Questions - OCR - A-Level Maths

Stage	State	A ction	Working	M inimax
\multirow{3}{*}{1}	0	0	4	4
	1	0	3	3
	2	0	2	2
\multirow{9}{*}{2}	\multirow{3}{*}{0}	0	$\max ( 6,4 ) = 6$	\multirow{3}{*}{3}
		1	$\max ( 2,3 ) = 3$
		2	$\max ( 3,2 ) = 3$
	\multirow{3}{*}{1}	0	$\max ( 2,4 ) =$	\multirow{3}{*}{}
		1	$\max ( 4,3 ) =$
		2	$\max ( 5,2 ) =$
	\multirow{3}{*}{2}	0	max(2,	\multirow{3}{*}{}
		1	max(3,
		2	max(4,
\multirow{3}{*}{3}	\multirow{3}{*}{0}	0	max(5,	\multirow{3}{*}{}
		1	max(5,
		2	max(2,

On the insert, complete the last two columns of the table.
State the minimax value and write down the minimax route.
Complete the diagram on the insert to show the network that is represented by the table.

OCR D2 2007 June Q5

13 marks Moderate -0.5

5 Answer this question on the insert provided. The network represents a system of pipes through which fluid can flow from a source, S , to a sink, T .
\includegraphics[max width=\textwidth, alt={}, center]{09d4aacd-026b-4d81-a826-3d3f29f9c105-5_1310_1301_447_424} The arrows are labelled to show excess capacities and potential backflows (how much more and how much less could flow in each pipe). The excess capacities and potential backflows are measured in litres per second. Currently the flow is 6 litres per second, all flowing along a single route through the system.

Write down the route of the 6 litres per second that is flowing from $S$ to $T$.
What is the capacity of the pipe AG and in which direction can fluid flow along this pipe?
Calculate the capacity of the $\operatorname { cut } \mathrm { X } = \{ \mathrm { S } , \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \} , \mathrm { Y } = \{ \mathrm { F } , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { T } \}$.
Describe how a further 7 litres per second can flow from S to T and update the labels on the arrows to show your flow. Explain how you know that this is the maximum flow. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR Further Pure Core 2 2023 June Q4

4 marks Challenging +1.2

4 In this question you must show detailed reasoning. The region $R$ is bounded by the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }$, the $x$-axis and the lines with equations $x = \frac { 1 } { 2 }$ and $x = 1$ (see diagram). The units of the axes are cm .
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when $R$ is rotated by $2 \pi$ radians about the $x$-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

OCR Further Pure Core 2 2023 June Q5

7 marks Challenging +1.2

5 In this question you must show detailed reasoning.

Using the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, show that $\sinh 2 x \equiv 2 \sinh x \cosh x$.
Solve the equation $15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20$, giving all your answers in logarithmic form.

OCR Further Pure Core 2 2023 June Q10

7 marks Challenging +1.2

10 In this question you must show detailed reasoning. A region, $R$, of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. $R$ is modelled as being the region enclosed by two curves, $C _ { 1 }$ and $C _ { 2 }$. The polar equations of $C _ { 1 }$ and $C _ { 2 }$ are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \\ C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with $R$ indicated.
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint $R$. Each tin of paint can cover an area of $0.5 \mathrm {~m} ^ { 2 }$.
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint $R$ completely.

OCR Further Pure Core 2 2020 November Q8

9 marks Standard +0.8

8 In this question you must show detailed reasoning. The complex number $- 4 + i \sqrt { 48 }$ is denoted by $z$.

Determine the cube roots of $z$, giving the roots in exponential form. The points which represent the cube roots of $z$ are denoted by $A , B$ and $C$ and these form a triangle in an Argand diagram.
Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.

OCR M3 2006 January Q10

Standard +0.8

10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes

Write your name, centre number and candidate number in the spaces provided on the answer booklet.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
The acceleration due to gravity is denoted by $\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }$. Unless otherwise instructed, when a numerical value is needed, use $g = 9.8$.
You are permitted to use a graphical calculator in this paper.

The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 72.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
You are reminded of the need for clear presentation in your answers.

OCR FS1 AS 2021 June Q1

8 marks Moderate -0.8

1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the $X$ th weekday of September.

State an assumption needed for $X$ to be well modelled by a geometric distribution.
Find $\mathrm { P } ( X = 11 )$.
Find $\mathrm { P } ( X \leqslant 8 )$.
Find $\operatorname { Var } ( X )$.
Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.

OCR FS1 AS 2021 June Q2

8 marks Standard +0.3

2 The probability distribution for the discrete random variable $W$ is given in the table.

$w$	1	2	3	4
$\mathrm { P } ( W = w )$	0.25	0.36	$x$	$x ^ { 2 }$

Show that $\operatorname { Var } ( W ) = 0.8571$.
Find $\operatorname { Var } ( 3 W + 6 )$. In this question you must show detailed reasoning.
The random variable $T$ has a binomial distribution. It is known that $\mathrm { E } ( T ) = 5.625$ and the standard deviation of $T$ is 1.875 . Find the values of the parameters of the distribution.

OCR FS1 AS 2021 June Q4

9 marks Standard +0.8

4 The table shows the results of a random sample drawn from a population which is thought to have the distribution $\mathrm { U } ( 20 )$. \end{table}

OCR FP1 AS 2017 December Q1

4 marks Moderate -0.3

1 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c c } - 3 & 3 & 2 \\ 5 & - 4 & - 3 \\ - 1 & 1 & 1 \end{array} \right)$.

Find $\mathbf { A } ^ { - 1 }$.
Solve the simultaneous equations $$\begin{aligned} - 3 x + 3 y + 2 z & = 12 a \\ 5 x - 4 y - 3 z & = - 6 \\ - x + y + z & = 7 \end{aligned}$$ giving your solution in terms of $a$.

OCR FP1 AS 2017 December Q2

9 marks Standard +0.3

2 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z - ( 3 + 2 \mathrm { i } ) | = 2$ and $\arg ( z - ( 3 + 2 \mathrm { i } ) ) = \frac { 5 \pi } { 6 }$ respectively.

Sketch $C _ { 1 }$ and $C _ { 2 }$ on a single Argand diagram.
Find, in surd form, the number represented by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.
Indicate, by shading, the region of the Argand diagram for which $$| z - ( 3 + 2 i ) | \leqslant 2 \text { and } \frac { 5 \pi } { 6 } \leqslant \arg ( z - ( 3 + 2 i ) ) \leqslant \pi$$

OCR FP1 AS 2017 December Q3

8 marks Standard +0.3

3 Two lines, $l _ { 1 }$ and $l _ { 2 }$, have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2

OCR FP1 AS 2017 December Q5

7 marks Standard +0.8

5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $l _ { 1 }$ and $l _ { 2 }$.
$Q$ is a point on $l _ { 1 }$ which is 12 metres away from $P . R$ is the point on $l _ { 2 }$ such that $Q R$ is perpendicular to $l _ { 1 }$.
Determine the length $Q R$. 4 In this question you must show detailed reasoning.
The distinct numbers $\omega _ { 1 }$ and $\omega _ { 2 }$ both satisfy the quadratic equation $4 x ^ { 2 } + 4 x + 17 = 0$.
Write down the value of $\omega _ { 1 } \omega _ { 2 }$.
$A , B$ and $C$ are the points on an Argand diagram which represent $\omega _ { 1 } , \omega _ { 2 }$ and $\omega _ { 1 } \omega _ { 2 }$. Find the area of triangle $A B C$. 5 In this question you must show detailed reasoning.
The equation $x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Using the identity $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma$ find the value of $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.
Given that $\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112$ find a cubic equation whose roots are $\alpha ^ { 3 } , \beta ^ { 3 }$ and $\gamma ^ { 3 }$.

OCR FP1 AS 2017 December Q6

5 marks

6 Prove by induction that $n ! \geqslant 6 n$ for $n \geqslant 4$.

OCR FP1 AS 2017 December Q7

7 marks Standard +0.8

7 A transformation is equivalent to a shear parallel to the $x$-axis followed by a shear parallel to the $y$-axis and is represented by the matrix $\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)$. Find in terms of $s$ the matrices which represent each of the shears.

OCR FP1 AS 2017 December Q10

Standard +0.3

10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $l _ { 1 }$ and $l _ { 2 }$.
$Q$ is a point on $l _ { 1 }$ which is 12 metres away from $P . R$ is the point on $l _ { 2 }$ such that $Q R$ is perpendicular to $l _ { 1 }$.
Determine the length $Q R$. 4 In this question you must show detailed reasoning.
The distinct numbers $\omega _ { 1 }$ and $\omega _ { 2 }$ both satisfy the quadratic equation $4 x ^ { 2 } + 4 x + 17 = 0$.
Write down the value of $\omega _ { 1 } \omega _ { 2 }$.
$A , B$ and $C$ are the points on an Argand diagram which represent $\omega _ { 1 } , \omega _ { 2 }$ and $\omega _ { 1 } \omega _ { 2 }$. Find the area of triangle $A B C$. 5 In this question you must show detailed reasoning.
The equation $x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Using the identity $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma$ find the value of $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.
Given that $\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112$ find a cubic equation whose roots are $\alpha ^ { 3 } , \beta ^ { 3 }$ and $\gamma ^ { 3 }$. 6 Prove by induction that $n ! \geqslant 6 n$ for $n \geqslant 4$. 7 A transformation is equivalent to a shear parallel to the $x$-axis followed by a shear parallel to the $y$-axis and is represented by the matrix $\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)$. Find in terms of $s$ the matrices which represent each of the shears. 8
(a) Find, in terms of $x$, a vector which is perpendicular to the vectors $\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)$ and $\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)$.
(b) Find the shortest possible vector of the form $\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)$ which is perpendicular to the vectors $\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)$ and $\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)$.
Vector $\mathbf { v }$ is perpendicular to both $\left( \begin{array} { c } - 1 \\ 1 \\ 1 \end{array} \right)$ and $\left( \begin{array} { c } 1 \\ p \\ p ^ { 2 } \end{array} \right)$ where $p$ is a real number. Show that it is impossible for $\mathbf { v }$ to be perpendicular to the vector $\left( \begin{array} { c } 1 \\ 1 \\ p - 1 \end{array} \right)$. \section*{OCR} Oxford Cambridge and RSA

OCR FS1 AS 2017 December Q1

8 marks Moderate -0.3

1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that $5 \%$ of letters receive a favourable reply.

Bill sends a letter to each of 40 potential sponsors. Assuming that the number $N$ of favourable responses can be modelled by a binomial distribution, find the mean and variance of $N$.
Gill sends one letter at a time to potential sponsors. $L$ is the number of letters she sends, up to and including the first letter that receives a favourable response.
(a) State two assumptions needed for $L$ to be well modelled by a geometric distribution.
(b) Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a $90 \%$ chance of receiving at least one favourable reply.

OCR FS1 AS 2017 December Q2

7 marks Moderate -0.3

2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.

All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
4 cards are chosen at random. Find the probability that at least three consonants ( $\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }$ ) are on the cards chosen.

OCR FS1 AS 2017 December Q3

7 marks Standard +0.3

3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.

Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of $t$ minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.

OCR FS1 AS 2017 December Q4

10 marks Standard +0.3

4 The discrete random variable $X$ has the distribution $\mathrm { U } ( n )$.

Use the results $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ and $\mathrm { E } ( X ) = \frac { n + 1 } { 2 }$ to show that $\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)$. It is given that $\mathrm { E } ( X ) = 13$.
Find the value of $n$.
Find $\mathrm { P } ( X < 7.5 )$. It is given that $\mathrm { E } ( a X + b ) = 10$ and $\operatorname { Var } ( a X + b ) = 117$, where $a$ and $b$ are positive.
Calculate the value of $a$ and the value of $b$.

OCR FS1 AS 2017 December Q5

8 marks Moderate -0.5

5 A shop manager recorded the maximum daytime temperature $T ^ { \circ } \mathrm { C }$ and the number $C$ of ice creams sold on 9 summer days. The results are given in the table and illustrated in the scatter diagram.

$T$	17	21	25	26	27	27	29	30	30
$C$	21	16	20	38	32	37	35	39	42

\includegraphics[max width=\textwidth, alt={}]{64d7ed6d-fadd-4c59-afb0-97d1788ba369-3_661_1189_1320_431}

$$n = 9 , \Sigma t = 232 , \Sigma c = 280 , \Sigma t ^ { 2 } = 6130 , \Sigma c ^ { 2 } = 9444 , \Sigma t c = 7489$$

State, with a reason, whether one of the variables $C$ or $T$ is likely to be dependent upon the other.
Calculate Pearson's product-moment correlation coefficient $r$ for the data.
State with a reason what the value of $r$ would have been if the temperature had been measured in ${ } ^ { \circ } \mathrm { F }$ rather than ${ } ^ { \circ } \mathrm { C }$.
Calculate the equation of the least squares regression line of $c$ on $t$.
The regression line is drawn on the copy of the scatter diagram in the Printed Answer Booklet. Use this diagram to explain what is meant by "least squares".

OCR FS1 AS 2017 December Q6

9 marks Standard +0.3

6 Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.

Arlosh calculates Spearman's rank correlation coefficient $r _ { s }$ for the critics' ratings. He calculates that $\Sigma d ^ { 2 } = 72$. Show that this value must be incorrect.
Arlosh checks his working with Sarah, whose answer $r _ { s } = \frac { 29 } { 35 }$ is correct. Find the correct value of $\Sigma d ^ { 2 }$.
Carry out an appropriate two-tailed significance test of the value of $r _ { s }$ at the $5 \%$ significance level, stating your hypotheses clearly. Each critic gives a score out of 100 to each film. Desi uses these scores to calculate Pearson's product-moment correlation coefficient. She carries out a two-tailed significance test of this value at the $5 \%$ significance level.
Explain with a reason whether you would expect the conclusion of Desi's test to be the same as the result of the test in part (iii).

OCR FS1 AS 2017 December Q7

11 marks Standard +0.3

7 Josh is investigating whether sticking pins into a map at random, while blindfolded, provides a random sample of regions of the map. Josh divides the map into 49 squares of equal size and asks each of 98 friends to stick a pin into the map at random, while blindfolded. He then notes the number of pins in each square. To analyse the results he groups the squares as shown in the diagram.

D	D	D	D	D	D	D
D	C	C	C	C	C	D
D	C	B	B	B	C	D
D	C	B	A	B	C	D
D	C	B	B	B	C	D
D	C	C	C	C	C	D
D	D	D	D	D	D	D

The results are summarised in the table.

Region	A	B	C	D
Number of squares	1	8	16	24
Number of pins	6	21	33	38

Test at the 10\% significance level whether the use of pins in this way provides a random sample of regions of the map.
What can be deduced from considering the different contributions to the test statistic? \section*{OCR} \section*{Oxford Cambridge and RSA}

OCR FM1 AS 2017 December Q1

4 marks Moderate -0.8

1 A climber of mass 65 kg climbs from the bottom to the top of a vertical cliff which is 78 m in height. The climb takes 90 minutes so the velocity of the climber can be neglected.

Calculate the work done by the climber in climbing the cliff.
Calculate the average power generated by the climber in climbing the cliff.

\(w\)	1	2	3	4
\(\mathrm { P } ( W = w )\)	0.25	0.36	\(x\)	\(x ^ { 2 }\)

\(T\)	17	21	25	26	27	27	29	30	30
\(C\)	21	16	20	38	32	37	35	39	42

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