Practice Paper Pure 1 (Edexcel International A Level Maths: Pure 4)

Practice Paper Questions

1a
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3 marks

Simplify the following expressions, giving your answers in the form a x to the power of n where a and n are rational numbers and any fractions are in lowest terms.

open parentheses 8 x squared close parentheses to the power of negative 1 third end exponent cross times 1 fourth x to the power of negative 1 third end exponent

1b
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3 marks

open parentheses 2 over 9 x to the power of 1 half end exponent cross times 1 over 18 x to the power of negative 3 over 4 end exponent close parentheses to the power of negative 1 fourth end exponent

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2a
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2 marks

Write the quadratic function y equals 4 x squared plus 8 x minus 5 in the form y equals a left parenthesis x plus b right parenthesis squared plus c where a, b and c are integers to be found.

2b
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1 mark

Write down the minimum point on the graph of y equals 4 x squared plus 8 x minus 5.

2c
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3 marks

Sketch the graph of space y equals 4 x squared plus 8 x minus 5, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

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3
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3 marks

Solve the inequality open parentheses x plus 2 close parentheses squared space greater than space 5.

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4a
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2 marks

The circle sector O A B is shown in the diagram below.

The angle at the centre is straight pi over 3 radians , and the line segments O C and B C have lengths of 8 cm and pcm respectively.

Additionally, C D is parallel to A B, so that A D equals B C and O D equals O C.

q11a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-medium

Show that the area of the sector O A B is  straight pi over 6 open parentheses p plus 8 close parentheses squared cm2.

4b
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2 marks

Show that the area of the triangle O C D is 16 square root of 3 cm2.

4c
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4 marks

Given that the area of the shaded shape A B C D is open parentheses fraction numerator 50 straight pi over denominator 3 end fraction minus 16 square root of 3 close parentheses cm2.  find the value of p.

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5
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7 marks

Solve the simultaneous equations

            x space plus space y space equals space 4
x squared long dash space 4 x space long dash space 3 y space equals space 0

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6a
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6 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis. The graph intersects the coordinate axes at the two marked points A left parenthesis 0 comma 6 right parenthesis and B left parenthesis 3 comma 0 right parenthesis.  The graph has two asymptotes as shown, with equations  y equals fraction numerator space 20 over denominator 3 end fraction   and  x equals fraction numerator space 10 over denominator 3 end fraction .

q7a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-hard

In separate diagrams, sketch the curves with equation

(i)

y equals straight f left parenthesis 5 x right parenthesis

(ii)
space y equals negative straight f left parenthesis x right parenthesis

On each diagram, give the coordinates of the images of points A and B under the given transformation, as well as stating the equations of the transformed asymptotes.

6b
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1 mark

The graph of y equals a straight f left parenthesis x right parenthesis has an asymptote with equation y equals 2.  Find the value of a.

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7a
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2 marks

The graph below shows part of the curve with equation y space equals space sin space open parentheses x space plus space k close parentheses,  where the angle is measured in radians and k is a constant.

screenshot-2023-07-22-at-8-00-27-pm

A student states that there are an infinite number of possible values for k. Is the student correct?  You must explain your answer fully.

7b
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2 marks

Another student claims that the curve could also be the graph of the equation y space equals space cos space open parentheses x space plus space k close parentheses. Find a value for k to show that the student is correct.

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8
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5 marks

The line l subscript 1 has equation 3 x minus 2 y plus 10 equals 0 and crosses the x-axis at point A

The line l subscript 2 is perpendicular to l subscript 1 and crosses the x-axis at (9, 0).

l subscript 2 crosses the y-axis at point B.

Find the area of the triangle O A B, where O is the origin.

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9a
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3 marks

The line l passes through the points open parentheses p comma 2 p close parentheses and open parentheses 3 p comma 9 p close parentheses.

Find an equation for the line l

9b
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2 marks

l intercepts the y-axis at (0, 3).

Find the value of p.

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10a
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3 marks

On the same diagram, sketch the graphs of space y equals x cubed minus 2 x squared minus 8 x and space y equals 1 over x.

10b
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1 mark

Use your graph to determine the number of solutions to the equation

x cubed minus 2 x squared minus 8 x equals 1 over x.

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11a
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2 marks

The equation 4 k minus 6 k x minus x squared equals 0 has two distinct real roots, alpha space and space beta.
k is a negative constant and 0 less than alpha less than beta.

Sketch the graph of y equals 4 k minus 6 k x minus x squared, labelling the points where the graph crosses the coordinate axes.

11b
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3 marks

Find the possible values of k.

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12a
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7 marks

A curve has equation y equals 5 minus open parentheses x minus 3 close parentheses squared.  

A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6.  

C is the point of intersection of the tangents to the curve at A and B

Find the coordinates of point C.

12b
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2 marks

Calculate the area of triangle ABC.

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13a
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3 marks

Use calculus to work out the following integrals:

integral space open parentheses 3 x space minus space fraction numerator square root of 4 x space end root space plus space 5 x to the power of 4 over denominator x end fraction close parentheses space d x

13b
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3 marks

integral space open parentheses fraction numerator 3 x over denominator square root of x end fraction space plus space open parentheses 2 x space plus space 1 close parentheses squared close parentheses space d x

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