Questions - OCR PURE - A-Level Maths

OCR PURE Q4

4

Expand $( 1 + x ) ^ { 4 }$.
Use your expansion to determine the exact value of $1002 ^ { 4 }$.

OCR PURE Q5

5 The function f is defined by $\mathrm { f } ( x ) = ( x + a ) ( x + 3 a ) ( x - b )$ where $a$ and $b$ are positive integers.

On the axes in the Printed Answer Booklet, sketch the curve $y = \mathrm { f } ( x )$.
On your sketch show, in terms of $a$ and $b$, the coordinates of the points where the curve meets the axes. It is now given that $a = 1$ and $b = 4$.
Find the total area enclosed between the curve $y = \mathrm { f } ( x )$ and the $x$-axis.

OCR PURE Q6

6 In this question you must show detailed reasoning.

Solve the inequality $x ^ { 2 } + x - 6 > 0$, giving your answer in set notation.
Solve the equation $x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0$.
Find the exact solution of the equation $\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }$.

OCR PURE Q7

7 Determine the points of intersection of the curve $3 x y + x ^ { 2 } + 14 = 0$ and the line $x + 2 y = 4$.

OCR PURE Q8

8 The histogram shows information about the lengths, $l$ centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{4c6b7c92-2fc9-4d4f-a199-8e70f34e5eed-5_904_1284_488_242} The number of worms in the sample with lengths in the class $3 \leqslant l < 4$ is 30 .

Find the number of worms in the sample with lengths in the class $0 \leqslant l < 2$.
Find an estimate of the number of worms in the sample with lengths in the range $4.5 \leqslant l < 5.5$.

OCR PURE Q9

9 A researcher is studying changes in behaviour in travelling to work by people who live outside London, between 2001 and 2011. He chooses the 15 Local Authorities (LAs) outside London with the largest decreases in the percentage of people driving to work, and arranges these in descending order. The table shows the changes in percentages from 2001 to 2011 in various travel categories, for these Local Authorities.

Local Authority	Work mainly at or from home	Underground, metro, light rail, tram	Train	Bus, minibus or coach	Driving a car or van	Passenger in a car or van	Bicycle	On foot
Brighton and Hove	3.2	0.1	1.5	0.8	-8.2	-1.5	2.1	2.3
Cambridge	2.2	0.0	1.6	1.2	-7.4	-1.0	3.1	0.6
Elmbridge	2.9	0.4	4.1	0.2	-6.6	-0.7	0.3	-0.3
Oxford	2.0	0.0	0.6	-0.4	-5.2	-1.1	2.2	2.1
Epsom and Ewell	1.6	0.4	3.9	1.1	-5.2	-0.9	0.0	-0.6
Watford	0.7	2.0	3.1	0.4	-4.5	-1.2	0.0	-0.1
Tandridge	3.3	0.2	4.0	-0.1	-4.5	-1.1	0.0	-1.3
Mole Valley	3.3	0.1	1.9	0.3	-4.4	-0.7	0.2	-0.3
St Albans	2.3	0.3	3.4	-0.3	-4.3	-1.2	0.3	-0.2
Chiltern	2.9	1.4	1.4	0.1	-4.2	-0.6	-0.2	-0.8
Exeter	0.7	0.0	1.0	-0.6	-4.2	-1.5	1.7	3.4
Woking	2.1	0.1	3.7	0.0	-4.2	-1.3	-0.1	0.0
Reigate and Banstead	1.8	0.1	3.2	0.6	-4.1	-1.0	0.1	-0.2
Waverley	4.3	0.1	2.5	-0.5	-3.9	-0.9	-0.3	-0.9
Guildford	2.7	0.1	2.4	0.2	-3.6	-1.2	0.0	-0.3

Explain why these LAs are not necessarily the 15 LAs with the largest decreases in the percentage of people driving to work.
The researcher wants to talk to those LAs outside London which have been most successful in encouraging people to change to cycling or walking to work.
Suggest four LAs that he should talk to and why.
The researcher claims that Waverley is the LA outside London which has had the largest increase in the number of people working mainly at or from home.
Does the data support his claim? Explain your answer.
Which two categories have replaced driving to work for the highest percentages of workers in these LAs? Support your answer with evidence from the table.
The researcher suggested that there would be strong correlation between the decrease in the percentage driving to work and the increase in percentage working mainly at or from home. Without calculation, use data from the table to comment briefly on this suggestion.

OCR PURE Q10

10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

Use a binomial model to test the manufacturer's claim, at the $2.5 \%$ significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.

OCR PURE Q1

1 In the triangle $A B C , A B = 3 , B C = 4$ and angle $A B C = 30 ^ { \circ }$. Find the following.

The area of the triangle.
The length $A C$.
The angle $A C B$.

OCR PURE Q2

2 The number of people, $n$, living in a small town is changing over time. In an attempt to predict the future growth of the town, a researcher uses the following model for $n$ in terms of $t$, where $t$ is the time in years from the start of the research.
$n = 12500 + \frac { 5000 } { t }$, for $t \geqslant 1$
Find the rate of change of $n$ when $t = 5$.

OCR PURE Q3

3 The diagram shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x )$ is a cubic polynomial in $x$. This diagram is repeated in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}

State the values of $x$ for which $\mathrm { f } ( x ) < \frac { 1 } { 2 }$, giving your answer in set notation.
On the diagram in the Printed Answer Booklet, draw the graph of $y = \mathrm { f } ( - x )$.
Explain how you can tell that $\mathrm { f } ( x )$ cannot be expressed as the product of three real linear factors.

OCR PURE Q4

4

Simplify $2 \binom { 6 } { - 3 } - 3 \binom { - 1 } { 2 }$.
The vector $\mathbf { a }$ is defined by $\mathbf { a } = r \binom { 6 } { - 3 } + s \binom { - 1 } { 2 }$, where $r$ and $s$ are constants. Determine two pairs of values of $r$ and $s$ such that $\mathbf { a }$ is parallel to the $y$-axis and $| \mathbf { a } | = 3$.

OCR PURE Q5

5 The fuel consumption of a car, $C$ miles per gallon, varies with the speed, $v$ miles per hour. Jamal models the fuel consumption of his car by the formula
$C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 }$, for $0 \leqslant v \leqslant 80$.

Suggest a reason why Jamal has included an upper limit in his model.
Determine the speed that gives the maximum fuel consumption. Amaya's car does more miles per gallon than Jamal's car. She proposes to model the fuel consumption of her car using a formula of the form
$C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 } + k$, for $0 \leqslant v \leqslant 80$, where $k$ is a positive constant.
Give a reason why this model is not suitable.
Suggest a different change to Jamal's formula which would give a more suitable model.

OCR PURE Q6

6 The power output, $P$ watts, of a certain wind turbine is proportional to the cube of the wind speed $v \mathrm {~ms} ^ { - 1 }$. When $v = 3.6 , P = 50$.
Determine the wind speed that will give a power output of 225 watts.

OCR PURE Q7

7 The relationship between the variables $P$ and $Q$ is modelled by the formula
$Q = a P ^ { n }$
where $a$ and $n$ are constants.
Some values of $P$ and $Q$ are obtained from an experiment.

State appropriate quantities to plot so that the resulting points lie approximately in a straight line.
Explain how to use such a graph to estimate the value of $n$.

OCR PURE Q8

8

Prove that the following statement is not true. $$p \text { is a positive integer } \Rightarrow 2 ^ { p } \geqslant p ^ { 2 }$$
Prove that the following statement is true.
$m$ and $n$ are consecutive positive odd numbers $\Rightarrow m n + 1$ is the square of an even number

OCR PURE Q9

9 In this question you must show detailed reasoning.
Find the equation of the straight line with positive gradient that passes through $( 0,2 )$ and is a tangent to the curve $y = x ^ { 2 } - x + 6$.

OCR PURE Q10

10 Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town B .

In the Printed Answer Booklet, complete the two-way frequency table.
\multirow{2}{*}{} Town
A B C Total
adult
child
Total
One of the people is chosen at random.
1. Find the probability that this person is an adult from town A .
2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 . Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756 Jane used the same figures to write down the following sample of five student numbers.
  114, 287, 562, 81 and 817
1. State, with a reason, which one of these samples is not random.
2. Explain why Jane omitted the number 859 from her sample.

OCR PURE Q11

11 A student is investigating changes in the number of residents in Local Authorities in the SouthEast Region between 2001 and 2011. The scatter diagram shows the number $x$ of residents in these Local Authorities in the age group 8 to 9 in 2001 and the number $y$ of residents in the same Local Authorities in the age group 18 to 19 in 2011.
\includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-07_1662_1406_431_210}

Suggest a reason why the student is comparing these two age groups in 2001 and 2011. The student notices that most of the data points are close to the line $y = x$.
1. Explain what this suggests about the residents in these Local Authorities.
2. The student says that correlation does not imply causation, so there is no causal link between the values of $x$ and the values of $y$. Explain whether or not they are correct.
Some of these Local Authorities contain universities.
1. On the diagram in the Printed Answer Booklet, circle three points that are likely to represent Local Authorities containing universities.
2. Give a reason for your choice of points in part (c)(i). Assume that the proportion of residents in age group 8 to 9 in 2001 was roughly the same in each Local Authority in the South-East. The Local Authority in this region with the largest population is Medway.
On the diagram in the Printed Answer Booklet, label clearly with the letter $M$ the point that corresponds to Medway.

OCR PURE Q12

12 The variable $X$ has the distribution $\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)$. The probabilities $\mathrm { P } ( X = r )$ for $r = 0$ to 50 are given by the terms of the expansion of $( a + b ) ^ { n }$ for specific values of $a , b$ and $n$.

State the values of $a$, $b$ and $n$. A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the $5 \%$ significance level.
Write down suitable hypotheses for the test.
Determine the rejection region for the test, showing the values of any relevant probabilities.

OCR PURE Q13

13

The probability distribution of a random variable $X$ is shown in the table, where $p$ is a constant.
$x$ 0 1 2 3
$P ( X = x )$ $\frac { 1 } { 12 }$ $\frac { 1 } { 4 }$ $p$ $3 p$
Two values of $X$ are chosen at random. Determine the probability that their product is greater than their sum.
A random variable $Y$ takes $n$ values, each of which is equally likely. Two values, $Y _ { 1 }$ and $Y _ { 2 }$, of $Y$ are chosen at random. It is given that $\mathrm { P } \left( Y _ { 1 } = Y _ { 2 } \right) = 0.02$.
Find $\mathrm { P } \left( Y _ { 1 } > Y _ { 2 } \right)$.

OCR PURE Q1

1 Find the term in $x ^ { 3 }$ in the binomial expansion of $( 3 - 2 x ) ^ { 5 }$.

OCR PURE Q2

2 In this question you must show detailed reasoning.
The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2$, where $a$ is a constant. You are given that $( x - 2 )$ is a factor of $\mathrm { f } ( x )$.

Find the value of $a$.
Find all the factors of $\mathrm { f } ( x )$.

OCR PURE Q3

3 The diagram in the Printed Answer Booklet shows part of the graph of $y = x ^ { 2 } - 4 x + 3$.

It is required to solve the equation $x ^ { 2 } - 3 x + 1 = 0$ graphically by drawing a straight line with equation $y = m x + c$ on the diagram, where $m$ and $c$ are constants. Find the values of $m$ and $c$.
Use the graph to find approximate values of the roots of the equation $x ^ { 2 } - 3 x + 1 = 0$.
By shading, or otherwise, indicate clearly the regions where all of the following inequalities are satisfied. You should use the values of $m$ and $c$ found in part (a).
$x \geqslant 0$
$x \leqslant 4$
$y \leqslant x ^ { 2 } - 4 x + 3$
$y \geqslant m x + c$

OCR PURE Q4

4 In this question you must show detailed reasoning. Solve the following equations, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.

$2 \tan x + 1 = 4$
$5 \sin x - 1 = 2 \cos ^ { 2 } x$

OCR PURE Q5

5 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 3 x$. The curve passes through the point (6, 20).

Determine the equation of the curve.
Hence determine $\int _ { 1 } ^ { p } y \mathrm {~d} x$ in terms of the constant $p$.

\(x\)	0	1	2	3
\(P ( X = x )\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(3 p\)

Questions — OCR PURE (137 questions)

Browse by module

Browse by board

OCR PURE Q4

OCR PURE Q5

OCR PURE Q6

OCR PURE Q7

OCR PURE Q8

OCR PURE Q9

OCR PURE Q10

OCR PURE Q1

OCR PURE Q2

OCR PURE Q3

OCR PURE Q4

OCR PURE Q5

OCR PURE Q6

OCR PURE Q7

OCR PURE Q8

OCR PURE Q9

OCR PURE Q10

OCR PURE Q11

OCR PURE Q12

OCR PURE Q13

OCR PURE Q1

OCR PURE Q2

OCR PURE Q3

OCR PURE Q4

OCR PURE Q5

\multirow{2}{*}{}	Town
	A	B	C	Total
adult
child
Total