Practice Paper 2 (Pure & Statistics) (OCR A Level Maths: Pure)

Practice Paper Questions

1
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4 marks

Differentiate  fraction numerator 5 x to the power of 7 over denominator sin space 2 x space space end fraction with respect to x.

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2
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2 marks

Part of the graph of y equals tan space straight theta is shown below, where straight theta is measured in radians.

q5-10-1-solving-equations-medium-a-level-maths-pureExplain why the change of sign rule would fail if attempting to locate a root of the function space straight f open parentheses straight theta close parentheses equals tan space straight theta using the values of θ = 1.55 and θ = 1.65.

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3
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3 marks

The sum of the first ten terms in an arithmetic series is 40.  The sum of the first twenty terms in the same series is 280.  Find the first term, a, and the common difference, d, of the series.

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4a
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3 marks

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.
At a certain point along the tank the height of water is measured relative to the calm water level which has a height of 0 space c m.
The height of water in the tank is modelled by the function

h left parenthesis t right parenthesis equals 12 space cos left parenthesis 20 t right parenthesis degree space space space space space space space space space t greater or equal than 0

where h cm is the height of water and t seconds is the time after the peak of the first wave passes the measuring point.

Sketch a graph of h spaceagainst t spacefor 0 less or equal than t less or equal than 54.

4b
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4 marks
(i)
What is the maximum height the water reaches according to the model?

(ii)
How frequent are the waves generated by the tank?

(iii)
How often is the water at its calm level?

(iv)
When will the peak of the 12th wave pass the measuring point?
4c
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1 mark

Comment on the suitability of using this model to simulate actual sea waves.

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5
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3 marks

The line L is perpendicular to the line with equation y minus 1 third x plus 2 over 3 equals 0, and passes through the origin.

Find the equation of the line L.

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6a
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2 marks

The points A left parenthesis negative 3 comma space 6 right parenthesis comma space B left parenthesis 5 comma space minus 4 right parenthesis and C left parenthesis 6 comma space 5 right parenthesis lie on a circle.

Show that ∠ACB=90°.

6b
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1 mark

Deduce a geometrical property of the line segment AB.

6c
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4 marks

Hence find the equation of the circle.

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7a
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4 marks

The energy company PowerX operate a wind turbine.
Engineers from PowerX model the power output, P kW (kiloWatts), of the wind turbine over a twelve-hour period according to the function

space space P open parentheses t close parentheses equals 50 plus 10 sin square root of 10 t end root space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 12

where t is time measured in hours.  The graph of y equals P left parenthesis t right parenthesis is shown below.

q3-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

Using the trapezium rule, with six strips of width 1, estimate the total power generated by the wind turbine in the first six hours.
You may use the table below to help.

Time (t)

0

1

2

3

4

5

6

Power (P)

             

7b
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1 mark

Briefly explain why it is difficult for the engineers from PowerX to determine whether the total amount of power found, using the method in part (a), is an over- or under- estimate.

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8a
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4 marks

Given that straight f open parentheses x close parentheses equals sin space x

Show that

straight f apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of open parentheses sin space x space open parentheses fraction numerator cos space h space minus 1 over denominator h end fraction close parentheses plus cos space x space open parentheses fraction numerator sin space h over denominator h end fraction space close parentheses close parentheses

8b
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3 marks

Hence prove that straight f apostrophe open parentheses x close parentheses equals cos space x .

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9
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3 marks

Given that left parenthesis a x plus b y right parenthesis left parenthesis 2 x plus y right parenthesis left parenthesis x minus 3 y right parenthesis equals 8 x cubed plus c x squared y plus d x y squared minus 9 y cubed, where a comma space b comma space c and d are constants, find the values of a comma space b comma space c and d.

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10a
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2 marks

The diagram below shows a cube whose vertices are O, A, B, C, D, E, F and G.

q9-11-1-vectors-in-2-dimensions-easy-a-level-maths-pure

a, b and c are the vectors stack O A with rightwards arrow on top comma space stack O B with rightwards arrow on top and stack O C with rightwards arrow on top respectively.

Find vectors stack O E with rightwards arrow on top and stack A G with rightwards arrow on top.

10b
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2 marks

Let P be a point on OE, and let Q be a point on AG.

Explain why the vectors stack O P with rightwards arrow on top and stack O Q with rightwards arrow on top can be expressed in the forms 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O P with rightwards arrow on top end cell equals cell lambda stack O E space with rightwards arrow on top space end cell row cell stack O Q with rightwards arrow on top end cell equals cell bold a plus mu stack A G with rightwards arrow on top end cell end table

where lambda and mu  are constants with 0 less or equal than lambda less or equal than 1 and 0 less or equal than mu less or equal than 1 

10c
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4 marks

By solving the equation stack O P with rightwards arrow on top equals stack O Q with rightwards arrow on top, using your results from (a) and (b), show that the diagonals OE and AG intersect each other, and determine the ratios into which they are cut by the point of intersection.

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11a
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1 mark

An estate agent, Terry, claims that there is a correlation between the value of a house (in £1000s) and the distance between that house and the nearest nightclub (miles).

Terry has a database containing over 100 houses and he takes a random sample of seven houses to investigate his claim. The scatter graph below shows the results:

q11-ocr-a-level-maths-statistics-practice-paper

Terry calculates the product moment correlation coefficient as r space equals space 0.837. Using the scatter graph, explain how you know Terry's PMCC value is incorrect.

11b
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3 marks

Terry corrects his mistake and calculates the correct PMCC as r space equals space minus 0.837.
The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, r, for a sample of size 7.

One tail 10% 5% 2.5% 1% 0.5% One tail
Two tail 20% 10% 5% 2% 1% Two tail
  0.5509 0.6694 0.7545 0.8329 0.8745  

(i)
Write down suitable null and alternative hypotheses for a two-tailed test to investigate Terry's claim.
(ii)
Test Terry's claim using a 5% level of significance.
11c
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1 mark

State, giving a reason, whether the conclusion to the test would be different if a 1% level of significance had been used.

11d
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1 mark

Suggest one way in which Terry could improve his investigation.

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12a
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2 marks

N’Oréal, a hair and beauty company, release an advertising campaign for its new product called ‘Face Amazifier’. The small print at the bottom of the advert says the following

‘55% of 25 existing customers agree that the product makes your face feel more amazing.’

The Advertising Standards Agency get complaints that the advert is not representative.

Explain the meaning of ‘bias’ in relation to the choice of sample N’Oréal has used.

12b
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2 marks

Suggest two ways N’Oréal could improve their sample to make it more representative.

12c
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3 marks
(i)
Show that 55% of 25 customers is not a valid statistic for N’Oréal to use.

(ii)
State the two closest possible alternative figures that would be valid.

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13a
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6 marks

The mean time that teenagers in the UK spend on social media is 132 minutes per day and the standard deviation is known to be 24 minutes.  Mr Headnovel, a teacher in the UK, claims that the students at his school spend more time on social media than the country’s average.  He takes a random sample of 15 students and calculates the mean time spent on social media to be 144 minutes.

Stating your hypotheses clearly, test Mr Headnovel’s claim using a 5% level of significance.

13b
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2 marks

State two assumptions you had to make about the times that teenagers in Mr Headnovel’s school spend on social media?

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14a
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4 marks

A bag contains 12 orange marbles, 8 purple marbles and 5 red marbles.  A marble is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second marble is then taken from the bag and its colour is recorded.

Draw a tree diagram to represent this information.

14b
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4 marks

Find the probability that:

(i)
both marbles are different colours

(ii)
the second marble is purple, given that both marbles are different colours.

 

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15a
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3 marks

The table below shows an extract from the large data set for the year 2011.
The figures shown are the number of people travelling to work by train in 7 randomly selected local authorities in the East of England.

Local Authority Number of people travelling to work by train
Cambridge 2 760
East Hertfordshire 9 383
Great Yarmouth 248
King's Lynn and West Norfolk 976
Stevenage 2 919
Watford 4 897
Waveney 517


Find the median, the upper and lower quartiles, and interquartile range.

15b
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2 marks

An outlier is defined as any data value that falls either more than
1.5 cross times (interquartile range) above the upper quartile or less than
1.5 cross times (interquartile range) below the lower quartile.

Find the boundaries (fences) at which outliers are defined.

15c
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2 marks

Explain why, in this case, there are no outliers.

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16a
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1 mark

Leonardo has constructed a biased spinner with six sectors labelled 0,1, 1, 2, 3 and 5.  The probability of the spinner landing on each of the six sectors is shown in the following table:

number on sector 0 1 1 2 3 5
probability 6 over 20 p 3 over 20 5 over 20 3 over 20 1 over 20


Find the value of p.

16b
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3 marks

Leonardo is playing a game with his biased spinner.  The score for the game, X, is the number which the spinner lands on after being spun.

 Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

16c
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2 marks

Complete the following cumulative probability function table for X:

Score bold italic x 0 1 2 3 5
bold P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis 6 over 20       1
16d
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2 marks

Find the probability that X is

(i)
no more than 1

(ii)
at least 3.

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17
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6 marks

The independent random variables X  and have probability distributions

                             straight P left parenthesis X equals x right parenthesis equals p comma space space space space space space x equals 1 comma 2 comma 3 comma 5 comma 8 comma 11 space

                         straight P left parenthesis Y equals y right parenthesis equals q over y comma space space space space space space y equals 1 comma 3 comma 6 space

where p and q are constants.

 Find  straight P left parenthesis X greater than Y right parenthesis.

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