Practice Paper Pure 2 (Edexcel International A Level Maths: Pure 1)

Practice Paper Questions

1a
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3 marks

Find the first three terms in the expansion of left parenthesis 2 minus 3 x right parenthesis to the power of 7.

1b
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

         left parenthesis 1 minus 2 x right parenthesis left parenthesis 2 minus 3 x right parenthesis to the power of 7 almost equal to 128 minus 1600 x plus 8736 x squared 
                 

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2a
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2 marks

The first k terms of a series are given by sum from r equals 1 to k of space open parentheses 7 plus 5 r close parentheses.

Show that this is an arithmetic series, and determine its first term and common difference.

2b
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3 marks

Given that   sum from r equals 1 to k of space open parentheses 7 plus 5 r close parentheses equals 1190 ,

(i)
Show that open parentheses 5 k plus 119 close parentheses open parentheses k minus 20 close parentheses equals 0.
(ii)
Hence find the value of k.

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3a
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5 marks

The trapezium rule is to be used to estimate the integral

            integral subscript negative 2 end subscript superscript 0 space x cubed plus 8 space d x

By completing the table of values below, use the trapezium rule to estimate the integral given above. 

 x

-2

-1.5

-1

-0.5

0

y space equals space x cubed plus 8 

 

 

 

 

 

3b
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3 marks
i)
By finding the exact value of the integral above, find the percentage error of your estimate from part (a).
ii)
Explain how you could increase the accuracy of your trapezium rule estimate.

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4a
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2 marks

straight f left parenthesis x right parenthesis equals 3 x to the power of 4 plus x cubed minus 12 x squared minus 49 x minus 15

Show that straight f left parenthesis x right parenthesis equals left parenthesis 3 x plus 1 right parenthesis left parenthesis a x cubed plus b x squared plus c x plus d right parenthesis where a comma space b comma space c and d are constants to be found.

4b
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5 marks

Given that left parenthesis x minus 3 right parenthesis is a factor of straight f left parenthesis x right parenthesis, factorise straight f left parenthesis x right parenthesis completely.

4c
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2 marks

Hence show that the equation straight f left parenthesis x right parenthesis equals 0 space has exactly 2 real roots.

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5a
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2 marks

June is to start training in order to run a marathon.
For the first week of training she will run a total of 3 miles.
Each subsequent week she’ll increase the total number of miles run by x miles.
June intends training for y weeks and will run a total distance of 570 miles during the training period.

Write down an expression in x for the number of miles June will run in the 10th week of training.

5b
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2 marks

Write down an equation in x and y for the total distance June will run during the training period.

5c
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2 marks

Given that June runs twice as far in week 4 than in week 2, find the values of x and y.

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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 angle A C B equals 90 degree. 

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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6 marks

The second and fifth terms of a geometric series are 13.44 and 5.67 respectively.  The series has first term a and common ratio r.

By first determining the values of a and r, calculate the sum to infinity of the series.

7b
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2 marks

Calculate the difference between the sum to infinity of the series and the sum of the first 20 terms of the series. Give your answer accurate to 2 decimal places.

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8a
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3 marks

Find the x-coordinates of the stationary points on the graph with equation

y equals x cubed minus 6 x squared plus 9 x minus 1.

8b
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2 marks

Find the nature of the stationary points found in part (a).

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9a
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2 marks

The diagram below shows a sketch of the curves with equations

  space y equals x cubed plus 3 spaceand y equals negative x cubed plus 2 x plus 3

q11-8-2-further-integration-hard-a-level-maths-pure-screenshot

Find the x-coordinates of the points of intersection of the two graphs.

9b
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5 marks

Use calculus to find the total shaded area enclosed by the two graphs.

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10
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4 marks

The set of numbers S is defined as all positive integers greater than 5 and less than 10.

Prove by exhaustion that the square of all values in S differ from a multiple of 5 by 1.

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11
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5 marks

Solve the equation  3 sinopen parentheses 2 x plus 30 degree close parenthesesequalstanopen parentheses 2 x plus 30 degree close parentheses   for  negative 180 degree less or equal than space x less or equal than 180 degree, giving your answers to 1 decimal place where appropriate.

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12
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5 marks

Solve the equation  2 sin2 thetaequalscos thetaplus 1 for  negative 180 degree less or equal than theta less or equal than 180 degree

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