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

Practice Paper Questions

1
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1 mark

The graph of y space equals space arccos space x is shown.

q1-aqa-a-level-pure-maths-practice-paper-set-a-stat

State the maximum value of y on the graph.

Circle your answer.

1 straight pi 2 2 straight pi

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2
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1 mark

Simplify fully

fraction numerator open parentheses 3 x minus 15 close parentheses open parentheses x plus 5 close parentheses over denominator open parentheses 5 plus x close parentheses open parentheses 5 minus x close parentheses end fraction space space space for space x not equal to plus-or-minus 5

Circle your answer.

negative 3 3 fraction numerator 3 x minus 15 over denominator 5 minus x end fraction fraction numerator 15 minus 3 x over denominator 5 minus x end fraction

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3
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1 mark

Given straight f left parenthesis x right parenthesis space equals space 4 x cubed obtain

limit as h rightwards arrow 0 of fraction numerator straight f open parentheses x plus h close parentheses minus straight f open parentheses x close parentheses over denominator h end fraction

Circle your answer.

fraction numerator 4 h cubed over denominator h end fraction 12 x squared x to the power of 4 fraction numerator 4 open parentheses x plus h close parentheses cubed minus 4 x cubed over denominator h end fraction

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4a
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3 marks

Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 5 minus 2 x right parenthesis to the power of 4.

4b
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2 marks

Use your answer to part (a) to estimate left parenthesis 4.5 right parenthesis to the power of 4.

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5
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4 marks

A sector of a circle,O P Q, is such that it has radius 3.4 cm and the angle at its centre, O, is fraction numerator 3 straight pi over denominator 4 end fraction radians.

(i)

Find the length of the arc P Q.

(ii)
Find the area of the sector O P Q.

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6
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3 marks

Given that theta is small, write an approximation in terms of theta for

                                             2sin thetapluscos theta minustan2 theta

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7a
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2 marks

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150π cm3. To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by S equals 2 pi r squared plus space fraction numerator 300 pi over denominator r end fraction.

7b
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4 marks

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

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8
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4 marks

The two parallel sides of a trapezium have lengths of open parentheses 4 plus square root of 3 close parentheses cm and open parentheses 6 minus 3 square root of 3 close parentheses cm, and the area of the trapezium is open parentheses 10 square root of 3 minus 6 close parentheses cm2.   Showing clear algebraic working, determine the perpendicular distance between the two parallel sides of the trapezium.  Give your answer in the form a square root of b where a and b are integers.

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9a
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1 mark

Frankie opens a savings account with £400.

Compound interest is paid at an annual rate of 3%.

Show that at the end of the first year Frankie has £412 in the savings account.

9b
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1 mark

At the start of the second year, and each subsequent year, Frankie adds another £400 to the savings account.

Write down the amount of interest the £400 invested at the start of year 2 will earn by the start of year 3.

9c
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2 marks

Explain why the amount of money in the savings account, in pounds, at the end of year 2 can be written as

         left parenthesis 400 cross times 1.03 right parenthesis cross times 1.03 plus 400 cross times 1.03.

9d
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2 marks

Hence show that after n years, the amount in pounds in Frankie’s savings account will be

         400 left parenthesis 1.03 plus 1.03 squared plus 1.03 cubed plus midline horizontal ellipsis plus 1.03 to the power of n right parenthesis.

9e
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2 marks

Show that the sum of the geometric series 1.03 plus 1.03 squared plus 1.03 cubed plus midline horizontal ellipsis plus 1.03 to the power of n is given by

         103 over 3 open parentheses 1.03 to the power of n space minus 1 close parentheses

9f
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2 marks

Hence find the amount of money in Frankie’s savings account at the end of 12 years.

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10
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6 marks

Use integration by parts to find, in terms of e, the exact value of

integral subscript 0 superscript 1 left parenthesis 5 x minus 4 right parenthesis e to the power of 3 x end exponent space straight d x

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11
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5 marks

A curve has the equation y equals x open parentheses x plus 6 close parentheses squared plus 4 open parentheses 3 x plus 11 close parentheses.

The point Popen parentheses x comma space y close parentheses is the stationary point of the curve.

Find the coordinates of P and determine its nature.

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12a
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2 marks

Given  y equals x cubed minus 6 x squared show that fraction numerator straight d squared y over denominator straight d x squared end fraction space equals 6 x minus 12.

12b
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2 marks

Solve the inequality  fraction numerator straight d squared y over denominator straight d x squared end fraction space less than 0  and hence determine the set of values for which the graph of  y equals x cubed minus 6 x squared space is concave.

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13
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1 mark

Aurora has collected data from 40 similar-sized beaches to investigate any correlation between the slope of the beach and the frequency of waves.

She calculates the correlation coefficient.

Which of the following statements best describes her answer of 13 over 7?

Tick (✓) one box.

Definitely incorrect square
Probably incorrect square
Probably correct square
Definitely correct square

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14
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1 mark

The random variable X is such that X tilde B open parentheses n comma space p close parentheses.

The mean value of X is 48.

The variance of X is 36.

Find p.

Circle your answer.

0.875 0.125 0.75 0.25

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15
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3 marks

q14-aqa-a-level-maths-practice-paper-set-a-statistics

The large data set contains data on 407 Ford cars from 2002. A selection of data relating to the CO2 and CO emissions (g/km) of Ford cars from 2002 is given above.

The large data set includes data on a variety of variables including propulsion and body type. Roughly three quarters of the Ford cars in the data set are petrol cars.

(i)
State which of the two samples above is more representative and why.
(ii)
If a stratified sample by body type had been used to select the data in the two samples above, roughly how many Ford cars of each type would you expect there to have been in the complete data set?

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16a
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4 marks

As part of an experiment, 15 maths teachers are asked to solve a riddle and their times, in minutes, are recorded:

8            12         19         20         20

21         22         23         23         23

25         26         27         37         39

An outlier is an observation which lies more than plus-or-minus 2 standard deviations away from the mean.

Show that there is exactly one outlier.

16b
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2 marks

State, with a reason, whether the mean or the median would be the most suitable measure of central tendency for these data.

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17a
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3 marks

A and B are two events with straight P open parentheses A close parentheses space equalsx  and  straight P open parentheses B close parentheses equalsy,  where xnot equal to0.  Given that A and B are independent, find the following probabilities in terms of x and y:

(i)
straight P open parentheses B vertical line A to the power of apostrophe close parentheses

(ii)
straight P open parentheses A intersection B vertical line A close parentheses

 

17b
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4 marks

A group of 18- to 25-year-olds, and a group of people over 65 years old, were asked whether they would prefer to holiday in Ibiza or Skegness. The following two-way table shows part of the results of the survey:

  Ibiza Skegness total
18-25     99
over 65     45
total 64 80 144

(i)

Find P(over 65) and P(Ibiza)

(ii)

Given that for the people in the sample the events ‘is over 65’ and ‘prefers to holiday in Ibiza’ are independent, find the missing values and complete the table of survey results.

 

 

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18a
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1 mark

The IQ of a student at Calculus High can be modelled as a random variable with the distribution straight N left parenthesis 126 comma 50 right parenthesis . The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

Write suitable null and alternative hypotheses to test the headteacher’s suspicion.

18b
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5 marks

The headteacher selects 10 students and asks them to complete an IQ test.  Their scores are:

127, 127, 129, 130, 130, 132, 132, 132, 133, 138

Test, at the 5% level of significance, whether there is evidence to support the headteacher’s suspicion.

18c
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1 mark

It was later discovered that the 10 students used in the sample were all in the same advanced classes.

Comment on the validity of the conclusion of the test based on this information.

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19a
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2 marks

The random variable X has the probability function

straight P open parentheses X equals x close parentheses equals space open curly brackets table row cell k x space space space space space space space space space space space space space space x equals 1 comma 3 comma 5 comma 7 end cell row cell 0 space space space space space space space space space space space space space space space space otherwise end cell end table close space

 Find the value of k.

19b
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2 marks

Find straight P left parenthesis X greater than 3 right parenthesis.

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20
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3 marks

The random variable X tilde B left parenthesis 50 comma 0.75 right parenthesis.  Find:

(i)
straight P left parenthesis 40 greater than X greater or equal than 30 right parenthesis

(ii)
straight P left parenthesis X less than 29 space space o r space space X greater than 38 right parenthesis

(iii)
straight P left parenthesis X less or equal than 52 space space a n d space space X greater than 31 right parenthesis

 

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21a
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2 marks

It is known that customers have an 83% success rate when attempting to order bundles of 5 or more festival tickets from a particular website. A second website claims that its customers have a greater probability of success.

State suitable null and alternative hypotheses to test the second website's claim.

21b
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3 marks

In one day, 1159 out of 1358 bundles of 5 or more tickets are successfully ordered from the second website.

Test the second website's claim using a 1% significance level.

21c
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3 marks

It is suggested that instead of using a 1% significance level for the test, the critical region X greater or equal than 1150 could have been used for testing the second website's claim against observed data for the number of successes out of 1358 attempted orders.

Find the probability of incorrectly rejecting the null hypothesis if that critical region is used.

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22a
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3 marks

The following histogram shows the distribution of weights, in grams, of a population of dormice in the UK:q1-very-hard-4-3-normal-distributions-edexcel-a-level-maths-statistics

(i)
Explain why it would be appropriate to use a normal distribution to model the distribution of weights of the dormouse population.

(ii)
Explain how the histogram could be altered so as to better approximate the smooth curve of the normal distribution.
22b
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3 marks

The mean and standard deviation for the weights of the dormouse population are calculated to be 20.5 g and 2.6 g respectively.  A normal curve is drawn corresponding to these values.

Write down the values of the weight that correspond to the line of symmetry and the points of inflection of the normal curve.

22c
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4 marks

Use the properties of the normal distribution to determine whether each of the following statements is likely to be true. In each case give a reason for your answer.

(i)
84% of the dormice have a weight that is less than 23.1 g.

(ii)
More than 99% of the dormice have a weight that is between 15.3 g and  28.3 g.

(iii)
18.5% of the dormice have a weight that is either less than 17.9 g or greater than 25.7 g.

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