Questions - OCR MEI M1 - A-Level Maths

Find the distance from A to B .
$T$ seconds after leaving A, the car is at a point C which is a distance of 10 m from B .
Find the value of $T$.
Find the displacement from A to C .

OCR MEI M1 2009 June Q2

2 A small box has weight $\mathbf { W } \mathrm { N }$ and is held in equilibrium by two strings with tensions $\mathbf { T } _ { 1 } \mathrm {~N}$ and $\mathbf { T } _ { 2 } \mathrm {~N}$. This situation is shown in Fig. 2 which also shows the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$ that are horizontal and vertically upwards, respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-2_259_629_1795_758} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The tension $\mathbf { T } _ { 1 }$ is $10 \mathbf { i } + 24 \mathbf { j }$.

Calculate the magnitude of $\mathbf { T } _ { 1 }$ and the angle between $\mathbf { T } _ { 1 }$ and the vertical. The magnitude of the weight is $w \mathrm {~N}$.
Write down the vector $\mathbf { W }$ in terms of $w$ and $\mathbf { j }$. The tension $\mathbf { T } _ { 2 }$ is $k \mathbf { i } + 10 \mathbf { j }$, where $k$ is a scalar.
Find the values of $k$ and of $w$.

OCR MEI M1 2009 June Q3

3 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-3_588_1091_351_529} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

How can you tell from the sketch that the acceleration is not modelled as being constant for $0 \leqslant t \leqslant 4$ ? The velocity of the sprinter, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, for the time interval $0 \leqslant t \leqslant 4$ is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 }$$
Find the acceleration that the model predicts for $t = 4$ and comment on what this suggests about the running of the sprinter.
Calculate the distance run by the sprinter from $t = 1$ to $t = 4$.

OCR MEI M1 2009 June Q4

4 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of $32 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-4_570_757_447_694} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Write down an expression, in terms of $\alpha$ and $t$, for the horizontal distance of the particle from O at time $t$ seconds after it is projected.
Show that $\cos \alpha = 0.28$.
Calculate the greatest height reached by the particle.

OCR MEI M1 2009 June Q5

5 The position vector of a toy boat of mass 1.5 kg is modelled as $\mathbf { r } = ( 2 + t ) \mathbf { i } + \left( 3 t - t ^ { 2 } \right) \mathbf { j }$ where lengths are in metres, $t$ is the time in seconds, $\mathbf { i }$ and $\mathbf { j }$ are horizontal, perpendicular unit vectors and the origin is O .

Find the velocity of the boat when $t = 4$.
Find the acceleration of the boat and the horizontal force acting on the boat.
Find the cartesian equation of the path of the boat referred to $x$ - and $y$-axes in the directions of $\mathbf { i }$ and $\mathbf { j }$, respectively, with origin O . You are not required to simplify your answer. Section B (36 marks)

OCR MEI M1 2009 June Q6

6 An empty open box of mass 4 kg is on a plane that is inclined at $25 ^ { \circ }$ to the horizontal.
In one model the plane is taken to be smooth.
The box is held in equilibrium by a string with tension $T \mathrm {~N}$ parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-5_314_575_621_785} \captionsetup{labelformat=empty} \caption{Fig. 6.1}

\end{figure}

Calculate $T$. A rock of mass $m \mathrm {~kg}$ is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
Find $m$. In a refined model the plane is rough.
The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of $P \mathrm {~N}$ inclined at $15 ^ { \circ }$ to the plane, as shown in Fig. 6.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-5_369_561_1653_790} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
\end{figure}
Draw a diagram showing all the forces acting on the box.
Calculate $P$.
Calculate the normal reaction of the plane on the box.

OCR MEI M1 2014 June Q1

1 Fig. 1 shows the velocity-time graph of a cyclist travelling along a straight horizontal road between two sets of traffic lights. The velocity, $v$, is measured in metres per second and the time, $t$, in seconds. The distance travelled, $s$ metres, is measured from when $t = 0$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{63a2dc41-5e8b-4275-8653-ece5067c4306-2_732_1116_513_477} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Find the values of $s$ when $t = 4$ and when $t = 18$.
Sketch the graph of $s$ against $t$ for $0 \leqslant t \leqslant 18$.

OCR MEI M1 2014 June Q2

2 The unit vectors $\mathbf { i }$ and $\mathbf { j }$ shown in Fig. 2 are in the horizontal and vertically upwards directions. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{63a2dc41-5e8b-4275-8653-ece5067c4306-2_132_145_1726_968} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Forces $\mathbf { p }$ and $\mathbf { q }$ are given, in newtons, by $\mathbf { p } = 12 \mathbf { i } - 5 \mathbf { j }$ and $\mathbf { q } = 16 \mathbf { i } + 1.5 \mathbf { j }$.

Write down the force $\mathbf { p } + \mathbf { q }$ and show that it is parallel to $8 \mathbf { i } - \mathbf { j }$.
Show that the force $3 \mathbf { p } + 10 \mathbf { q }$ acts in the horizontal direction.
A particle is in equilibrium under forces $k \mathbf { p } , 3 \mathbf { q }$ and its weight $\mathbf { w }$. Show that the value of $k$ must be - 4 and find the mass of the particle.

OCR MEI M1 2014 June Q3

3 Fig. 3 shows a smooth ball resting in a rack. The angle in the middle of the rack is $90 ^ { \circ }$. The rack has one edge at angle $\alpha$ to the horizontal. The weight of the ball is $W \mathrm {~N}$. The reaction forces of the rack on the ball at the points of contact are $R \mathrm {~N}$ and $S \mathrm {~N}$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{63a2dc41-5e8b-4275-8653-ece5067c4306-3_314_460_484_813} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Draw a fully labelled triangle of forces to show the forces acting on the ball. Your diagram must indicate which angle is $\alpha$.
Find the values of $R$ and $S$ in terms of $W$ and $\alpha$.
On the same axes draw sketches of $R$ against $\alpha$ and $S$ against $\alpha$ for $0 ^ { \circ } \leqslant \alpha \leqslant 90 ^ { \circ }$. For what values of $\alpha$ is $R < S$ ?

OCR MEI M1 2014 June Q4

4 Fig. 4 illustrates a situation in which a film is being made. A cannon is fired from the top of a vertical cliff towards a ship out at sea. The director wants the cannon ball to fall just short of the ship so that it appears to be a near-miss. There are actors on the ship so it is important that it is not hit by mistake. The cannon ball is fired from a height 75 m above the sea with an initial velocity of $20 \mathrm {~ms} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal. The ship is 90 m from the bottom of the cliff. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{63a2dc41-5e8b-4275-8653-ece5067c4306-3_337_1242_1717_406} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

The director calculates where the cannon ball will hit the sea, using the standard projectile model and taking the value of $g$ to be $10 \mathrm {~ms} ^ { - 2 }$. Verify that according to this model the cannon ball is in the air for 5 seconds. Show that it hits the water less than 5 m from the ship.
Without doing any further calculations state, with a brief reason, whether the cannon ball would be predicted to travel further from the cliff if the value of $g$ were taken to be $9.8 \mathrm {~ms} ^ { - 2 }$.

OCR MEI M1 2014 June Q5

5 In a science fiction story a new type of spaceship travels to the moon. The journey takes place along a straight line. The spaceship starts from rest on the earth and arrives at the moon's surface with zero speed. Its speed, $v$ kilometres per hour at time $t$ hours after it has started, is given by $$v = 37500 \left( 4 t - t ^ { 2 } \right) .$$

Show that the spaceship takes 4 hours to reach the moon.
Find an expression for the distance the spaceship has travelled at time $t$. Hence find the distance to the moon.
Find the spaceship's greatest speed during the journey. Section B (36 marks)

OCR MEI M1 2014 June Q6

6 In this question the origin is a point on the ground. The directions of the unit vectors $\left( \begin{array} { l } 1
0
0 \end{array} \right) , \left( \begin{array} { l } 0
1
0 \end{array} \right)$ and $\left( \begin{array} { l } 0
0
1 \end{array} \right)$ are
\includegraphics[max width=\textwidth, alt={}, center]{63a2dc41-5e8b-4275-8653-ece5067c4306-5_398_689_434_689} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
Alesha's mass, including her equipment, is 100 kg .
Initially, her position vector is $\left( \begin{array} { r } - 75
90
750 \end{array} \right) \mathrm { m }$ and her velocity is $\left( \begin{array} { r } - 5
0
- 10 \end{array} \right) \mathrm { ms } ^ { - 1 }$.
1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of $\left( \begin{array} { r } 0
0
- 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0
0
880 \end{array} \right) \mathrm { N }$ and $\left( \begin{array} { r } 50
- 20
0 \end{array} \right) \mathrm { N }$ are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is $1.14 \mathrm {~ms} ^ { - 2 }$. One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time $t$ seconds after the start of the dive according to this model. Verify that when $t = 30$ she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

OCR MEI M1 2015 June Q1

1 Fig. 1 shows four forces acting at a point. The forces are in equilibrium. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-2_401_645_397_719} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Show that $P = 14$. Find $Q$, giving your answer correct to 3 significant figures.

OCR MEI M1 2015 June Q2

2 Fig. 2 shows a 6 kg block on a smooth horizontal table. It is connected to blocks of mass 2 kg and 9 kg by two light strings which pass over smooth pulleys at the edges of the table. The parts of the strings attached to the 6 kg block are horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-2_344_1143_1352_443} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Draw three separate diagrams showing all the forces acting on each of the blocks.
Calculate the acceleration of the system and the tension in each string.

OCR MEI M1 2015 June Q3

3 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors $\binom { 1 } { 0 }$ and $\binom { 0 } { 1 }$ are in the directions east and north. At time $t$ hours the position vectors of two hikers, Ashok and Kumar, are given by: $$\begin{array} { l l } \text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t ,
\text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } . \end{array}$$

Prove that the two hikers meet and give the coordinates of the point where this happens.
Compare the speeds of the two hikers.

OCR MEI M1 2015 June Q4

4 Fig. 4 illustrates a straight horizontal road. A and B are points on the road which are 215 metres apart and M is the mid-point of AB . When a car passes A its speed is $12 \mathrm {~ms} ^ { - 1 }$ in the direction AB . It then accelerates uniformly and when it reaches $B$ its speed is $31 \mathrm {~ms} ^ { - 1 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-3_138_1152_1247_459} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Find the car's acceleration.
Find how long it takes the car to travel from A to B .
Find how long it takes the car to travel from A to M .
Explain briefly, in terms of the speed of the car, why the time taken to travel from A to M is more than half the time taken to travel from A to B .

OCR MEI M1 2015 June Q5

5 A golf ball is hit at an angle of $60 ^ { \circ }$ to the horizontal from a point, O , on level horizontal ground. Its initial speed is $20 \mathrm {~ms} ^ { - 1 }$. The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time $t \mathrm {~s}$ the horizontal and vertical components of its displacement from O are denoted by $x \mathrm {~m}$ and $y \mathrm {~m}$.

Write down equations for $x$ and $y$ in terms of $t$.
Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 } .$$
Find the range of the golf ball.
A bird is hovering at position $( 20,16 )$. Find whether the golf ball passes above it, passes below it or hits it.

OCR MEI M1 2015 June Q6

6 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.

From A to B the road is horizontal.
Between B and C, it goes up a hill with a uniform slope of $1.5 ^ { \circ }$ to the horizontal.
Between C and D the road goes down a hill with a uniform slope of $3 ^ { \circ }$ to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
From D to E the road is again horizontal.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_241_1134_808_450} \captionsetup{labelformat=empty} \caption{Fig. 6.1}

\end{figure} The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.

Between A and B , Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Show that the resistance to the car's motion is 100 N . Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .
They stop at B and then Martin tries to push the car up the hill BC. Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.
Find the acceleration of the car when Martin is pushing it and Carol is standing outside.
While between B and C , Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of $30 ^ { \circ }$ to the line of the car. This is illustrated in Fig. 6.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_218_426_2133_831} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
\end{figure} Find the acceleration of the car.
At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of $8 \mathrm {~ms} ^ { - 1 }$ they can get the engine to start. Does the car reach this speed before it reaches D ?

OCR MEI M1 2016 June Q1

1 Fig. 1 shows a block of mass $M \mathrm {~kg}$ being pushed over level ground by means of a light rod. The force, $T \mathrm {~N}$, this exerts on the block is along the line of the rod. The ground is rough.
The rod makes an angle $\alpha$ with the horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_307_876_621_593} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Draw a diagram showing all the forces acting on the block.
You are given that $M = 5 , \alpha = 60 ^ { \circ } , T = 40$ and the acceleration of the block is $1.5 \mathrm {~ms} ^ { - 2 }$. Find the frictional force.

OCR MEI M1 2016 June Q2

2 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_117_1162_1486_438} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, $t$, is measured in seconds. The particle has constant acceleration, $a \mathrm {~ms} ^ { - 2 }$. Initially it is at the point O and has velocity $u \mathrm {~ms} ^ { - 1 }$.
When $t = 2$, the particle is at A where OA is 12 m . The particle is also at A when $t = 6$.

Write down two equations in $u$ and $a$ and solve them.
The particle changes direction when it is at B . Find the distance AB .

OCR MEI M1 2016 June Q3

3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_330_809_525_628} \captionsetup{labelformat=empty} \caption{Fig. 3.1}

\end{figure}

Write down two equations of motion, one for each block.
Find the value of $a$. The table is now tilted at an angle of $\theta$ to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_410_727_1324_669} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure}
The system is now in equilibrium. Find the value of $\theta$.

OCR MEI M1 2016 June Q4

4 A particle is initially at the origin, moving with velocity $\mathbf { u }$. Its acceleration $\mathbf { a }$ is constant. At time $t$ its displacement from the origin is $\mathbf { r } = \binom { x } { y }$, where $\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }$.

Write down $\mathbf { u }$ and $\mathbf { a }$ as column vectors.
Find the speed of the particle when $t = 2$.
Show that the equation of the path of the particle is $y = 3 x - x ^ { 2 }$.

OCR MEI M1 2016 June Q5

5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-4_400_1221_1078_411} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

The stone is thrown from point O on level ground. Its initial velocity is $15 \mathrm {~ms} ^ { - 1 }$ in the horizontal direction and $8 \mathrm {~ms} ^ { - 1 }$ in the vertical direction.
The pigeon is at point P which is 4 m above the ground.
The house is 22.5 m from O .
The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.

Show that the stone does not reach the height of the pigeon. Determine whether the stone hits the window.