Questions - OCR - A-Level Maths

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.

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 $\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)$.
1. Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
2. 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 $\beta + \frac { 1 } { \beta }$.
  1. Show that $p = \frac { 5 } { 6 }$.
  2. Find the value of $q$. 9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)$.
    1. Find, in terms of $a$, the determinant of $\mathbf { M }$.
    2. Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
    3. 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.
    4. Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.
    5. Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
    6. 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 FP3 2007 June Q7

10 marks Standard +0.3

Show that $\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1$.
Write down the seven roots of the equation $z ^ { 7 } = 1$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$ and show their positions in an Argand diagram.
Hence express $z ^ { 7 } - 1$ as the product of one real linear factor and three real quadratic factors.

OCR FP3 2013 June Q2

9 marks Challenging +1.2

Write down the operation table and, assuming associativity, show that $G$ is a group.
State the order of each element.
Find all the proper subgroups of $G$. The group $H$ consists of the set $\{ 1,3,7,9 \}$ with the operation of multiplication modulo 10 .
Explaining your reasoning, determine whether $H$ is isomorphic to $G$.

OCR FP3 2016 June Q7

12 marks Challenging +1.2

Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
Hence show that, for $\sin 2 \theta \neq 0$, $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$

OCR D1 2006 June Q6

16 marks Standard +0.3

Calculate the shortest distance that the mole must travel if it starts and ends at vertex $A$.
The pipe connecting $B$ to $H$ is removed for repairs. By considering every possible pairing of odd vertices, and showing your working clearly, calculate the shortest distance that the mole must travel to pass along each pipe on this reduced network, starting and finishing at $A$.

OCR H240/01 2020 November Q11

10 marks Challenging +1.2

1. Show that the $x$-coordinate of $A$ satisfies the equation $\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0$.
2. Hence determine the equation of the tangent to the circle at $A$ which passes through $P$. [4] A second tangent is drawn from $P$ to meet the circle at a second point $B$. The equation of this tangent is of the form $y = n x + 2$, where $n$ is a constant less than 1 .
Determine the exact value of $\tan A P B$.

OCR H240/02 2018 June Q6

13 marks Moderate -0.3

Find the $x$-coordinate of the point where the curve crosses the $x$ axis.
The points $A$ and $B$ lie on the curve and have $x$ coordinates 2 and 4. Show that the line $A B$ is parallel to the $x$-axis.
Find the coordinates of the turning point on the curve.
Determine whether this turning point is a maximum or a minimum.

OCR PURE Q6

7 marks Standard +0.3

Show that the equation $6 \cos ^ { 2 } \theta = \tan \theta \cos \theta + 4$ can be expressed in the form $6 \sin ^ { 2 } \theta + \sin \theta - 2 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-4_446_1150_1119_338} The diagram shows parts of the curves $y = 6 \cos ^ { 2 } \theta$ and $y = \tan \theta \cos \theta + 4$, where $\theta$ is in degrees. Solve the inequality $6 \cos ^ { 2 } \theta > \tan \theta \cos \theta + 4$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.

OCR Further Statistics AS 2024 June Q4

12 marks Challenging +1.2

Find the probability that 4 telephone calls are received in a randomly chosen one-minute period.
A sample of 10 independent observations of $X$ is obtained. Find the expected number of these 10 observations that are in the interval $2 < X < 8$. It is also known that $P ( X + Y = 4 ) = \frac { 27 } { 8 } P ( X = 2 ) \times P ( Y = 2 )$.
Determine the possible values of $\mathrm { E } ( Y )$.
Explain where in your solution to part (c) you have used the assumption that telephone calls and e-mails are received independently of one another.

OCR Further Mechanics AS 2019 June Q5

14 marks Standard +0.8

By considering forces on $R$, express $T _ { 2 }$ in terms of $m _ { 2 }$.
Show that
1. $T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)$,
2. $\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }$.
Deduce that, in the case where $m _ { 1 }$ is much bigger than $m _ { 2 } , \omega \approx 3.5$. In a different case, where $m _ { 1 } = 2.5$ and $m _ { 2 } = 2.8 , P$ slows down. Eventually the system comes to rest with $P$ and $R$ hanging in equilibrium.
Find the total energy lost by $P$ and $R$ as the angular velocity of $P$ changes from the initial value of $\omega \mathrm { rads } ^ { - 1 }$ to zero.

OCR Further Pure Core 2 2019 June Q6

6 marks Standard +0.8

Show that the motion of the particle can be modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when $t = 0$.
Find $v$ in terms of $t$.
Find the velocity of the particle when $t = 2$. When $t = 2$ the force acting in the positive $x$-direction is replaced by a constant force of magnitude $\frac { 1 } { 2 } \mathrm {~N}$ in the same direction.
Refine the differential equation given in part (a) to model the motion for $t \geqslant 2$.
Use the refined model from part (d) to find an exact expression for $v$ in terms of $t$ for $t \geqslant 2$. $6 \quad A$ is a fixed point on a smooth horizontal surface. A particle $P$ is initially held at $A$ and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point $B$ whose displacement is 0.2 m from $A$. The point $M$ is halfway between $A$ and $B$. The displacement of $P$ from $M$ at time $t$ seconds after release is denoted by $x \mathrm {~m}$.
1. On the axes provided in the Printed Answer Booklet, sketch a graph of $x$ against $t$ for $0 \leqslant t \leqslant 0.4$.
2. Find the displacement of $P$ from $M$ at 0.75 seconds after release.

OCR Further Pure Core 2 2019 June Q9

11 marks Challenging +1.2

Find the exact area enclosed by the curve.
Show that the greatest value of $r$ on the curve is $\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }$.

OCR Further Pure Core 2 2022 June Q8

7 marks Challenging +1.2

Show that $\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta$, where $a$ is an integer to be determined.
Hence show that $\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }$, where $b$ and $c$ are integers to be determined.

OCR Further Mechanics 2024 June Q7

14 marks Standard +0.8

Show that $B$ 's motion can be modelled by the differential equation $\frac { 1 } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dx } } = - 4$.
1. Solve the differential equation in part (a) to find the particular solution for $v$ in terms of $x$ and $u$.
2. By considering the behaviour of $v$ as $x \longrightarrow \infty$ describe one feature of the model that is not realistic. At the instant when $B$ reaches the point $A$, where $\mathrm { x } = \mathrm { X }$, its speed is $V \mathrm {~ms} ^ { - 1 }$. The work done by the resistance as $B$ moves from $O$ to $A$ is denoted by $W \mathrm {~J}$.
1. Use the formula $\mathrm { W } = \int \mathrm { F } \mathrm { dx }$ to determine an expression for $W$ in terms of $X$ and $u$.
2. Explain the relevance of the sign of your answer in part (c)(i).
3. By writing your answer to part (c)(i) in terms of $V$ and $u$ show how the quantity $W$ relates to the energy of $B$.

OCR Further Pure Core 1 2023 June Q9

14 marks Challenging +1.8

9 In this question you must show detailed reasoning.

Use de Moivre's theorem to determine constants $A$, $B$ and $C$ such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by $\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1$.
Show that $\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$. \includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$ for $0 \leqslant x < 1$ and the asymptote $x = 1$. The region $R$ is the unbounded region between the curve, the $x$-axis, the line $x = 0$ and the line $x = 1$. You are given that the area of $R$ is finite.
Determine the exact area of $R$.

OCR Further Pure Core 1 2023 June Q6

4 marks Standard +0.8

6 In this question you must show detailed reasoning. The power output, $p$ watts, of a machine at time $t$ hours after it is switched on can be modelled by the equation $\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )$ for $t \geqslant 0$. Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to $\mathbf { 2 }$ decimal places.

OCR D2 2007 January Q6

12 marks Moderate -0.5

6 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a maximin problem.

Stage	State	Action	Working	Maximin
\multirow{2}{*}{1}	0	0	4	4
	1	0	3	3
\multirow{6}{*}{2}	0	0	$\min ( 6,4 ) = 4$	\multirow{2}{*}{}
		1	$\min ( 2,3 ) = 2$
	\multirow{2}{*}{1}	0	$\min ( 2,4 ) =$	\multirow{2}{*}{}
		1	$\min ( 4,3 ) =$
	\multirow{2}{*}{2}	0	min(2,	\multirow{2}{*}{}
		1	min(3,
\multirow{3}{*}{3}	\multirow{3}{*}{0}	0	min(5,	\multirow{3}{*}{}
		1	$\min ( 5$,
		2	$\min ( 2$,

Complete the last two columns of the table in the insert.
State the maximin value and write down the maximin route.

OCR D1 2006 January Q1

5 marks Easy -1.2

1 Answer this question on the insert provided.

\includegraphics[max width=\textwidth, alt={}]{8f17020a-14bf-4459-9241-1807b954a629-2_956_1203_349_493}

This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight.

OCR D1 2006 January Q2

6 marks Moderate -0.8

2 Answer this question on the insert provided.

\includegraphics[max width=\textwidth, alt={}]{8f17020a-14bf-4459-9241-1807b954a629-2_659_1136_1720_530}

This diagram shows part of a network. There are other arcs connecting $D$ and $E$ to other parts of the network. Apply Dijkstra's algorithm starting from $A$, as far as you are able, showing your working. Note: you will not be able to give permanent labels to all the vertices shown.

OCR D1 2007 January Q5

16 marks Moderate -0.3

5 Answer part (i) of this question on the insert provided. Rhoda Raygh enjoys driving but gets extremely irritated by speed cameras.
The network represents a simplified map on which the arcs represent roads and the weights on the arcs represent the numbers of speed cameras on the roads. The sum of the weights on the arcs is 72 . \includegraphics[max width=\textwidth, alt={}, center]{8a1232ae-6a6e-4afb-8757-fffe4fc9570f-05_874_1484_664_333}

Rhoda lives at Ayton ( $A$ ) and works at Kayton ( $K$ ). Use Dijkstra's algorithm on the diagram in the insert to find the route from $A$ to $K$ that involves the least number of speed cameras and state the number of speed cameras on this route.
In her job Rhoda has to drive along each of the roads represented on the network to check for overhanging trees. This requires finding a route that covers every arc at least once, starting and ending at Kayton (K). Showing all your working, find a suitable route for Rhoda that involves the least number of speed cameras and state the number of speed cameras on this route.
If Rhoda checks the roads for overhanging trees on her way home, she will instead need a route that covers every arc at least once, starting at Kayton and ending at Ayton. Calculate the least number of speed cameras on such a route, explaining your reasoning.

OCR D1 2009 January Q3

23 marks Moderate -0.3

3 Answer this question on the insert provided. \includegraphics[max width=\textwidth, alt={}, center]{43fe5fd5-4b98-4c3a-90ca-a1bd5cf065fe-3_492_1006_356_568}

This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight.
Use your answer to part (i) to find the weight of a minimum spanning tree for the network with vertex $E$, and all the arcs joined to $E$, removed. Hence find a lower bound for the travelling salesperson problem on the original network.
Show that the nearest neighbour method, starting from vertex $A$, fails on the original network.
Apply the nearest neighbour method, starting from vertex $B$, to find an upper bound for the travelling salesperson problem on the original network.
Apply Dijkstra's algorithm to the copy of the network in the insert to find the least weight path from $A$ to $G$. State the weight of the path and give its route.
The sum of the weights of all the arcs is 300 . Apply the route inspection algorithm, showing all your working, to find the weight of the least weight closed route that uses every arc at least once. The weights of least weight paths from vertex $A$ should be found using your answer to part (v); the weights of other such paths should be determined by inspection.

OCR D1 2009 January Q4

12 marks Easy -1.2

4 Answer this question on the insert provided. The list of numbers below is to be sorted into decreasing order using shuttle sort. $$\begin{array} { l l l l l l l l l } 21 & 76 & 65 & 13 & 88 & 62 & 67 & 28 & 34 \end{array}$$

How many passes through shuttle sort will be required to sort the list? After the first pass the list is as follows. $$\begin{array} { l l l l l l l l l } 76 & 21 & 65 & 13 & 88 & 62 & 67 & 28 & 34 \end{array}$$
State the number of comparisons and the number of swaps that were made in the first pass.
Write down the list after the second pass. State the number of comparisons and the number of swaps that were used in making the second pass.
Complete the table in the insert to show the results of the remaining passes, recording the number of comparisons and the number of swaps made in each pass. You may not need all the rows of boxes printed. When the original list is sorted into decreasing order using bubble sort there are 30 comparisons and 17 swaps.
Use your results from part (iv) to compare the efficiency of these two methods in this case. Katie makes and sells cookies. Each batch of plain cookies takes 8 minutes to prepare and then 12 minutes to bake. Each batch of chocolate chip cookies takes 12 minutes to prepare and then 12 minutes to bake. Each batch of fruit cookies takes 10 minutes to prepare and then 12 minutes to bake. Katie can only bake one batch at a time. She has the use of the kitchen, including the oven, for at most 1 hour.
[0pt]

Each batch of cookies must be prepared before it is baked. By considering the maximum time available for baking the cookies, explain why Katie can make at most 4 batches of cookies. [2] Katie models the constraints as $$\begin{gathered} x + y + z \leqslant 4 \\ 4 x + 6 y + 5 z \leqslant 24 \\ x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$ where $x$ is the number of batches of plain cookies, $y$ is the number of batches of chocolate chip cookies and $z$ is the number of batches of fruit cookies that Katie makes.
Each batch of cookies that Katie prepares must be baked within the hour available. By considering the maximum time available for preparing the cookies, show how the constraint $4 x + 6 y + 5 z \leqslant 24$ was formed.
In addition to the constraints, what other restriction is there on the values of $x , y$ and $z$ ? Katie will make $\pounds 5$ profit on each batch of plain cookies, $\pounds 4$ on each batch of chocolate chip cookies and $\pounds 3$ on each batch of fruit cookies that she sells. Katie wants to maximise her profit.
Write down an expression for the objective function to be maximised. State any assumption that you have made.
Represent Katie's problem as an initial Simplex tableau. Perform one iteration of the Simplex algorithm, choosing to pivot on an element from the $x$-column. Show how each row was obtained. Write down the number of batches of cookies of each type and the profit at this stage. After carrying out market research, Katie decides that she will not make fruit cookies. She also decides that she will make at least twice as many batches of chocolate chip cookies as plain cookies.
Represent the constraints for Katie's new problem graphically and calculate the coordinates of the vertices of the feasible region. By testing suitable integer-valued coordinates, find how many batches of plain cookies and how many batches of chocolate chip cookies Katie should make to maximise her profit. Show your working.

OCR D1 2010 January Q1

11 marks Standard +0.3

1 Answer this question on the insert provided. \includegraphics[max width=\textwidth, alt={}, center]{e1495f6b-c09f-46a1-a6f8-02354e28887a-02_533_1353_342_395}

Apply Dijkstra's algorithm to the copy of this network in the insert to find the least weight path from $A$ to $F$. State the route of the path and give its weight.
Apply the route inspection algorithm, showing all your working, to find the weight of the least weight closed route that uses every arc at least once. Write down a closed route that has this least weight. An extra arc is added, joining $B$ to $E$, with weight 2 .
Write down the new least weight path from $A$ to $F$. Explain why the new least weight closed route, that uses every arc at least once, has no repeated arcs.

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