vertex.]
6. The solid T is suspended from a point P on the circumference of the base of C . Find the acute angle between the axis of symmetry of T and the vertical. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-4_668_262_255_904} \captionsetup{labelformat=empty} \caption{Fig. 3}
  \end{figure} One end of a light elastic string, of natural length 2.7 m and modulus of elasticity 54 N , is attached to a fixed point L . The other end of the string is attached to a particle P of mass 2.5 kg . One end of a second light elastic string, of natural length 1.7 m and modulus of elasticity 8.5 N , is attached to P . The other end of this second string is attached to a fixed point M , which is 6 m vertically below L . This situation is shown in Fig. 3. The particle P is released from rest when it is 4.2 m below L . Both strings remain taut throughout the subsequent motion. At time $t \mathrm {~s}$ after P is released from rest, its displacement below L is $x \mathrm {~m}$.
7. Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 10 ( x - 4 )$.
8. Write down the value of $x$ when P is at the centre of its motion.
9. Find the amplitude and the period of the oscillations.
10. Find the velocity of P when $t = 1.2$.

OCR MEI M3 2016 June Q4

18 marks Challenging +1.2

4 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 . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles $\alpha$ and $\beta$ with the upward vertical as shown in Fig. 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-5_414_399_434_833} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} The speed of P at A is $\sqrt { \frac { 17 a g } { 3 } }$. The speed of P at B is $\sqrt { 5 a g }$ and $\cos \beta = \frac { 2 } { 3 }$.

Show that $\cos \alpha = \frac { 1 } { 3 }$. On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
Show that when the string becomes slack, the speed of the combined particle is $\sqrt { \frac { a g } { 3 } }$. The mass of the particle Q is $k m$.
Find the value of $k$.
Find, in terms of $m$ and $g$, the instantaneous change in the tension in the string as a result of the collision.

Edexcel M3 Q1

7 marks Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-2_341_652_217_621} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point $O$ which lies on a smooth plane inclined at an angle $\alpha$ to the horizontal where $\tan \alpha = \frac { 3 } { 4 }$ as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from $O$ down the line of greatest slope of the plane.

Find the tension in the spring.
Find the initial acceleration of the particle.

Edexcel M3 Q2

7 marks Standard +0.3

2. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis under the action of a single force directed away from the origin $O$. When $P$ is $x$ metres from $O$, the magnitude of the force is $3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}$ and $P$ has a speed of $v \mathrm {~ms} ^ { - 1 }$. Given that when $x = 1 , P$ is moving away from $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,

find an expression for $v ^ { 2 }$ in terms of $x$,
show that when $x = 4 , P$ has a speed of $7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 1 decimal place.

Edexcel M3 Q3

10 marks Standard +0.3

3. A particle is performing simple harmonic motion along a straight line between the points $A$ and $B$ where $A B = 8 \mathrm {~m}$. The period of the motion is 12 seconds.

Find the maximum speed of the particle in terms of $\pi$. The points $P$ and $Q$ are on the line $A B$ at distances of 3 m and 6 m respectively from $A$.
Find, correct to 3 significant figures, the time it takes for the particle to travel directly from $P$ to $Q$.
(6 marks)

Edexcel M3 Q4

11 marks Standard +0.3

4. Whilst in free-fall a parachutist falls vertically such that his velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when he is $x$ metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where $k$ is a constant.
Given that he experiences an acceleration of $\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }$,

show that $f = g \mathrm { e } ^ { - \frac { 2 x } { k } }$. After falling a large distance, his velocity is constant at $49 \mathrm {~ms} ^ { - 1 }$.
Find the value of $k$.
Hence, express $f$ in the form ( $\lambda - \mu v ^ { 2 }$ ) where $\lambda$ and $\mu$ are constants which you should find.
(4 marks)

Edexcel M3 Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-3_588_291_1126_662} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height $2 r$ and base radius $r$, to one of the plane surfaces of a cylinder of height $h$ and base radius $r$ as shown in Figure 2. Using this model,

show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle $\alpha$ to the horizontal. Given that the firework does not slip or topple and that $h = 4 r$,
Find, correct to the nearest degree, the maximum value of $\alpha$.

Edexcel M3 Q6

13 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-4_437_364_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The two ends of a light inextensible string of length $3 a$ are attached to fixed points $Q$ and $R$ which are a distance of $a \sqrt { } 3$ apart with $R$ vertically below $Q$. A particle $P$ of mass $m$ is attached to the string at a distance of $2 a$ from $Q$. $P$ is given a horizontal speed, $u$, such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

Show that $\angle P R Q$ is a right angle.
Find $\angle P Q R$ in degrees.
Find, in terms of $a , g , m$ and $u$, the tension in the section of string
1. $P Q$,
2. $P R$.
Show that $u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }$.

Edexcel M3 Q7

14 marks Challenging +1.2

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane and the coefficient of friction between the particle and the plane is $\frac { 10 } { 49 }$. The particle is projected from $O$ along the plane with an initial speed of $5 \mathrm {~ms} ^ { - 1 }$.

Show that the greatest distance from $O$ which the particle reaches is 1.84 m .
Find, correct to 2 significant figures, the speed at which the particle returns to $O$.

Edexcel M3 Q1

7 marks Moderate -0.3

A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, $s$ metres, of a point $P$ on the airbag $t$ seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

find, correct to the nearest centimetre, the maximum displacement of $P$,
find the initial velocity of $P$,
find the acceleration of $P$ in terms of $t$.
Explain why this model is unlikely to be realistic.

Edexcel M3 Q2

8 marks Standard +0.3

2. A particle $P$ is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point $A$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-2_410_570_1210_735} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} When a horizontal force of magnitude 20 N is applied to $P$, it rests in equilibrium with the string making an angle of $30 ^ { \circ }$ with the vertical and $A P = 1.2 \mathrm {~m}$ as shown in Figure 1.

Find the tension in the string.
Find the elastic potential energy stored in the string.

Edexcel M3 Q3

8 marks Challenging +1.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-3_513_570_196_625} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle of mass $m$ is suspended at a point $A$ vertically below a fixed point $O$ by a light inextensible string of length $a$ as shown in Figure 2. The particle is given a horizontal velocity $u$ and subsequently moves along a circular arc until it reaches the point $B$ where the string becomes slack. Given that the point $B$ is at a height $\frac { 1 } { 2 } a$ above the level of $O$,

show that $\angle B O A = 120 ^ { \circ }$,
show that $u ^ { 2 } = \frac { 7 } { 2 } g a$.

Edexcel M3 Q4

12 marks Standard +0.3

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs $6 \frac { 1 } { 4 }$ hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
(6 marks)
Given that a ship is not ready to enter the harbour until 5 p.m.,
find, to the nearest minute, how long the ship must wait before it can enter the harbour.

Edexcel M3 Q5

13 marks Standard +0.8

5. (a) Use integration to show that the centre of mass of a uniform solid right circular cone of height $h$ is $\frac { 3 } { 4 } h$ from the vertex of the cone.
(6 marks) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius $r$ so that the remaining solid consists of a hemisphere of radius $r$ and a cone of height $r$ and base radius $r$ as shown in Figure 3.
(b) Find the distance of the centre of mass of the paperweight from its vertex.
(7 marks)

Edexcel M3 Q6

13 marks Standard +0.8

6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is $\frac { 2 } { 5 }$.

Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
(5 marks)
The owner of the track decides to bank the corner at an angle of $25 ^ { \circ }$ in order to enable the cars to travel more quickly.
Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
(8 marks)

Edexcel M3 Q7

14 marks Standard +0.3

7. A particle is travelling along the $x$-axis. At time $t = 0$, the particle is at $O$ and it travels such that its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at a distance $x$ metres from $O$ is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is $a \mathrm {~ms} ^ { - 2 }$.

Show that $a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }$.
(4 marks) The points $A$ and $B$ lie on the $x$-axis. Given that the particle travels $d$ metres from $O$ to $A$ in $T$ seconds and 4 metres from $A$ to $B$ in 9 seconds,
show that $d = 1.5$,
find $T$.

Edexcel M3 Q1

7 marks Standard +0.3

A light elastic string has natural length $a$ and modulus of elasticity 4 mg . One end of the string is attached to a fixed point $A$ and a particle of mass $m$ is attached to the other end.

The particle is released from rest at $A$ and falls vertically until it comes to rest instantaneously at the point $B$. Find the distance $A B$ in terms of $a$.
(7 marks)

Edexcel M3 Q2

7 marks Standard +0.3

2. A particle $P$ of mass 0.25 kg is moving on a horizontal plane. At time $t$ seconds the velocity, $\mathbf { v } \mathrm { ms } ^ { - 1 }$, of $P$ relative to a fixed origin $O$ is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in the horizontal plane.

Find the acceleration of $P$ in terms of $t$.
Find, correct to 3 significant figures, the magnitude of the resultant force acting on $P$ when $t = 1$.
(4 marks)

Edexcel M3 Q3

9 marks Moderate -0.5

3. A coin of mass 5 grams is placed on a vinyl disc rotating on a record player. The distance between the centre of the coin and the centre of the disc is 0.1 m and the coefficient of friction between the coin and the disc is $\mu$. The disc rotates at 45 revolutions per minute around a vertical axis at its centre and the coin moves with it and does not slide. By modelling the coin as a particle and giving your answers correct to an appropriate degree of accuracy, find

the speed of the coin,
the horizontal and vertical components of the force exerted on the coin by the disc. Given that the coin is on the point of moving,
show that, correct to 2 significant figures, $\mu = 0.23$.

Edexcel M3 Q4

11 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-3_447_506_205_657} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A stand used to reach high shelves in a storeroom is in the shape of a frustum of a cone. It is modelled as a uniform solid formed by removing a right circular cone of height $2 h$ from a similar cone of height $3 h$ and base radius $3 r$ as shown in Figure 1.

Show that the centre of mass of the stand is a distance of $\frac { 33 } { 76 } h$ from its larger plane face.
(7 marks)
The stand is stored hanging in equilibrium from a point on the circumference of the larger plane face. Given that $h = 2 r$,
find, correct to the nearest degree, the acute angle which the plane faces of the stand make with the vertical.
(4 marks)

Edexcel M3 Q5

12 marks Standard +0.8

5. A particle of mass 0.8 kg is moving along the positive $x$-axis at a speed of $5 \mathrm {~ms} ^ { - 1 }$ away from the origin $O$. When the particle is 2 metres from $O$ it becomes subject to a single force directed towards $O$. The magnitude of the force is $\frac { k } { x ^ { 2 } } \mathrm {~N}$ when the particle is $x$ metres from $O$. Given that when the particle is 4 m from $O$ its speed has been reduced to $3 \mathrm {~ms} ^ { - 1 }$,

show that $k = \frac { 128 } { 5 }$,
find the distance of the particle from $O$ when it comes to instantaneous rest. (4 marks)

Edexcel M3 Q6

14 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_206_977_201_470} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a particle $P$ of mass $m$ which lies on a smooth horizontal table. It is attached to a point $A$ on the table by a light elastic spring of natural length $3 a$ and modulus of elasticity $\lambda$, and to a point $B$ on the table by a light elastic spring of natural length $2 a$ and modulus of elasticity $2 \lambda$. The distance between the points $A$ and $B$ is $7 a$.

Show that in equilibrium $A P = \frac { 9 } { 2 } a$. The particle is released from rest at a point $Q$ where $Q$ lies on the line $A B$ and $A Q = 5 a$.
Prove that the subsequent motion of the particle is simple harmonic with a period of $\pi \sqrt { \frac { 3 m a } { \lambda } }$.
(9 marks)

Edexcel M3 Q7

15 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_300_952_1201_497} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section $A B$ followed by a downhill section $B C$. The point $O$ is on the same horizontal level as $C$ and $B C$ is a circular arc of radius 30 m and centre $O$, such that $\angle B O C = 90 ^ { \circ }$. A skier of mass 60 kg is skiing at $12 \mathrm {~ms} ^ { - 1 }$ along $A B$.

Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at $B$.
(4 marks)
The skier subsequently leaves the slope at the point $P$.
Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point $D$ which is on the same horizontal level as $C$.

Edexcel M3 Q1

7 marks Moderate -0.3

The mechanism for releasing the ball on a pinball machine contains a light elastic spring of natural length 15 cm and modulus of elasticity $\lambda$.

The spring is held compressed to a length of 9 cm by a force of 4.5 N .

Find $\lambda$.
Find the work done in compressing the spring from a length of 9 cm to a length of 5 cm .
(4 marks)

Edexcel M3 Q2

7 marks Standard +0.8

2. A small bead $P$ is threaded onto a smooth circular wire of radius 0.8 m and centre $O$ which is fixed in a vertical plane. The bead is projected from the point vertically below $O$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in complete circles about $O$.

Suggest a suitable model for the bead.
Given that the minimum speed of $P$ is $60 \%$ of its maximum speed, use the principle of conservation of energy to show that $u = 7$.
(6 marks)

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