Questions - OCR MEI - A-Level Maths

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

7 marks Standard +0.3

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

8 marks Moderate -0.8

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

16 marks Moderate -0.3

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

6 marks Easy -1.2

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

8 marks Moderate -0.8

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

8 marks Moderate -0.3

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

7 marks Moderate -0.3

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

7 marks Moderate -0.8

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

18 marks Moderate -0.5

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 C2 Q1

14 marks Moderate -0.3

Use calculus to find, correct to 1 decimal place, the coordinates of the turning points of the curve $y = x ^ { 3 } - 5 x$. [You need not determine the nature of the turning points.]
Find the coordinates of the points where the curve $y = x ^ { 3 } - 5 x$ meets the axes and sketch the curve.
Find the equation of the tangent to the curve $y = x ^ { 3 } - 5 x$ at the point $( 1 , - 4 )$. Show that, where this tangent meets the curve again, the $x$-coordinate satisfies the equation $$x ^ { 3 } - 3 x + 2 = 0$$ Hence find the $x$-coordinate of the point where this tangent meets the curve again.

OCR MEI C2 Q2

12 marks Moderate -0.3

2 The equation of a cubic curve is $y = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that the tangent to the curve when $x = 3$ passes through the point $( - 1 , - 41 )$.
Use calculus to find the coordinates of the turning points of the curve. You need not distinguish between the maximum and minimum.
Sketch the curve, given that the only real root of $2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2 = 0$ is $x = 0.2$ correct to 1 decimal place. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b6ea89e3-a8a4-41a2-8ed5-eed6c2dfda7e-2_1017_935_285_638} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the cubic curve $y = \mathrm { f } ( x )$. The values of $x$ where it crosses the $x$-axis are - 5 , - 2 and 2 , and it crosses the $y$-axis at $( 0 , - 20 )$.

OCR MEI C2 Q4

11 marks Moderate -0.8

Differentiate $x ^ { 3 } - 3 x ^ { 2 } - 9 x$. Hence find the $x$-coordinates of the stationary points on the curve $y = x ^ { 3 } - 3 x ^ { 2 } - 9 x$, showing which is the maximum and which the minimum.
Find, in exact form, the coordinates of the points at which the curve crosses the $x$-axis.
Sketch the curve.

OCR MEI C2 Q5

12 marks Moderate -0.8

5 The equation of a curve is $\quad y = 7 + 6 x - x ^ { 2 }$.

Use calculus to find the coordinates of the turning point on this curve. Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
Find $\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x$, showing your working.
The curve and the line $y = 12$ intersect at $( 1,12 )$ and $( 5,12 )$. Using your answer to part (ii), find the area of the finite region between the curve and the line $y = 12$.

OCR MEI M1 2015 June Q1

5 marks Moderate -0.8

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

8 marks Standard +0.3

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

7 marks Moderate -0.3

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

8 marks Moderate -0.8

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

8 marks Moderate -0.3

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

18 marks Standard +0.3

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 C2 Q1

4 marks Easy -1.2

1 The point $\mathrm { R } ( 6 , - 3 )$ is on the curve $y = \mathrm { f } ( x )$.

Find the coordinates of the image of R when the curve is transformed to $y = \frac { 1 } { 2 } \mathrm { f } ( x )$.
Find the coordinates of the image of R when the curve is transformed to $y = \mathrm { f } ( 3 x )$.