Practice Paper 1 (Pure) (OCR A Level Maths: Pure)

Practice Paper Questions

1
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4 marks

Solve the inequality x squared minus 5 x greater than 6.

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2
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3 marks

On the same diagram sketch the following position vectors

(i)
3i + 4j,
(ii)
-5i,
(iii)

-8i - 6j.

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3
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4 marks

The position vector stack A B with rightwards arrow on top is given by stack A B with rightwards arrow on top equals 6 bold i plus 3 bold j.

(i)
Find the magnitude of stack A B with rightwards arrow on top, giving your answer in the form p square root of q where p and q are integers to be found.

(ii)
Find the angle between stack A B with rightwards arrow on top and the positive x-axis, giving your answer in degrees to one decimal place.

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4
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3 marks

State the range for the following functions based on the given domains.

(i)
straight f colon space x rightwards arrow from bar e to the power of x               x element of straight real numbers
(ii)
straight f colon space x rightwards arrow from bar x squared space plus space 1        x element of straight real numbers
(iii)
straight f colon space x rightwards arrow from bar 1 over x               x element of straight real numbers

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5a
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3 marks

The graph of y space equals space straight f left parenthesis x right parenthesis is shown below.

q5a-2-8-hard-aqa-a-level-maths

(i)
Use the graph to write down the domain and range of straight f left parenthesis x right parenthesis.
(ii)
Write down the equation of the dotted line on the graph.
5b
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2 marks

On the diagram above sketch the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis.

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6a
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2 marks

Lloyd is to start training in order to run a marathon.

For the first week of training he will run a total of 2 miles.

Each subsequent week he’ll increase the total number of miles run by 3 miles.

He intends training for 15 weeks.

Calculate how far Lloyd will run during his eighth week of training

6b
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2 marks

Work out how much further Lloyd will run in his last week of training compared to his first.

6c
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3 marks

Find the total number of miles Lloyd will run across all 15 weeks of his training schedule.

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7a
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2 marks

Show that the point (0 , π) lies on the curve with equation

3 tan space y equals 2 x y

7b
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction.

7c
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2 marks

Find the gradient at the point (0 ,π).

7d
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3 marks

Hence find an equation of the tangent to the curve at the point (0, π).

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8
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5 marks

Prove, from first principles, that the derivative of space 2 x cubed spaceis 6 x squared.

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9
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4 marks

A geometric series has first term a, and its terms are connected by the relationship u subscript n plus 4 end subscript equals 9 u subscript n for all n greater or equal than 1.

Given that all the terms of the series are positive, show that the sum of the first twelve terms of the series may be written in the form

            S subscript 12 equals k a left parenthesis square root of n plus 1 right parenthesis

where k and n are positive integers and square root of n is a surd.

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10
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8 marks

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

(i)
Given that the initial volume of water in the container is 4000 litres and that after 10 minutes the volume of water in the container has dropped by 30%, write down and solve a differential equation connecting the volume, V, of water in the container to the time, t.

 

(ii)
What does your solution predict will happen to the volume of water in the container after a very long time?

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11a
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4 marks

Show that 5 space sin space theta plus 12 space cos space theta can be written in the form R space sin open parentheses theta plus straight alpha degree close parentheses where R greater than 0 and 0 degree less than alpha less than 90 degree.

11b
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4 marks

Sketch the graph of y equals 5 spacesin x plus 12 spacecos x for 0 degree less or equal than space x less or equal than 360 degree.
Label any points where the graph intercepts the coordinate axes.

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12a
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5 marks

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

q5a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-hard

Given that the area of triangle O A Bequals 5.64 cm2, find the area of the shaded segment. Give your answer correct to 3 significant figures.

12b
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3 marks

Find the perimeter of the sector O A B, giving your answer correct to 3 significant figures.

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13a
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2 marks

Show that the equation space x cubed minus 5 x equals 2 spacecan be rewritten as

x equals 1 fifth left parenthesis x cubed minus 2 right parenthesis.

13b
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3 marks

Starting with x subscript 0 equals 1, use the iterative formula

x subscript n plus 1 end subscript equals 1 fifth open parentheses space x subscript n superscript 3 minus 2 close parentheses

to find values for x subscript 1 comma x subscript 2and x subscript 3, giving each to four decimal places where appropriate.

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14
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5 marks

Use the substitution  u equals 2 plus ln space x space space to show that

integral fraction numerator 1 over denominator x open parentheses 2 plus ln space x close parentheses cubed end fraction space straight d x equals fraction numerator negative 1 over denominator 2 open parentheses ln space x space plus 2 close parentheses squared end fraction plus c

where c is the constant of integration.

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15
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6 marks

Use integration by parts to find an expression for

integral negative 3 x sin space x space straight d x.

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16
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5 marks

Use a suitable substitution to find the following

integral 8 x space sin space left parenthesis 3 x squared plus 1 right parenthesis space space straight d x

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17
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3 marks

Show that

fraction numerator straight d over denominator straight d x end fraction open square brackets a to the power of k x end exponent close square brackets equals k a to the power of k x end exponent ln space a

where a and k are constants.

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18
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7 marks

Find the coordinates of the stationary points, and their nature, on the graph with equation  y equals e to the power of 0.4 x end exponent left parenthesis x squared plus 3 x minus 4 right parenthesis.

Find the coordinates of any points of inflection, determine if these are also stationary points and find the intervals in which the graph is convex and concave.

Give values to three significant figures.

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