Questions - OCR - A-Level Maths

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 H240/03 2019 June Q5

9 marks Challenging +1.2

Prove that $( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = \frac { 1 + \cos \theta } { 1 - \cos \theta }$.
Hence solve, for $0 < \theta < 2 \pi , 3 ( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = 2 \sec \theta$.
\includegraphics[max width=\textwidth, alt={}]{7d1b7598-8f97-43a0-8366-efa8192d549e-06_574_695_306_258}
The diagram shows part of the curve $y = \frac { 2 x - 1 } { ( 2 x + 3 ) ( x + 1 ) ^ { 2 } }$.
Find the exact area of the shaded region, giving your answer in the form $p + q \ln r$, where $p$ and $q$ are positive integers and $r$ is a positive rational number.

OCR H240/03 2020 November Q9

13 marks Standard +0.3

For the motion before $B$ hits the ground, show that the acceleration of $B$ is $0.48 \mathrm {~ms} ^ { - 2 }$.
For the motion before $B$ hits the ground, show that the tension in the string is 23.3 N .
Determine the value of $\mu$. After $B$ hits the ground, $A$ continues to travel up the plane before coming to instantaneous rest before it reaches $P$.
Determine the distance that $A$ travels from the instant that $B$ hits the ground until $A$ comes to instantaneous rest.
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-09_917_784_244_242} The diagram shows a wall-mounted light. It consists of a rod $A B$ of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at $A$, and a lamp of mass 0.5 kg fixed at $B$. The system is held in equilibrium by a chain $C D$ whose end $C$ is attached to the midpoint of $A B$. The end $D$ is fixed to the wall a distance 0.4 m vertically above $A$. The rod $A B$ makes an angle of $60 ^ { \circ }$ with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.
By taking moments about $A$, determine the tension in the chain.
1. Determine the magnitude of the force exerted on the rod at $A$.
2. Calculate the direction of the force exerted on the rod at $A$.
Suggest one improvement that could be made to the model to make it more realistic.

OCR H240/03 2022 June Q5

14 marks Standard +0.8

Show that the $x$-coordinate of $P$ satisfies the equation $$4 x ^ { 3 } + 3 x - 3 = 0 .$$
Show by calculation that the $x$-coordinate of $P$ lies between 0.5 and 1 .
Show that the iteration $$x _ { n + 1 } = \frac { 3 - 4 x _ { n } ^ { 3 } } { 3 }$$ cannot converge to the $x$-coordinate of $P$ whatever starting value is used.
Use the Newton-Raphson method, with initial value 0.5 , to determine the coordinates of $P$ correct to $\mathbf { 5 }$ decimal places.

OCR H240/03 2022 June Q10

8 marks Challenging +1.2

Find the acceleration of $Q$ while $P$ and $B$ are in contact.
Determine the coefficient of friction between $P$ and $B$.
Given that the coefficient of friction between $B$ and the horizontal surface is $\frac { 5 } { 49 }$, determine the least possible value for the mass of $B$.
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-09_634_625_255_246} A uniform rod $A B$ of mass 4 kg and length 3 m rests in a vertical plane with $A$ on rough horizontal ground. A particle of mass 1 kg is attached to the rod at $B$. The rod makes an angle of $60 ^ { \circ }$ with the horizontal and is held in limiting equilibrium by a light inextensible string $C D . D$ is a fixed point vertically above $A$ and $C D$ makes an angle of $60 ^ { \circ }$ with the vertical. The distance $A C$ is $x \mathrm {~m}$ (see diagram).
Find, in terms of $g$ and $x$, the tension in the string. The coefficient of friction between the rod and the ground is $\frac { 9 \sqrt { 3 } } { 35 }$.
Determine the value of $x$.

OCR PURE 2018 May Q7

7 marks Standard +0.3

$\overrightarrow { A C }$
$\overrightarrow { O P }$
(ii) Hence prove that the diagonals of a parallelogram bisect one another.

OCR PURE 2066 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 Statistics AS Specimen Q5

7 marks Moderate -0.3

The random variable $X$ has the distribution $\operatorname { Geo } ( 0.6 )$.
(a) Find $\mathrm { P } ( X \geq 8 )$.
(b) Find the value of $\mathrm { E } ( X )$.
(c) Find the value of $\operatorname { Var } ( X )$.
The random variable $Y$ has the distribution $\operatorname { Geo } ( p )$. It is given that $\mathrm { P } ( Y < 4 ) = 0.986$ correct to 3 significant figures. Use an algebraic method to find the value of $p$. Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
State these two assumptions.
For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution $\operatorname { Po } ( 0.8 )$.
(a) Write down an expression for the probability that, in a given one minute period, exactly $r$ cars pass Sabrina's house.
(b) Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house.
Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house.
The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution $\operatorname { Po } ( 1.5 )$. Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition.

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 \end{itemize} $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ 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 $x \mathrm {~m}$.
On the axes provided in the Printed Answer Booklet, sketch a graph of $x$ against $t$ for $0 \leqslant t \leqslant 0.4$.
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 Pure Core 2 Specimen Q7

7 marks Challenging +1.2

Use the Maclaurin series for $\sin x$ to work out the series expansion of $\sin x \sin 2 x \sin 4 x$ up to and including the term in $x ^ { 3 }$.
Hence find, in exact surd form, an approximation to the least positive root of the equation $2 \sin x \sin 2 x \sin 4 x = x$.

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.

Questions — OCR (4619 questions)

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OCR FP3 2013 June Q2

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OCR H240/03 2020 November Q9

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OCR PURE 2018 May Q7

OCR PURE 2066 Q6

OCR Further Statistics AS 2024 June Q4

OCR Further Statistics AS Specimen Q5

OCR Further Mechanics AS 2019 June Q5

OCR Further Pure Core 2 2019 June Q6

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OCR Further Pure Core 1 2023 June Q6

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