Practice Paper 3 (Pure & Statistics) (AQA A Level Maths: Statistics)

Practice Paper Questions

1
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1 mark

Given that

integral subscript 0 superscript 8 space straight g left parenthesis x right parenthesis space straight d x space equals space 5

deduce the value of

integral subscript 0 superscript 8 space left parenthesis g left parenthesis x right parenthesis space plus space 1 right parenthesis space d x

Circle your answer.

5 13 6 –3

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2
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1 mark

Given that

5 space cos space alpha space long dash space 12 space sin space alpha space identical to space R space cos left parenthesis theta space plus space alpha right parenthesis

find the value of R.

Circle your answer.

17 13 5 12

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3
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1 mark

Determine which one of these graphs represents a function whose inverse function exists.

Tick (✓) one box.

q4-aqa-a-level-practice-paper-maths-set-b-pure

q4-1-aqa-a-level-practice-paper-maths-set-b-pure

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4a
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2 marks

straight f left parenthesis x right parenthesis equals 4 x cubed plus 4 x squared minus 23 x minus 30

Use the factor theorem to show that left parenthesis x plus 2 right parenthesis spaceis a factor of space straight f left parenthesis x right parenthesis.

4b
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4 marks

Factorise space straight f left parenthesis x right parenthesis completely.

4c
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2 marks

Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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5a
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1 mark

In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

B equals 16 e to the power of 0.85 t end exponent

B is the number of birds after t years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

5b
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2 marks

According to this model, how many birds will be in the reserve after 3 years?

5c
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2 marks

How long will it take for the population of birds within the reserve to reach 500?

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6a
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2 marks

Write  open parentheses 3 over 5 close parentheses to the power of x in the form e to the power of k x end exponent, giving the value of k to three significant figures.

6b
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2 marks

Write open parentheses begin inline style 4 over 7 end style close parentheses to the power of 3 t end exponent  in the form e to the power of k t end exponent, giving the value of k to three significant figures.
State, and justify, whether this would represent exponential growth or decay.

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7a
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4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis. The marked point B left parenthesis 4 comma 8 right parenthesis lies on the graph, and the graph meets the origin at the marked point A.

q5a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-medium

In separate diagrams, sketch the curves with equation

(i)

y equals negative straight f left parenthesis x right parenthesis

(ii)
y equals straight f left parenthesis 4 x right parenthesis

On each diagram, give the coordinates of the images of points A and B under the given transformation.

7b
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2 marks

On the graph of y equals a straight f left parenthesis x right parenthesis the image of one of the two marked points has a y spacecoordinate of 4. Find the value of a.

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8
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2 marks

Show that, for all values of k,

      C presuperscript k subscript 1 space equals k

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9
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2 marks

In the expansion of left parenthesis a minus x right parenthesis to the power of 4, the coefficient of the x squared term is 96.
Given that a greater than 0, find the value of a.

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10
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3 marks

Given that C presuperscript n subscript 3 equals 35 find the value of n.

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11a
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3 marks

A geometric sequence has first term 900 and a common ratio r where space r greater than 0.  The 18th term of the sequence is 18.

Show that r satisfies the equation 17 space log r plus log 50 equals 0.

11b
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1 mark

Hence or otherwise find the value of r correct to 3 significant figures.

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12a
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3 marks

A geometric series has first term 9, and the sum of the first three terms of the series is 19.  The common ratio of the series is r.

Show that 9 r squared plus 9 r minus 10 equals 0.

12b
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2 marks

Find the two possible values of r.

12c
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3 marks

Given that the series converges, find the sum to infinity of the series.

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13
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5 marks

Show that

sin 3 theta plussin theta identical to 4 sin theta minus 4 spacesin3 theta

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14
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1 mark

The probabilities of events A comma space B and C are related, as shown in the Venn diagram below.

q13-aqa-a-level-maths-practice-paper-set-b-pure

Find P open parentheses A union C close parentheses.

Circle your answer.

0 0.36 0.71 0.35

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15
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1 mark

The table below shows the heights of sunflowers at the start of each week, for 9 weeks.

Week 1 2 3 4 5 6 7 8 9
Height (cm) 2 8 15 20 26 34 44 48 53

Calculate the standard deviation of these heights, correct to 1 decimal place.

Circle your answer.

17.1 292.2 27.8 14.7

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16a
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4 marks

The cumulative frequency diagram below shows completion times for 100 competitors at the 2019 Rubik’s cube championships.  The quickest completion time was 9.8 seconds and the slowest time was 52.4 seconds.q2-medium-2-3-working-with-data-edexcel-a-level-maths-statistics

The grid below shows a box plot of the 2020 championship data.  Draw a box plot on the grid to represent the 2019 championship data. q2a-medium-2-3-working-with-data-edexcel-a-level-maths-statistics

16b
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3 marks
(i)
Compare the distribution of completion times for the 2019 and 2020 championships.

(ii)
Given that the 2020 championships happened after the global pandemic, during which many competitors spent months at home, interpret your findings from part (b)(i).

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17a
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2 marks

240 students are surveyed regarding their belief in supernatural creatures.  144 say they believe in unicorns open parentheses U close parentheses .  75 say they believe in vampires open parentheses V close parentheses.  Of those who believe in vampires, 27 also believe in unicorns.

Draw a two-way table to show this information.

17b
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4 marks

One student is chosen at random.  Find:

(i)
straight P open parentheses U to the power of apostrophe close parentheses

(ii)
P open parentheses U to the power of apostrophe intersection space V to the power of apostrophe close parentheses

(iii)
straight P open parentheses U vertical line V close parentheses

(iv)
straight P open parentheses V vertical line U close parentheses

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18a
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4 marks

Adrenaline is a new rollercoaster at a theme park. It is known that the time a customer spends in the queue follows a normal distribution with a variance of 52 minutes².  The mean time spent in a queue for other rollercoasters is 41 minutes.  The manager of the theme park wants to use a hypothesis test to investigate whether the mean time in the queue for Adrenaline is different to the mean time for the other rollercoasters.  She takes a sample of 10 customers over a period of several days and records their times spent in the queue for Adrenaline.

Find the critical region for the test at the 10% level of significance. State your hypotheses clearly.

18b
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3 marks

The queuing times for the 10 people in the sample are:

38        49        40        39        49

39        59        32        55        41

State the conclusion of the test in context.

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19
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5 marks

A company wants to survey 15% of its staff to find out whether employees would like to continue working from home after the Covid-19 pandemic. The company's 580 members of staff are grouped by job as follows: 295 engineers, 11 managers, 154 office staff and 120 apprentices.

Suggest a suitable sampling method and explain how the company can use this method to obtain its sample.

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20a
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1 mark

A teacher, Ms Pearman, claims that there is a positive correlation between the number of hours spent studying for a test and the percentage scored on it.

Write down suitable null and alternative hypotheses to test Ms Pearman's claim.

20b
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2 marks

Ms Pearman takes a random sample of 25 students and gives them a week to prepare for a test. She records the percentage they score in the test, s %, and the amount of revision they did, h hours.

Ms Pearson calculates the product moment correlation coefficient for these data as r space equals space 0.874.

Given that the p-value for the test statistic r space equals space 0.874 is 0.0217, test at the 5% level of significance whether Ms Pearman's claim is justified.

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21a
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3 marks

A machine is used to fill bags of potatoes for a supermarket chain.  The weight, W kg, of potatoes in the bags is normally distributed with mean 3 kg and standard deviation sigma kg. 

Given that 7% of the bags contain a weight of potatoes that is at least 50 g more than  the mean, find:

straight P left parenthesis 2.9 less or equal than W less or equal than 3.1 right parenthesis.

21b
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4 marks

Twelve of the bags of potatoes are chosen at random.

Find the probability that not more than one of the bags will contain less than 2.96 kg of potatoes.

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22a
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2 marks

In the town of Wooster, Ohio, it is known that 90% of the residents prefer the locally produced Woostershire brand sauce when preparing a Caesar salad.  The other 10% of residents prefer another well-known brand.

 30 residents are chosen at random by a pollster.  Let the random variable X represent the number of those 30 residents that prefer Woostershire brand sauce.

 Suggest a suitable distribution for X and comment on any necessary assumptions.

22b
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3 marks

Find the probability that

(i)
90% or more of the residents chosen prefer Woostershire brand sauce

(ii)
none of the residents chosen prefer the other well-known brand.
22c
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2 marks

The pollster knows that there is a greater than 97% chance of at least k of the 30 residents preferring Woostershire brand sauce, where k is the largest possible value that makes that statement true.

Find the value of k.

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23a
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3 marks

Vauxhall state that 8.5% of their cars produce less than 100 g/km of carbon dioxide emissions.

Elaine would like to investigate this claim using the large data set by testing the following hypotheses:

straight H subscript 0 space colon space p space equals space 0.085 space space space space space space space space space space space space space straight H subscript 1 space colon space p space not equal to space 0.085

There are 1069 Vauxhall cars in the large data set. The critical regions are stated to be X space less or equal than space 75 and X space greater or equal than space 107.

Calculate the actual level of significance based on these critical regions.

23b
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2 marks

73 Vauxhall cars out 1069 in the large data set have less than 100 g/km of carbon dioxide emissions.

State a conclusion to the hypothesis test for this value, giving a reason for your answer.

23c
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1 mark

With reference to the large data set, state one limitation of your conclusion.

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