Questions - OCR - A-Level Maths

By considering the acceleration of $P$, show that $T = 31.5 L$.
Write down another relationship between $T$ and $L$.
Find the value of $T$ and the value of $L$.
Find the angle that the string makes with the downwards vertical through $O$.

OCR Further Mechanics 2018 March Q4

Challenging +1.2

4 A ball $B$ of mass 1.7 kg is connected to one end of a light elastic spring of natural length 1.2 m . The other end of the spring is attached to a point $O$ on the ceiling of a large room. The modulus of elasticity of the spring is 50 N . The ball is held 3.2 m vertically below $O$ and projected upwards with an initial speed of $0.5 \mathrm {~ms} ^ { - 1 }$. In order to model the motion of $B$ (before any collision with the ceiling) the following assumptions are made.

Air resistance is ignored.
$B$ is small.
The fully compressed length of the spring is negligible.
1. Determine whether, according to the model, $B$ reaches $O$.
2. Without doing any further calculations, explain whether the answer to part (i) could change in each of the following different cases.
  (a) A new model is used in which air resistance is taken into account.
  (b) The spring is replaced by an elastic string with the same natural length and modulus of elasticity.
  (c) $\quad B$ is initially projected downwards rather than upwards.

OCR Further Mechanics 2018 March Q5

Standard +0.3

5 A simple pendulum consists of a small sphere of mass $m$ connected to one end of a light rod of length $h$. The other end of the rod is freely hinged at a fixed point. When the sphere is pulled a short distance to one side and released from rest the pendulum performs oscillations. The time taken to perform one complete oscillation is called the period and is denoted by $P$.

Assuming that $P = k m ^ { \alpha } h ^ { \beta } g ^ { \gamma }$, where $g$ is the acceleration due to gravity and $k$ is a dimensionless constant, find the values of $\alpha , \beta$ and $\gamma$. A student conducts an experiment to investigate how $P$ varies as $h$ varies. She measures the value of $P$ for various values of $h$, ensuring that all other conditions remain constant. Her results are summarised in the table below.
$h ( \mathrm {~m} )$ 0.40 2.50 3.60
$P ( \mathrm {~s} )$ 1.27 2.17 3.81
Show that these results are not consistent with the answers to part (i).
The student later realises that she has recorded one of her values of $P$ incorrectly.
- Identify the incorrect value.
- Estimate the correct value that she should have recorded.

OCR Further Mechanics 2018 March Q6

Standard +0.8

6 A particle $P$ of mass 2.5 kg strikes a rough horizontal plane. Immediately before $P$ strikes the plane it has a speed of $6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its direction of motion makes an angle of $30 ^ { \circ }$ with the normal to the plane at the point of impact. The impact may be assumed to occur instantaneously. The coefficient of restitution between $P$ and the plane is $\frac { 2 } { 3 }$. The friction causes a horizontal impulse of magnitude 2 Ns to be applied to $P$ in the plane in which it is moving.

Calculate the velocity of $P$ immediately after the impact with the plane.
$\quad P$ loses about $x \%$ of its kinetic energy as a result of the impact. Find the value of $x$.

OCR Further Mechanics 2018 March Q7

5 marks Challenging +1.2

7 A smooth track $A B$ is in the shape of an arc of a circle with centre $O$ and radius 1.4 m . The track is fixed in a vertical plane with $A$ above the level of $B$ and a point $C$ on the track vertically below $O$. Angle $A O C$ is $60 ^ { \circ }$ and angle $C O B$ is $30 ^ { \circ }$. Point $C$ is 2.5 m vertically above the point $F$, which lies in a horizontal plane. A particle of mass 0.4 kg is placed at $A$ and projected down the track with an initial velocity of $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The particle first hits the plane at point $H$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_767_1265_488_415}

Find the magnitude of the contact force between the particle and the track when the particle is at $B$. [5]
Find the distance $F H$.

OCR Further Mechanics 2018 March Q8

Challenging +1.2

8 A piston of mass 1.5 kg moves in a straight line inside a long straight horizontal cylinder. At time $t \mathrm {~s}$ the displacement of the piston from its initial position at one end of the cylinder is $x \mathrm {~m}$ and its velocity is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_168_805_1726_630} The piston starts moving when $t = 2$ and is brought to rest when it reaches the other end of the cylinder. While the piston is in motion it is acted on by a force of magnitude $\frac { 6 } { t ^ { 2 } } \mathrm {~N}$ in the positive $x$ direction, and also by a force of magnitude $\frac { 3 v } { t } \mathrm {~N}$ resisting the motion.

Show that, while the piston is in motion, $\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 2 v } { t } = \frac { 4 } { t ^ { 2 } }$. The piston reaches the other end of the cylinder when $t = 20$.
Find the speed of the piston immediately before it is brought to rest.
Show that the piston travels a distance of 5.61 m , correct to 3 significant figures. \section*{OCR} \section*{Oxford Cambridge and RSA}

OCR Further Discrete 2018 March Q1

Moderate -0.5

1 The masses, in kg , of ten bags are given below. $$\begin{array} { l l l l l l l l l l } 8 & 10 & 10 & 12 & 12 & 12 & 13 & 15 & 18 & 18 \end{array}$$

Use first-fit decreasing to pack the bags into crates that can hold a maximum of 50 kg each. Only two crates are available, so only some of the bags will be packed. Each bag is given a value.
Bag A B C D E F G H I J
Mass (kg) 8 10 10 12 12 12 13 15 18 18
Value 6 3 3 2 4 5 4 6 4 4
Find a packing into two crates so that the total value of the bags in the crates is at least 32 .

OCR Further Discrete 2018 March Q2

Challenging +1.2

2 A linear programming problem is $$\begin{array} { l l } \text { Maximise } & P = 4 x - y - 2 z \\ \text { subject to } & x + 5 y + 3 z \leqslant 60 \\ & 2 x - 5 y \leqslant 80 \\ & 2 y + z \leqslant 10 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$

Use the simplex algorithm to solve the problem. In the case when $z = 0$ the feasible region can be represented graphically.
\includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-2_636_1619_1564_230} The vertices of the feasible region are $( 0,0 ) , ( 40,0 ) , ( 46.67,2.67 ) , ( 35,5 )$ and $( 0,5 )$, where non-integer values are given to 2 decimal places. The linear programming problem is given the additional constraint that $x$ and $y$ are integers.
Use branch-and-bound, branching on $x$ first, to show that the optimum solution with this additional constraint is $x = 45 , y = 2$. 350 people are at a TV game show. 21 of the 50 are there to take part in the game show and the others are friends who are in the audience, 22 are women and 20 are from London, 2 are women from London who are there to take part in the game show and 15 are men who are not from London and are friends who are in the audience.
Deduce how many of the 50 people are in two of the categories 'there to take part in the game show', 'is a woman' and 'is from London', but are not in all three categories. The 21 people who are there to take part in the game show are moved to the stage where they are seated in two rows of seats with 20 seats in each row. Some of the seats are empty.
Show how the pigeonhole principle can be used to show that there must be at least one pair of these 21 people with no empty chair between them. The 21 people are split into three sets of 7 . In each round of the game show, three of the people are chosen. The three people must all be from the same set of 7 but once two people have played in the same round they cannot play together in another round. For example, if A plays with B and C in round 1 then A cannot play with B or with C in any other round.
By first considering how many different rounds can be formed using the first set of seven people, deduce how many rounds there can be altogether.

OCR Further Discrete 2018 March Q4

Challenging +1.8

4 The graph below connects nine vertices A, B, ..., H, I.
\includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-3_543_693_1347_680}

(a) Show that the minimum sum of the degrees of each pair of non-adjacent vertices is 9 .
(b) Explain what you can deduce from the result in part (a).
Use Kuratowski's theorem to prove that the graph is non-planar.
Prove that there is no subgraph of the graph that is isomorphic to $\mathrm { K } _ { 4 }$, without using subdivision or contraction.

OCR Further Discrete 2018 March Q5

Moderate -0.5

5 The diagram represents a map of seven locations (A to G) and the direct road distances (km) between some of them.
\includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-4_382_771_356_644} A delivery driver needs to start from his depot at D , make deliveries at each of $\mathrm { A } , \mathrm { B } , \mathrm { F }$ and G , and finish at D .

Write down a route from A to G of length 70 km . The table shows the length of the shortest path between some pairs of places.
D A B F G
D -
A - 70
B - 84
F 84 -
G 70 -
(a) Complete the table.
(b) Use the nearest neighbour method on the table, starting at D , to find the length of a cycle through $\mathrm { D } , \mathrm { A } , \mathrm { B } , \mathrm { F }$ and G , ignoring possibly repeating E and C .
By first considering the table with the row and column for D removed, find a lower bound for the distance that the driver must travel.
What can you conclude from your previous answers about the distance that the driver must travel? A new road is constructed between D and F . Using this road the driver starts from D , makes the deliveries and returns to D having travelled just 172 km .
Find the length of the new road if
(a) the driver does not return to D until all the deliveries have been made,
(b) the driver uses the new road twice in making the deliveries.

OCR Further Discrete 2018 March Q6

Standard +0.8

6 The activities involved in a project, their durations, immediate predecessors and the number of workers required for each activity are shown in the table.

Activity	Duration (hours)	Immediate predecessors	Number of workers
A	6	-	2
B	4	-	1
C	4	-	1
D	2	A	2
E	3	A, B	1
F	4	C	1
G	3	D	1
H	3	E, F	2

Model the project using an activity network.
- Calculate the early and late event times.
- Calculate the independent and interfering float for each activity.
- Draw a cascade chart for the project, showing each activity starting at its earliest possible start time.
- Construct a schedule to show how three workers can complete the project in the minimum possible time.

OCR Further Discrete 2018 March Q7

Challenging +1.8

7 Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form ( $a , b$ ), where $a =$ points won by Alix and $b =$ points won be Ben. Each is trying to maximise the points they win. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Ben}

		Card X	Card Y	Card Z
	Card P	$( 4,4 )$	$( 5,9 )$	$( 1,7 )$
\multirow[t]{2}{*}{Alix}	Card Q	$( 3,5 )$	$( 4,1 )$	$( 8,2 )$
	Card R	$( x , y )$	$( 2,2 )$	$( 9,4 )$

\end{table}

Explain why the table cannot be reduced through dominance no matter what values $x$ and $y$ have.
Show that the game is not stable no matter what values $x$ and $y$ have.
Find the Nash equilibrium solutions for the various values that $x$ and $y$ can have. \section*{OCR} \section*{Oxford Cambridge and RSA}

OCR Further Additional Pure 2018 March Q1

Standard +0.3

1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$

OCR Further Additional Pure 2018 March Q2

Standard +0.8

2 Four points $A , B , C$ and $D$ have coordinates $( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )$ and $( 0,3,7 )$ respectively. Find the volume of tetrahedron $A B C D$.

OCR Further Additional Pure 2018 March Q3

Challenging +1.2

3 The surface $S$ has equation $z = \frac { x } { y } \sin y + \frac { y } { x } \cos x$ where $0 < x \leqslant \pi$ and $0 < y \leqslant \pi$.

Find
- $\frac { \partial z } { \partial x }$,
- $\frac { \partial z } { \partial y }$.
- Determine the equation of the tangent plane to $S$ at the point $A$ where $x = y = \frac { 1 } { 4 } \pi$. Give your answer in the form $a x + b y + c z = d$ where $a , b , c$ and $d$ are exact constants.
- Write down a normal vector to $S$ at $A$.

OCR Further Additional Pure 2018 March Q4

Hard +2.3

(a) Find all the quadratic residues modulo 11.
(b) Prove that the equation $y ^ { 5 } = x ^ { 2 } + 2017$ has no solution in integers $x$ and $y$.
In this question you must show detailed reasoning. The numbers $M$ and $N$ are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that $M$ is divisible by $N$.

OCR Further Additional Pure 2018 March Q5

Challenging +1.8

(a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that $X _ { 0 } = 12$ and $X _ { 1 } = 1$.
(b) Show that the sequence $\left\{ X _ { n } \right\}$ approaches a geometric sequence as $n$ increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that $X _ { n }$ represents the profit (in $\pounds$ ) at the end of year $n$.
(a) Determine the number of years taken for the projected profit to exceed one million pounds.
(b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
(a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
(b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
(c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next $X _ { n }$ in the process.

OCR Further Additional Pure 2018 March Q6

Challenging +1.8

6 In this question you must show detailed reasoning. It is given that $I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t$ for integers $n \geqslant 0$.

Show that $I _ { 1 } = \frac { 7 } { 3 }$.
Prove that, for $n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }$. The curve $C$ is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve $C$ is rotated through $2 \pi$ radians about the $x$-axis, a surface of revolution is formed with surface area $A$.
Determine
- the values of the integers $k$ and $m$ such that $A = k \pi I _ { m }$,
- the exact value of $A$.

OCR Further Additional Pure 2018 March Q7

Challenging +1.8

7 The set $M$ contains all matrices of the form $\mathbf { X } ^ { n }$, where $\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)$ and $n$ is a positive integer.

Show that $M$ contains exactly 12 elements.
Deduce that $M$, together with the operation of matrix multiplication, form a cyclic group $G$.
Determine all the proper subgroups of $G$. \section*{END OF QUESTION PAPER}

OCR FP1 AS 2018 March Q1

The complex number 3-4i is denoted by $z _ { 1 }$. Write $z _ { 1 }$ in modulus-argument form, giving your angle in radians to 3 significant figures.
The complex number $z _ { 2 }$ has modulus 6 and argument - 2.5 radians. Express $z _ { 1 } z _ { 2 }$ in modulus-argument form with the angle in radians correct to 3 significant figures.

OCR FP1 AS 2018 March Q2

Moderate -0.5

2 In this question you must show detailed reasoning.
The quadratic equation $3 x ^ { 2 } - 7 x + 5 = 0$ has roots $\alpha$ and $\beta$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Hence find the values of the following expressions.
(a) $\frac { 1 } { \alpha } + \frac { 1 } { \beta }$
(b) $\alpha ^ { 2 } + \beta ^ { 2 }$
$3 \quad l _ { 1 }$ and $l _ { 2 }$ are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }

OCR FP1 AS 2018 March Q3

Standard +0.3

3
3
- 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1
3
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1
a
1 \end{array} \right) + \mu \left( \begin{array} { c } 2
2
- 3 \end{array} \right) \end{aligned}$$

Find the position vector of the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.
Determine the value of $a$.

OCR FP1 AS 2018 March Q4

Standard +0.8

4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which $| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }$ and $\operatorname { Re } ( z ) \geqslant 9$.

OCR FP1 AS 2018 March Q5

5 The matrix $\mathbf { A }$ is given by $\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)$ and the matrix $\mathbf { B }$ is given by $\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)$.

$\mathbf { A }$ represents a reflection. Write down the value of $\operatorname { det } \mathbf { A }$.
Hence find the possible values of $a$.
$\mathbf { r }$ is the position vector of a point $R$. Given that $\mathbf { A r } = \mathbf { r }$ describe the location of $R$.
$\mathbf { B }$ represents a rotation. Write down the value of $\operatorname { det } \mathbf { B }$.
Hence find the possible values of $b$.

OCR FP1 AS 2018 March Q6

Standard +0.3

6 The matrix $\mathbf { A }$ is given by $\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)$ and the matrix $\mathbf { B }$ is given by $\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)$.

Find the matrix $\mathbf { A B }$.
State the conditions on $a$ and $b$ for $\mathbf { A B }$ to be a singular matrix.
$P Q R S$ is a quadrilateral and the vertices $P , Q , R$ and $S$ are in clockwise order. A transformation, T , is represented by the matrix $\mathbf { A B }$.
State the effect on both the area and also the orientation of the image of $P Q R S$ under T in each of the following cases.
(a) $\quad a = 1$ and $b = 1$
(b) $\quad a = 2$ and $b = 3$

		Card X	Card Y	Card Z
	Card P	\(( 4,4 )\)	\(( 5,9 )\)	\(( 1,7 )\)
\multirow[t]{2}{*}{Alix}	Card Q	\(( 3,5 )\)	\(( 4,1 )\)	\(( 8,2 )\)
	Card R	\(( x , y )\)	\(( 2,2 )\)	\(( 9,4 )\)

\(h ( \mathrm {~m} )\)	0.40	2.50	3.60
\(P ( \mathrm {~s} )\)	1.27	2.17	3.81

Bag	A	B	C	D	E	F	G	H	I	J
Mass (kg)	8	10	10	12	12	12	13	15	18	18
Value	6	3	3	2	4	5	4	6	4	4

Questions — OCR (4619 questions)

Browse by module

Browse by board

OCR Further Mechanics 2018 March Q3

OCR Further Mechanics 2018 March Q4

OCR Further Mechanics 2018 March Q5

OCR Further Mechanics 2018 March Q6

OCR Further Mechanics 2018 March Q7

OCR Further Mechanics 2018 March Q8

OCR Further Discrete 2018 March Q1

OCR Further Discrete 2018 March Q2

OCR Further Discrete 2018 March Q4

OCR Further Discrete 2018 March Q5

OCR Further Discrete 2018 March Q6

OCR Further Discrete 2018 March Q7

OCR Further Additional Pure 2018 March Q1

OCR Further Additional Pure 2018 March Q2

OCR Further Additional Pure 2018 March Q3

OCR Further Additional Pure 2018 March Q4

OCR Further Additional Pure 2018 March Q5

OCR Further Additional Pure 2018 March Q6

OCR Further Additional Pure 2018 March Q7

OCR FP1 AS 2018 March Q1

OCR FP1 AS 2018 March Q2

OCR FP1 AS 2018 March Q3

OCR FP1 AS 2018 March Q4

OCR FP1 AS 2018 March Q5

OCR FP1 AS 2018 March Q6