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

Practice Paper Questions

1
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1 mark

straight f left parenthesis x right parenthesis space= arctan x

State the maximum possible domain of straight f.

Tick (✓) one box.

open curly brackets x element of straight real numbers space colon space minus 1 less or equal than x less or equal than 1 close curly brackets square
open curly brackets x element of straight real numbers space colon space minus straight pi over 2 less or equal than x less or equal than straight pi over 2 close curly brackets square
open curly brackets x element of straight real numbers space colon space minus straight pi less or equal than x less or equal than straight pi close curly brackets square
x element of straight real numbers square

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2
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1 mark

Find the value of the below expression.

fraction numerator 120 factorial over denominator 118 factorial space cross times space 4 factorial end fraction

Circle your answer.

15 over 59 Undefined 595 70210

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3
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1 mark

Determine which one of the formulae below defines an increasing sequence for n space greater or equal than space 1.

Tick (✓) one box.

u subscript n space equals space 2 to the power of n minus 1 end exponent square
u subscript n equals cos space open parentheses 180 n degree close parentheses square
u subscript 1 equals 20 space space space u subscript n plus 1 end subscript space space space equals u subscript n minus 5 square
u subscript n space equals space 20 minus 3 n square

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4
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4 marks

Write down the inequalities that define the region R shown in the diagram below.

2-4-edexcel-alevel-maths-pure-q8medium

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5
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5 marks

The diagram below shows the sector of a circle O A B.

q6-aqa-a-level-maths-practice-paper-pure

(i)
Find the area of the sector O A B, giving your answer to 3 significant figures.
(ii)
Find the area of the triangle O A B, giving your answer to 3 significant figures.
(iii)
Find the area of the shaded segment, giving your answer to 3 significant figures.

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6
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4 marks

Prove that a triangle with side lengths of 8 cm, 6 cm and 10 cm must contain a right-angle.  You may use the diagram below to help.

1-1-maths-q9medium-1

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7a
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3 marks

Show that

fraction numerator 11 over denominator left parenthesis 2 x minus 3 right parenthesis left parenthesis x plus 4 right parenthesis end fraction

can be written in the form

fraction numerator A over denominator 2 x minus 3 end fraction plus fraction numerator B over denominator x plus 4 end fraction

7b
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4 marks

Hence find

integral fraction numerator 11 over denominator left parenthesis 2 x minus 3 right parenthesis left parenthesis x plus 4 right parenthesis end fraction space straight d x

writing your answer as a single logarithm.

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8a
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1 mark

Scientists introduced a small number of rare breed deer to a large wildlife sanctuary.

The population of deer, within the sanctuary, is modelled by

D equals 20 e to the power of 0.1 t end exponent

D is the number of deer after t years of first being introduced to the sanctuary.

Write down the number of deer the scientists introduced to the sanctuary.

8b
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2 marks

How many years does it take for the deer population to double?

8c
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1 mark

Give one criticism of the model for population growth.

8d
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2 marks

The scientists suggest that the population of deer are separated after either 25 years or when their population exceeds 400.
Find the earliest time the deer should be separated.

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9a
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2 marks

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years

 A model for the mass of carbon-14, y g, in an object originally containing 100 g,
at time t years is

y equals 100 e to the power of negative k t end exponent

where k is a constant.

Find the value of k, giving your answer to three significant figures.

9b
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2 marks

The object is considered as having no radioactivity once the mass of carbon-14 it contains falls below 0.5 g. Find out how old the object would have to be, to be considered non-radioactive.

9c
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2 marks

A different object currently contains 25g of carbon-14.
In 500 years’ time how much carbon-14 will remain in the object?

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10a
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2 marks

The curve C has equation

1 fifth x squared e to the power of y equals 5

Show that C intersects the x-axis at the points (-5 , 0) and (5 , 0).

10b
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2 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of x and y.

10c
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2 marks

Hence find the gradients of C at the two points where C intercepts the x-axis.

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11a
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4 marks

The curve C is defined by the equation ln space y equals 1 minus x y .

The point P(1 , 1) lies on C.

Show that

fraction numerator d y over denominator d x end fraction equals negative fraction numerator y squared over denominator 1 plus x y end fraction

11b
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3 marks

Find the gradient of the tangent to C at point P, and hence find the gradient of the normal to C at point P.

11c
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2 marks

Find the equation of the normal to C at point P, giving your answer in the form a x plus b y plus c equals 0, where a, b and c are integers to be found.

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12
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1 mark

Which of the options below best describes the correlation shown in the diagram below?

q16-aqa-a-level-maths-practice-paper-pure

Circle your answer.

Moderate
Positive
Strong
Positive
Moderate
Negative
Strong
Negative

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13
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1 mark

Mark is one of a team of people interviewing hockey fans as they exit a stadium.

He is asked to survey every 20th person who exits.

Identify the name of this type of sampling.

Circle your answer.

Simple Random Quota Stratified Systematic

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14a
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1 mark

The incomplete box plot below shows the carbon dioxide (CO2) emissions, measured in grams per kilometre, of a random sample of 34 vehicles taken from the large data set.

q17-aqa-a-level-maths-practice-paper-pure

Find the interquartile range.

14b
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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.

14c
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3 marks

The random sample of 34 cars has one outlier, a CO2 reading of 250 g/km.
Complete the box plot given that the maximum and minimum values should be located at the boundaries (fences) at which outliers are defined.
Any outliers should be indicated with a cross (cross times).

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15a
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2 marks

A snowboarder is trying to perform the Poptart trick.

The snowboarder has a success rate of 25% of completing the trick.

The snowboarder will model the number of times they can expect to successfully complete the Poptart trick, out of their next 12 attempts, using the random variable X tilde B left parenthesis 12 comma 0.25 right parenthesis.

(i)
Give a reason why the model is suitable in this case.
(ii)
Suggest a reason why the model may not be suitable in this case.
15b
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2 marks

Using the model, find the probability that the snowboarder

(i)
successfully completes the Poptart trick more than 3 times in their next 12 attempts
(ii)
fails to successfully complete the trick on any of their next 12 attempts.

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16
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6 marks

The random variable X tilde B left parenthesis 50 comma 0.85 right parenthesis. space spaceFind:

(i)
the largest value of q such that  straight P left parenthesis X less than q right parenthesis less than 0.16

(ii)
the largest value of r such that straight P left parenthesis X greater or equal than r right parenthesis greater than 0.977

(iii)
the smallest value of s such that  straight P left parenthesis X greater than s right parenthesis less than 0.025.

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17a
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2 marks

Aand B are two events with  straight P open parentheses A close parentheses space equals space 0.47 and  P open parentheses B close parentheses equals 0.31.  Given that Aand B are independent, write down

(I)
straight P open parentheses A vertical line B close parentheses

(ii)
P open parentheses B vertical line A to the power of apostrophe close parentheses
17b
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4 marks

A group of middle and senior school students were asked whether they preferred vinegar or ketchup as a topping on their chips. The following two-way table shows the results of the survey:

  vinegar ketchup total
middle 49 21 70
senior 63 27 90
total 112 48 160

(i)

Find and P(ketchup|middle) and P(middle|ketchup).

(ii)
Use your results from part (b)(i) to show that for the students in the sample ‘is in middle school’ and ‘prefers ketchup on chips’ are independent events.

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18a
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1 mark

The table below shows data from the United States regarding annual per capita chicken consumption (in pounds) and the unemployment rate ( % of population) between the years 2005 and 2014:

Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Chicken
consumption
(pounds)
86.4 86.9 85.5 83.8 80.0 82.8 83.3 80.8 82.3 83.8
Unemployment
rate (%)
5.08 4.62 4.62 5.78 9.25 9.63 8.95 8.07 7.38 6.17

The product moment correlation coefficient for these data is r space equals space minus 0.821.
The critical values for a 10% two-tailed test are ±0.5495.

State what is measured by the product moment correlation coefficient.

18b
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3 marks
(i)
Write down suitable null and alternative hypotheses for a two-tailed test of the correlation coefficient.
(ii)
Show that, at the 10% level of significance, there is evidence that the correlation coefficient is different from zero.
18c
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1 mark

A newspaper's headline states:

"Eating chicken is the secret to reducing the unemployment rate in the US!"

Explain whether this headline is fully justified.

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19a
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4 marks

Pizza Prince is a fast-food restaurant which is known for their Crown pizza.  The weights of Crown pizza are normally distributed with standard deviation 42 g.  It is thought that the mean weight,mu, is 350 g.

A restaurant inspector believes that the mean weight of the Crown pizza is less than
350 g.  She visits the restaurant over the period of a week, and samples and weighs five randomly selected Crown pizzas.  She uses the data to carry out a hypothesis test at the 5% level of significance.

She tests straight H subscript 0 ∶ mu equals 350  against  straight H subscript 1 ∶ mu less than 350.

When the inspector writes up her report, she can only find the values for four of the weights, these are shown below:

325.2              356.1              319.7              300.5

Given that the result of the hypothesis test is that there is insufficient evidence to reject straight H subscript 0 at the 5% level of significance, calculate the minimum possible value for the missing weight, w. Give your answer correct to 1 decimal place.

19b
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4 marks

The inspector remembers her assistant claiming that if she had used a 10% level of significance then the outcome to the hypothesis test would have been different.

Using this information, write down an inequality for w.

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20a
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4 marks

For the standard normal distribution Z tilde N left parenthesis 0 comma 1 squared right parenthesis, find:

(i)
straight P left parenthesis Z less than 1.5 right parenthesis

(ii)
straight P left parenthesis Z greater than negative 0.8 right parenthesis

(iii)
straight P left parenthesis negative 2.1 less than Z less than negative 0.3 right parenthesis
20b
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3 marks

The random variable X tilde N left parenthesis 2 comma space 0.1 squared right parenthesis.

By using the coding relationship between X and Z, re-express the probabilities from parts (a) (i), (ii) and (iii) in the forms straight P left parenthesis X less than a right parenthesisspace straight P left parenthesis X greater than b right parenthesis and straight P left parenthesis c less than X less than d right parenthesis space spacerespectively, where a, b, c and d are constants to be found.

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21a
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4 marks

The random variable X tilde N left parenthesis mu comma sigma squared right parenthesis. It is known that straight P left parenthesis X greater than 34.451 right parenthesis equals 0.001 and straight P left parenthesis X less than 14.792 right parenthesis equals 0.2

Use the relationship between X and the standard normal variable Z to show that the following simultaneous equations must be true:

mu plus 3.0902 sigma equals 34.451

mu minus 0.8416 sigma equals 14.792

21b
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2 marks

By solving the simultaneous equations in (a), determine the values of mu and sigma.

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