Practice Paper 1 (Pure) (AQA A Level Maths: Statistics)

Practice Paper Questions

1a
Sme Calculator
1 mark

The first three terms, in ascending powers of x, of the binomial expansion of left parenthesis 25 space minus space 4 x right parenthesis to the power of 3 over 2 end exponent  are given by

open parentheses 25 space minus space 4 x close parentheses to the power of 3 over 2 end exponent space almost equal to space k space minus space 30 x space plus space fraction numerator 6 x squared over denominator 5 end fraction

where k is a constant.

State the range of values of x for which this expansion is valid.

Circle your answer.

open vertical bar x close vertical bar less than 4 over 5 open vertical bar x close vertical bar less than 25 over 4 open vertical bar x close vertical bar less than 1 open vertical bar x close vertical bar less than 4 over 25
1b
Sme Calculator
1 mark

Find the value of k.

Circle your answer.

4 25 1 125

Did this page help you?

2
Sme Calculator
1 mark

A student is searching for a solution to the equation straight g left parenthesis x right parenthesis equals 0.

They correctly evaluate

straight g left parenthesis negative 1 right parenthesis space equals space minus 2 space space space and space space space space straight g left parenthesis 1 right parenthesis space equals space 2

and conclude that there must be a root between negative 1 and 1 due to the change of sign.

Select the function straight g left parenthesis x right parenthesis for which the conclusion is incorrect.

Circle your answer.

straight g left parenthesis x right parenthesis space equals space fraction numerator 2 plus 4 x over denominator x plus 2 end fraction straight g left parenthesis x right parenthesis space equals space 2 x cubed straight g left parenthesis x right parenthesis space equals 2 over x straight g left parenthesis x right parenthesis space equals 2 x

Did this page help you?

3
Sme Calculator
1 mark

The diagram shows a sector O A B of a circle with centre O and radius 2.

q3-aqa-a-level-practice-paper-maths-set-b-pure

The angle A O B is theta radians and the perimeter of the sector is 7.

Find the value of theta.

Circle your answer.

3.5 1.5 2 over 7 square root of 7

Did this page help you?

4a
Sme Calculator
3 marks

On the same axes, sketch the graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals open vertical bar g open parentheses x close parentheses close vertical bar where

straight f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers
straight g left parenthesis x right parenthesis space equals space 2 x space plus space 4 space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

4b
Sme Calculator
2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals open vertical bar g open parentheses x close parentheses close vertical bar.

Did this page help you?

5
Sme Calculator
2 marks

Use a counter-example to prove that the difference between any two square numbers is not always odd.

Did this page help you?

6
Sme Calculator
4 marks
(i)
Given that  straight f open parentheses x close parentheses equals 2 x squared plus 5, find straight f apostrophe open parentheses x close parentheses.

 

(ii)
Hence, or otherwise, find

integral fraction numerator 4 x over denominator 2 x squared plus 5 end fraction space straight d x.

Did this page help you?

7a
Sme Calculator
2 marks

A Fibonacci sequence can be expressed as the following recurrence relation

         u subscript n plus 2 end subscript equals u subscript n plus 1 end subscript plus u subscript n comma space space space space space space space space space space space space space n greater or equal than 1

Write down the first six terms of the Fibonacci sequence with u subscript 1 equals u subscript 2 equals 1.

7b
Sme Calculator
2 marks

Find

      sum from r equals 1 to 5 of u subscript r

with u subscript 1 equals 2 comma space u subscript 2 equals 4

Did this page help you?

8a
Sme Calculator
2 marks

A lifejacket falls over the side of a boat from a height of 3 m.
The height, h spacem, of the lifejacket above or below sea level open parentheses h equals 0 close parentheses, at time t seconds after falling, is modelled by the equation h equals 3 e to the power of negative 0.7 t end exponentcos 4 t .

The lifejacket reaches its furthest point below sea level after 0.742 seconds.
Find the total distance it has fallen, giving your answer to three significant figures.

8b
Sme Calculator
2 marks

Write down the value of t for the first three times the lifejacket is at sea level.

8c
Sme Calculator
3 marks
(i)
Find the value of  3 e to the power of negative 0.7 t end exponent  when  t equals 6.2.

(ii)
Hence justify why, from 6.2 seconds on, the lifejacket will always be within 4 centimetres of sea level.

Did this page help you?

9a
Sme Calculator
2 marks

Use the factor theorem to show that left parenthesis x plus 1 right parenthesis is a factor of  x cubed minus 3 x squared plus 4.

9b
Sme Calculator
2 marks

Hence, or otherwise, fully factorise x cubed minus 3 x squared plus 4.

9c
Sme Calculator
4 marks

Show that fraction numerator 19 minus 8 x over denominator x cubed minus 3 x squared plus 4 end fraction  can be written in the form fraction numerator A over denominator x plus 1 end fraction plus fraction numerator B over denominator x minus 2 end fraction plus C over open parentheses x minus 2 close parentheses squared  , where A comma B spaceand C are integers to be found.

Did this page help you?

10a
Sme Calculator
3 marks

The 4th and 8th terms of an arithmetic sequence are 20 and 64 respectively.
Find the first term and the common difference.

10b
Sme Calculator
2 marks

The 12th and 16th terms of an arithmetic sequence differ by 20.
Find the possible values of the common difference.

Did this page help you?

11
Sme Calculator
5 marks

The third term of an arithmetic series is 32.  The eleventh term is 0.  The sum of the first n terms is -44.

Find the value of n.

Did this page help you?

12a
Sme Calculator
3 marks

Traffic is monitored by three average speed cameras, along a stretch of the road where the speed limit is 30 mph.

A car passes the first camera at time zero.
The car’s speed and the time it passes each camera, are recorded.

The results are shown in the table below. 

Camera

1

2

3

Time (hours)

0

0.25

0.5

Speed (mph)

32

38

27

Use the trapezium rule with all the data in the table to estimate the distance between the first and last camera.
Give your estimate to three significant figures.

12b
Sme Calculator
2 marks

Is the car driving within the speed limit? You must show clearly how you achieve your answer.

Did this page help you?

13a
Sme Calculator
3 marks

The drilling rig AlphaBeta began leaking oil into the North Sea following a technical fault. It took 14 hours for engineers to trace and repair the fault.

During that time the rate of oil leaking, measured in tonnes per hour, was recorded every 2 hours.  The results are shown in the table below.

Time

0

2

4

6

8

10

12

14

Rate of leak

0

8

12

18

26

38

18

0

For safety reasons oil rigs are required to shut down and stop all operations until an inspection is carried out should the total amount of oil leaked during any incident exceed 250 tonnes.

Use all the data in the table to decide if the AlphaBeta rig should be shut down or not.

13b
Sme Calculator
1 mark

Explain why using the trapezium rule to estimate the total amount of oil spilled from AlphaBeta in the first 10 hours would be an over-estimate.

Did this page help you?

14
Sme Calculator
4 marks

Find the gradient of the curve with equation

3 x squared y plus 4 x minus y equals 39

at the point with coordinates (2 , 3).

Did this page help you?

15a
Sme Calculator
1 mark

Verify that the point A(1 , 1) lies on the curve with equation

ln open parentheses space x y close parentheses space plus x y squared equals 1.

15b
Sme Calculator
8 marks

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.
Find the area of the triangle OBC.

Did this page help you?

16a
Sme Calculator
4 marks

The functions f(x) and g(x) are defined as follows

 straight f left parenthesis x right parenthesis space equals space 3 x space plus space 5 space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space minus 2 x space space space space space space space space space space space space space space space x element of straight real numbers space

Find

(i)
fg(x)   

(ii)
gf(x

 

16b
Sme Calculator
2 marks

Solve the equation f(x) = g(x).

Did this page help you?

17
Sme Calculator
4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar 2 minus x cubed
(ii)
straight f colon x rightwards arrow from bar sin space x
(iii)
straight f colon x rightwards arrow from bar 1 over x squared
(iv)
straight f colon x rightwards arrow from bar In space x

Did this page help you?

18a
Sme Calculator
1 mark

The function straight f left parenthesis x right parenthesis is defined as

straight f colon x rightwards arrow from bar square root of 25 minus x squared end root space space space space space space space space space space space x element of straight real numbers comma space minus 5 less or equal than x less or equal than 5

Explain why the inverse of straight f left parenthesis x right parenthesis does not exist.

18b
Sme Calculator
2 marks

Suggest an adaption to the domain of straight f left parenthesis x right parenthesis so the following conditions are met:

  • the inverse of space straight f left parenthesis x right parenthesis spaceexists,
  • the graph of y space equals space straight f left parenthesis x right parenthesis lies in the first quadrant only, and,
  • the domain of straight f left parenthesis x right parenthesis is as large as possible.

State the range for your adapted straight f left parenthesis x right parenthesis.

18c
Sme Calculator
3 marks

The domain of straight f left parenthesis x right parenthesis is changed to negative 5 less or equal than space x less or equal than 0.

Find an expression for straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

Did this page help you?

19
Sme Calculator
3 marks

The diagrams below show the graphs of four different functions.

q5a-10-1-solving-equations-hard-a-level-maths-pure

q5-2-10-1-solving-equations-hard-a-level-maths-pure

Match each graph above with the correct statement below.

 

  1. The sign change rule with values of x equals 2 and x equals 4 would indicate a root but has failed due to the discontinuity (asymptote) at x equals 3.
  2. The sign change rule with values of x equals 1 and space x equals 5 spacewould indicate no root but has failed because there are two roots in the interval (1 , 5).
  3. The sign change rule with values ofspace x equals 3 space and x equals 5 would indicate no root but fail as there are two roots in the interval (3 , 5).
  4. The sign change rule with values of x equals 3 and x equals 5 would indicate no root but has failed to find the root as the graph has a turning point at x equals alpha.

Did this page help you?

20a
Sme Calculator
2 marks

The function space straight f left parenthesis x right parenthesis spaceis defined as

 space straight f open parentheses x close parentheses equals sin space 3 x minus ln space 2 x     x greater than 0, where x is in radians.

Find straight f apostrophe left parenthesis x right parenthesis.

20b
Sme Calculator
4 marks

Use the Newton-Raphson method with x subscript 0 equals 0.8 to find a root, α, of the equation straight f open parentheses x close parentheses equals 0, correct to four decimal places.

20c
Sme Calculator
1 mark

The graph of y equals straight f left parenthesis x right parenthesis has a local maximum point at x equals straight beta. Briefly explain why the Newton-Raphson method would fail if the exact value of β was used for x subscript 0.

Did this page help you?

21a
Sme Calculator
3 marks

Show that

integral tan space k x space straight d x equals 1 over k ln space open vertical bar sec space k x close vertical bar plus c

where k is a constant, and c is the constant of integration.

21b
Sme Calculator
5 marks

Use calculus to find the exact value of

integral subscript straight pi over 18 end subscript superscript straight pi over 9 end superscript fraction numerator cosec to the power of 2 space end exponent 3 theta space over denominator 3 space cot space 3 theta end fraction space straight d theta

writing your answer in the form  a ln b , where a and b are rational numbers to be found.

Did this page help you?