Questions - OCR - A-Level Maths

Number of attempts	1	2	3	4 or more
Frequency	20	19	13	3

Test at the $1 \%$ significance level whether the results are consistent with the distribution Geo(0.4).
[0pt]
Suggest which two probabilities should be changed, and in what way, to produce an improved model. (Numerical values are not required.) You should give a reason for your suggestion. [3]

OCR Further Statistics 2018 December Q8

8 A continuous random variable $X$ has probability density function given by the following function, where $a$ is a constant.
$\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise. } \end{array} \right\}$
The expected value of $X$ is 4 .

Show that $a = 6$. Five independent observations of $X$ are obtained, and the largest of them is denoted by $M$.
Find the cumulative distribution function of $M$. \section*{OCR} Oxford Cambridge and RSA

OCR Further Mechanics 2018 December Q1

1 A particle, $P$, of mass 2 kg moves in two dimensions. Its initial velocity is $\binom { - 19.5 } { - 60 } \mathrm {~ms} ^ { - 1 }$.

Calculate the initial kinetic energy of $P$. For $t \geqslant 0 , P$ is acted upon only by a variable force $\mathbf { F } = \binom { 4 t } { - 2 } \mathrm {~N}$, where $t$ is the time in seconds.
Find
- the velocity of $P$ in terms of $t$,
- the times when the power generated by $\mathbf { F }$ is zero.

OCR Further Mechanics 2018 December Q2

2 A car of mass 800 kg is driven with its engine generating a power of 15 kW .

The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
The car is next driven at constant speed up a straight road inclined at an angle $\theta$ to the horizontal. The resistance to motion is now modelled as being constant with magnitude 150 N . Given that $\sin \theta = \frac { 1 } { 20 }$, find the speed of the car.
The car is now driven at a constant speed of $30 \mathrm {~ms} ^ { - 1 }$ along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming that the resistance to motion of the car is three times the resistance to motion of the trailer, find
- the resistance to motion of the car,
- the magnitude of the tension in the towbar.

OCR Further Mechanics 2018 December Q3

3 Three particles, $A , B$ and $C$, of masses $2 \mathrm {~kg} , 3 \mathrm {~kg}$ and 5 kg respectively, are at rest in a straight line on a smooth horizontal plane with $B$ between $A$ and $C$. Collisions between $A$ and $B$ are perfectly elastic. The coefficient of restitution for collisions between $B$ and $C$ is $e$.
$A$ is projected towards $B$ with a speed of $5 u \mathrm {~ms} ^ { - 1 }$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-2_186_903_2330_251} Show that only two collisions occur.

OCR Further Mechanics 2018 December Q4

4 A particle $P$ of mass 8 kg moves in a straight line on a smooth horizontal plane. At time $t \mathrm {~s}$ the displacement of $P$ from a fixed point $O$ on the line is $x \mathrm {~m}$ and the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$. Initially, $P$ is at rest at $O$.
$P$ is acted on by a horizontal force, directed along the line away from $O$, with magnitude proportional to $\sqrt { 9 + v ^ { 2 } }$. When $v = 1.25$, the magnitude of this force is 13 N .

Show that $\frac { 1 } { \sqrt { 9 + v ^ { 2 } } } \frac { \mathrm {~d} v } { \mathrm {~d} t } = \frac { 1 } { 2 }$.
Find an expression for $v$ in terms of $t$ for $t \geqslant 0$.
Find an expression for $x$ in terms of $t$ for $t \geqslant 0$.

OCR Further Mechanics 2018 December Q5

5 One end of a light inextensible string of length 0.8 m is attached to a fixed point, $O$. The other end is attached to a particle $P$ of mass $1.2 \mathrm {~kg} . P$ hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below $O$ is $F$.
$P$ is projected horizontally with speed $3.5 \mathrm {~ms} ^ { - 1 }$. The string breaks when $O P$ makes an angle of $\frac { 1 } { 3 } \pi$ radians with the downwards vertical through $O$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-3_776_910_1242_244}

Find the magnitude of the tension in the string at the instant before the string breaks.
Find the distance between $F$ and the point where $P$ first hits the plane.

OCR Further Mechanics 2018 December Q6

6 This question is about modelling the relation between the pressure, $P$, volume, $V$, and temperature, $\theta$, of a fixed amount of gas in a container whose volume can be varied. The amount of gas is measured in moles; 1 mole is a dimensionless constant representing a fixed number of molecules of gas. Gas temperatures are measured on the Kelvin scale; the unit for temperature is denoted by K . You may assume that temperature is a dimensionless quantity. A gas in a container will always exert an outwards force on the walls of the container. The pressure of the gas is defined to be the magnitude of this force per unit area of the walls, with $P$ always positive. An initial model of the relation is given by $P ^ { \alpha } V ^ { \beta } = n R \theta$, where $n$ is the number of moles of gas present and $R$ is a quantity called the Universal Gas Constant. The value of $R$, correct to 3 significant figures, is $8.31 \mathrm { JK } ^ { - 1 }$.

Show that $[ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }$ and $[ R ] = \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 }$.
Hence show that $\alpha = 1$ and $\beta = 1$. 5 moles of gas are present in the container which initially has volume $0.03 \mathrm {~m} ^ { 3 }$ and which is maintained at a temperature of 300 K .
Find the pressure of the gas, as predicted by the model. An improved model of the relation is given by $\left( P + \frac { a n ^ { 2 } } { V ^ { 2 } } \right) ( V - n b ) = n R \theta$, where $a$ and $b$ are constants.
Determine the dimensions of $b$ and $a$. The values of $a$ and $b$ (in appropriate units) are measured as being 0.14 and $3.2 \times 10 ^ { - 5 }$ respectively.
Find the pressure of the gas as predicted by the improved model. Suppose that the volume of the container is now reduced to $1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 3 }$ while keeping the temperature at 300 K .
By considering the value of the pressure of the gas as predicted by the improved model, comment on the validity of this model in this situation.

OCR Further Mechanics 2018 December Q7

7 Particles $A , B$ and $C$ of masses $2 \mathrm {~kg} , 3 \mathrm {~kg}$ and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with $A$ at the point ( 0,0 ), $B$ at the point ( $0.6,0$ ) and $C$ at the point ( $0.4,0.2$ ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-5_304_666_434_251} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} $G$, which is the centre of mass of the frame, is at the point $( \bar { x } , \bar { y } )$.

- Show that $\bar { x } = 0.38$.
- Find $\bar { y }$.
- Explain why it would be impossible for the frame to be in equilibrium in a horizontal plane supported at only one point.
A rough plane, $\Pi$, is inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 3 } { 5 }$. The frame is placed on $\Pi$ with $A B$ vertical and $B$ in contact with $\Pi$. $C$ is in the same vertical plane as $A B$ and a line of greatest slope of $\Pi . C$ is on the down-slope side of $A B$. The frame is kept in equilibrium by a horizontal light elastic string whose natural length is $l \mathrm {~m}$ and whose modulus of elasticity is $g \mathrm {~N}$. The string is attached to $A$ at one end and to a fixed point on $\Pi$ at the other end (see Fig. 2). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-5_611_842_1649_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The coefficient of friction between $B$ and $\Pi$ is $\mu$.
Show that $l = 0.3$.
Show that $\mu \geqslant \frac { 14 } { 27 }$. \section*{OCR} Oxford Cambridge and RSA

OCR Further Discrete 2018 December Q1

1 Arif and Bindiya play a game as follows.

They each secretly choose a positive integer from $\{ 2,3,4,5 \}$.
They then reveal their choices. Let Arif's choice be $A$ and Bindiya's choice be $B$.
If $A ^ { B } \geqslant B ^ { A }$, Arif wins $B$ points and Bindiya wins $- 4 - B$ points.
If $A ^ { B } < B ^ { A }$, Arif wins $- 4 - A$ points and Bindiya wins $A$ points.
1. Assuming that each of the 16 possible outcomes is equally likely to be chosen, show that the average amount won by Arif is 0 .
2. 1. Describe how to convert this game to a zero-sum game.
  2. Construct the pay-off matrix for this zero-sum game, with Arif on rows.

OCR Further Discrete 2018 December Q2

2 Two simply connected graphs are shown below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{10ca0bf1-beaa-4460-8f28-08f0e4e44d5c-02_307_584_1151_301} \captionsetup{labelformat=empty} \caption{Graph 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{10ca0bf1-beaa-4460-8f28-08f0e4e44d5c-02_307_584_1151_1178} \captionsetup{labelformat=empty} \caption{Graph 2}

\end{figure}

1. Write down the orders of the vertices for each of these graphs.
2. How many ways are there to allocate these vertex degrees to a graph with vertices $\mathrm { P } , \mathrm { Q }$, $\mathrm { R } , \mathrm { S } , \mathrm { T }$ and U ?
3. Use the vertex degrees to deduce whether the graphs are Eulerian, semi-Eulerian or neither.
Show that graphs 1 and 2 are not isomorphic.
1. Write down a Hamiltonian cycle for graph 1.
2. Use Euler's formula to determine the number of regions for graph 1.
3. Identify each of these regions for graph 1 by listing the cycle that forms its boundary.

OCR Further Discrete 2018 December Q3

3 A set of ten cards have the following values:
$\begin{array} { l l l l l l l l l l } 13 & 8 & 4 & 20 & 12 & 15 & 3 & 2 & 10 & 8 \end{array}$ Kerenza wonders if there is a set of these cards with a total of exactly 50 .

Which type of problem (existence, construction, enumeration or optimisation) is this? The five cards $4,8,8,10$ and 20 have a total of 50.
How many ways are there to arrange three of these five cards (with the two 8 s being indistinguishable) so that the total of the numbers on the first two cards is less than the number on the third card?
How many ways are there to select (choose) three of the five cards so that the total of the numbers on the three cards is less than 25 ?
Show how quicksort works by using it to sort the original list of ten cards into increasing order.
You should indicate the pivots used and which values are already known to be in their correct position.

OCR Further Discrete 2018 December Q4

4 An algorithm is represented by the flow diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{10ca0bf1-beaa-4460-8f28-08f0e4e44d5c-04_1871_1719_293_173} The algorithm is applied with $n = 4$ and the table of inputs $\mathrm { d } ( i , j )$ : $$j = 1 \quad j = 2 \quad j = 3 \quad j = 4$$ $$\begin{aligned} & i = 1
& i = 2
& i = 3
& i = 4 \end{aligned}$$

0	3	5	2
3	0	4	3
5	4	0	1
2	3	1	0

An incomplete trace through the algorithm is shown below.

$n$	$i$	$j$	$\mathrm { d } ( i , j )$	A	$t$	$m$
4
					1
	1
				1		100
		1
			0
		2
			3
					2	3
		3
			5
		4
			2
					4	2
	4
				1, 4		100
		1
			2
		2
			3
					2	3
		3
			1
					3	1
		4
			0

The next box to be used is the box 'Let $i = t$ '.

Complete the trace in the Printed Answer Booklet. The table of inputs represents a weighted matrix for a network, where the weights represent distances.
1. State how the output of the algorithm relates to the network represented by the matrix.
2. How can the list A be used in the solution of the travelling salesperson problem on the network represented by the matrix?
3. Write down a limitation on the distances $\mathrm { d } ( i , j )$ for this algorithm.
Explain why the algorithm is finite for any table of inputs. Suppose that, for a problem with $n$ vertices, the run-time for the algorithm is given by $\alpha D + \beta T$, where $\alpha$ and $\beta$ are constants, $D$ is the number of times that a value of $\mathrm { d } ( i , j )$ is looked up and $T$ is the number of times that $t$ is updated.
Show how this means that the algorithm has $\mathrm { O } \left( n ^ { 2 } \right)$ complexity. A computer takes 3 seconds to run the algorithm for a problem with $n = 35$.
Use the complexity to calculate an approximate run-time for a problem with $n = 100$. The run-time using a second algorithm has $\mathrm { O } ( n ! )$ complexity.
A computer takes 2.8 seconds to run the second algorithm for a problem with $n = 35$.
Without performing any further calculations, give a reason why the first algorithm is likely to be more efficient than the second for a problem with $n = 100$.

OCR Further Discrete 2018 December Q5

5 A rapid transport system connects 8 stations using three railway lines.
The blue line connects A to B to C to D .

From	to	Travel time
A	B	5
B	C	3
C	D	9

The red line connects $B$ to $F$ to $E$ to $D$.

From	to	Travel time
B	F	2
F	E	3
E	D	2

The green line connects E to G to H to A .

From	to	Travel time
E	G	5
G	H	6
H	A	4

The travel times for the return journeys are the same as for the outward journeys (so, for example, the travel time from B to A is 5 minutes, the same as the time from A to B ).
All travel times include time spent stopped at stations.
No trains are delayed so the travel times are all correct.
1. (i) Model the blue, red and green lines, and the travel times above, as a network.
  (ii) Use Dijkstra's algorithm to find the quickest travel times from C to each of the other stations.
2. 1. Write down a route from A to D with travel time 12 minutes.
  2. Construct a table of quickest travel times.
3. Give a reason why the quickest journey from A to D may take longer than 12 minutes.

OCR Further Discrete 2018 December Q6

6 Jack is making pizzas for a party. He can make three types of pizza:

	Suitable for vegans	Suitable for vegetarians	Suitable for meat eaters
Type X			✓
Type Y		✓	✓
Type Z	✓	✓	✓

There is enough dough to make 30 pizzas.
Type Z pizzas use vegan cheese. Jack only has enough vegan cheese to make 2 type Z pizzas.
At least half the pizzas made must be suitable for vegetarians.
Jack has enough onions to make 50 type X pizzas or 20 type Y pizzas or 20 type Z pizzas or some mixture of the three types.

Suppose that Jack makes $x$ type X pizzas, $y$ type Y pizzas and $z$ type Z pizzas.

Formulate the constraints above in terms of the non-negative, integer valued variables $x , y$ and $z$, together with non-negative slack variables $s , t , u$ and $v$. Jack wants to find out the maximum total number of pizzas that he can make.

Set up an initial simplex tableau for Jack's problem.

Carry out one iteration of the simplex algorithm, choosing your pivot so that $x$ becomes a basic variable. When Jack carries out the simplex algorithm his final tableau is:

$P$	$x$	$y$	$z$	$s$	$t$	$u$	$v$	RHS
1	0	0	0	0	0	$\frac { 3 } { 7 }$	$\frac { 2 } { 7 }$	$28 \frac { 4 } { 7 }$
0	0	0	0	1	0	$- \frac { 3 } { 7 }$	$- \frac { 2 } { 7 }$	$1 \frac { 3 } { 7 }$
0	0	0	1	0	1	0	0	2
0	1	0	0	0	0	$\frac { 5 } { 7 }$	$\frac { 1 } { 7 }$	$14 \frac { 2 } { 7 }$
0	0	1	1	0	0	$- \frac { 2 } { 7 }$	$\frac { 1 } { 7 }$	$14 \frac { 2 } { 7 }$

Use this final tableau to deduce how many pizzas of each type Jack should make. Jack knows that some of the guests are vegans. He decides to make 2 pizzas of type $Z$.
1. Plot the feasible region for $x$ and $y$.
2. Complete the branch-and-bound formulation in the Printed Answer Booklet to find the number of pizzas of each type that Jack should make.
  You should branch on $x$ first. \section*{END OF QUESTION PAPER}

OCR Further Additional Pure 2018 December Q1

1 A surface has equation $z = x \tan y$ for $- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi$.

Find
- $\frac { \partial z } { \partial x }$,
- $\frac { \partial z } { \partial y }$.
- Find in cartesian form, the equation of the tangent plane to the surface at the point where $x = 1$ and $y = \frac { 1 } { 4 } \pi$.

OCR Further Additional Pure 2018 December Q2

2 A sequence $\left\{ u _ { n } \right\}$ is given by $u _ { n + 1 } = 4 u _ { n } + 1$ for $n \geqslant 1$ and $u _ { 1 } = 3$.

Find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Solve the recurrence system (*).
1. Prove by induction that each term of the sequence can be written in the form $( 10 m + 3 )$ where $m$ is an integer.
2. Show that no term of the sequence is a square number.

OCR Further Additional Pure 2018 December Q3

Show that $10 ^ { 2 } \equiv 6 ( \bmod 47 )$.
Determine the integer $r$, with $0 < r < 47$, such that $6 r \equiv 1 ( \bmod 47 )$.
Determine the least positive integer $n$ for which $10 ^ { n } \equiv 1$ or $- 1 ( \bmod 47 )$.

OCR Further Additional Pure 2018 December Q4

4 The set $L$ consists of all points $( x , y )$ in the cartesian plane, with $x \neq 0$. The operation ◇ is defined by $( a , b ) \diamond ( c , d ) = ( a c , b + a d )$ for $( a , b ) , ( c , d ) \in L$.

1. Show that $L$ is closed under ◇.
2. Prove that $\diamond$ is associative on $L$.
3. Find the identity element of $L$ under ◇ .
4. Find the inverse element of $( a , b )$ under ◇.
Find a subgroup of $( L , \diamond )$ of order 2.

OCR Further Additional Pure 2018 December Q5

5 Torque is a vector quantity that measures how much a force acting on an object causes that object to rotate. The torque (about the origin), $\mathbf { T }$, exerted on an object is given by $\mathbf { T } = \mathbf { p } \times \mathbf { F }$, where $\mathbf { F }$ is the force acting on the object and $\mathbf { p }$ is the position vector of the point at which $\mathbf { F }$ is applied to the object. The points $A$ and $B$, with position vectors $\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$ and $\mathbf { b } = 3 \mathbf { i } + 5 \mathbf { j } + \mathbf { k }$ are on the surface of a rock. The force $\mathbf { F } _ { 1 } = 6 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }$ is applied to the rock at $A$ while the force $\mathbf { F } _ { 2 } = - 7 \mathbf { i } - 10 \mathbf { j } + 2 \mathbf { k }$ is applied to the rock at $B$.

Find the torque (about the origin) exerted on the rock by $\mathbf { F } _ { 1 }$.
Determine which of the two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ exerts a torque (about the origin) of greater magnitude on the rock.
Show that the torque (about the origin) is the same as your answer to part (a) when $\mathbf { F } _ { 1 }$ acts on the rock at any point on the line $\mathbf { r } = \mathbf { a } + \lambda \mathbf { p }$, where $\mathbf { p }$ is a vector in the same direction as $\mathbf { F } _ { 1 }$. A third force $\mathbf { F } _ { 3 }$ is now applied to the rock at $A$, which exerts zero torque (about the origin).
Show that $\mathbf { F } _ { 3 }$ must act in the direction of the line through $A$ and the origin.

OCR Further Additional Pure 2018 December Q6

6 For positive integers $n$, the integrals $I _ { n }$ are given by $I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } \sqrt { 2 + x ^ { 2 } } \mathrm {~d} x$.

Show that $I _ { 1 } = 26 \sqrt { 3 }$.
Prove that, for $n \geqslant 3 , ( n + 2 ) I _ { n } = 3 \sqrt { 3 } \left( 27 \times 5 ^ { n - 1 } - 1 \right) - 2 ( n - 1 ) I _ { n - 2 }$.
Determine the exact value of $I _ { 5 }$ as a rational multiple of $\sqrt { 3 }$.

OCR Further Additional Pure 2018 December Q7

7 For each value of $t$, the surface $S _ { t }$ has equation $z = t x ^ { 2 } + y ^ { 2 } + 3 x y - y$.

Verify that there are no stationary points on $S _ { t }$ when $t = \frac { 9 } { 4 }$.
Determine, as $t$ varies, the nature of any stationary point(s) of $S _ { t }$.
(You do not have to find the coordinates of the stationary points.) \section*{OCR} Oxford Cambridge and RSA

OCR Pure 1 2018 March Q1

1 A circle with equation $x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = k$ has a radius of 4 .

Find the coordinates of the centre of the circle.
Find the value of the constant $k$.

OCR Pure 1 2018 March Q2

Given that $| n | = 5$, find the greatest value of $| 2 n - 3 |$, justifying your answer.
Solve the equation $| 3 x - 6 | = | x - 6 |$.

OCR Pure 1 2018 March Q3

3 The equation $k x ^ { 2 } + ( k - 6 ) x + 2 = 0$ has two distinct real roots. Find the set of possible values of the constant $k$, giving your answer in set notation.

\(n\)	\(i\)	\(j\)	\(\mathrm { d } ( i , j )\)	A	\(t\)	\(m\)
4
					1
	1
				1		100
		1
			0
		2
			3
					2	3
		3
			5
		4
			2
					4	2
	4
				1, 4		100
		1
			2
		2
			3
					2	3
		3
			1
					3	1
		4
			0

\(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
1	0	0	0	0	0	\(\frac { 3 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(28 \frac { 4 } { 7 }\)
0	0	0	0	1	0	\(- \frac { 3 } { 7 }\)	\(- \frac { 2 } { 7 }\)	\(1 \frac { 3 } { 7 }\)
0	0	0	1	0	1	0	0	2
0	1	0	0	0	0	\(\frac { 5 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)
0	0	1	1	0	0	\(- \frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)

Questions — OCR (4619 questions)

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OCR Further Statistics 2018 December Q7

OCR Further Statistics 2018 December Q8

OCR Further Mechanics 2018 December Q1

OCR Further Mechanics 2018 December Q2

OCR Further Mechanics 2018 December Q3

OCR Further Mechanics 2018 December Q4

OCR Further Mechanics 2018 December Q5

OCR Further Mechanics 2018 December Q6

OCR Further Mechanics 2018 December Q7

OCR Further Discrete 2018 December Q1

OCR Further Discrete 2018 December Q2

OCR Further Discrete 2018 December Q3

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OCR Further Discrete 2018 December Q5

OCR Further Discrete 2018 December Q6

OCR Further Additional Pure 2018 December Q1

OCR Further Additional Pure 2018 December Q2

OCR Further Additional Pure 2018 December Q3

OCR Further Additional Pure 2018 December Q4

OCR Further Additional Pure 2018 December Q5

OCR Further Additional Pure 2018 December Q6

OCR Further Additional Pure 2018 December Q7

OCR Pure 1 2018 March Q1

OCR Pure 1 2018 March Q2

OCR Pure 1 2018 March Q3