Practice Paper 1 (Pure) (Edexcel A Level Maths: Mechanics)

Practice Paper Questions

1
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4 marks

Given that x equals 1 half is a root of the function straight f left parenthesis x right parenthesis equals 2 x cubed plus left parenthesis p squared plus 1 right parenthesis x squared minus 11 x plus 4, find the possible values of space p.

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2a
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3 marks

Given that x is small, show that 

16 minus 7 sin x plus 2 tan 2 xnegative 18 cos xalmost equal to 9 x squared minus 3 x minus 2

2b
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3 marks

Hence, or otherwise, find approximate solution(s) to the equation

16 minus 7 sin x plus 2tan 2 x minus 18 cos x equals 0

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3
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3 marks

Show that the function straight f open parentheses x close parentheses equals 7 x squared minus 2 x left parenthesis x squared plus 5 right parenthesis is decreasing for all x element of straight real numbers.

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4a
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3 marks

In the binomial expansion of  square root of 4 plus begin inline style p over q end style x end root   where p less than 0 less than q, the coefficient of the x squared term is equal to the coefficient of the x cubed term.

Show that space p equals negative 8 q.

4b
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2 marks

Given further that p q equals negative 8 space find the values of p and q.

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5a
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6 marks

Given that straight f left parenthesis x right parenthesis equals x cubed minus left parenthesis 2 square root of 3 right parenthesis x squared plus 3 x

Sketch the graph of y equals straight f left parenthesis x right parenthesis , showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

5b
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3 marks

The functions straight g left parenthesis x right parenthesis and straight h left parenthesis x right parenthesis are defined by the equations

straight g left parenthesis x right parenthesis equals straight f left parenthesis negative x right parenthesis space
straight h left parenthesis x right parenthesis equals straight g left parenthesis x plus a right parenthesis

The graph of straight h left parenthesis x right parenthesis touches the x-axis at the point left parenthesis 5 comma 0 right parenthesis.  Find the value of a, giving your answer as an exact value.

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6
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5 marks

Solve the equation

8 spacecos4 theta minus 5 cos 2 theta minus 2 equals 0 space space space space space space space space space space 0 less or equal than theta less or equal than straight pi

State your answers as multiples of straight pi.

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7
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3 marks

Given that

tanopen parentheses 3 A degree minus 30 degree close parentheses equals negative fraction numerator square root of 3 over denominator 3 end fraction

find the values of A such that  negative 120 degree less or equal than A degree less or equal than 120 degree.

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8a
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4 marks

A manufacturer claims their flask will keep a hot drink warm for up to 7 hours.

In this sense, warm is considered to be 50 degree straight C or higher.

Assuming a hot drink is made at 85 degree straight C and its temperature inside the flask is 50 degree straight C after exactly 7 hours, find:

(i)
a linear model for the temperature of the drink inside the flask of the form T equals a plus b t, and

(ii)
an exponential model for the temperature of the drink inside the flask of the form T equals A e to the power of negative k t end exponent

where T degree straight C is the temperature of the drink in the flask after t hours and a comma b comma A spaceand k are constants.

8b
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2 marks

Compare the rate of change of the temperature of the drink inside the flask of both models after 3 hours.

8c
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1 mark

A user of the flask suggests that hot drinks are only kept warm for 5 hours.
Suggest a reason why the user’s experience may not be up to the claims of the manufacturer.

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9
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8 marks

The diagram below shows part of the graph of y equals x left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

Find the total area of the two shaded regions.

q7-8-1-integration-hard-a-level-maths-pure-screenshot

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10a
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2 marks

The diagram below shows part of the function y space equals space straight f left parenthesis x right parenthesis wherespace straight f left parenthesis x right parenthesis equals 3 x squared sin squared space x space minus 2.

q1a-10-1-solving-equations-hard-a-level-maths-pure

Correct to three significant figures, straight f open parentheses 0.9 close parentheses equals negative 0.509 andspace straight f left parenthesis 3.4 right parenthesis equals 0.265.

Explain why using the sign change rule with these values would not necessarily be helpful in finding the root close to x equals 0.98.

10b
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2 marks

Using suitable values of x, show that there is a root close to x equals 0.98.

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11
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3 marks

Without using a calculator, show that

log subscript 4 space 8 equals log subscript 9 space 27

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12
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3 marks

Prove that straight f left parenthesis x right parenthesis greater or equal than 0 for all values of x,  where begin mathsize 14px style straight f left parenthesis x right parenthesis equals fraction numerator 9 x squared plus 12 x plus 4 over denominator 5 end fraction end style.

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13a
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2 marks

A ball is dropped and bounces such that the height of each bounce is 80% of the previous bounce.

The first bounce of the ball reaches a height of 1.60 m.
Find the height the ball bounces on its 8th bounce 

13b
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2 marks

The ball is considered as no longer bouncing once its bounce height fails to reach 1% of its first bounce height.
Find the number of bounces the ball makes.

13c
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2 marks

Find the total distance travelled by the ball from when it first hits the ground to when it stops bouncing.

13d
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1 mark

Give one reason why this model may be unrealistic.

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14a
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2 marks

The alternating voltage,V , in a domestic electrical circuit, t seconds after it is switched on is modelled by the function

V equals 115 sin omega t plus 115 square root of 3 space end rootcos omega t

Express

115 spacesin omega t plus 115 square root of 3 spacecos omega t

in the form

R spacesin open parentheses omega t plus alpha close parentheses

where R spaceand alpha are constants to be found.  R greater than 0 and is α acute.

14b
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4 marks

In the UK, domestic electricity runs at a frequency,f , of 50 Hertz (Hz).

The constant omega, is given by omega equals 2 pi f.

(i)
Find the initial voltage when a domestic appliance (such as a kettle or TV) is switched on.

(ii)
Find the time at which the voltage first turns negative.
14c
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2 marks
(i)
Find the period of one cycle of voltage in the UK.

(ii)
In the US, the period of one cycle is  begin inline style 1 over 60 end styleseconds
Write down the frequency of US domestic electricity.

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15a
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1 mark

The diagram below shows a sketch of the curves with equations

y equals square root of 25 minus x squared space space end root     and      space y equals 6 x minus x squared minus 5

q9-8-2-further-integration-veryhard-a-level-maths-pure-screenshot

Show that the two curves intersect at the points (3, 4), (4, 3) and (5, 0).

15b
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7 marks

By using the substitution  x equals 5 sin space u, show that

integral square root of 25 minus x squared end root space straight d x equals fraction numerator 25 arcsin space open parentheses x over 5 close parentheses space plus x square root of 25 minus x squared end root over denominator 2 end fraction plus c

where c is the constant of integration.

15c
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5 marks

Using calculus, and your results from parts (a) and (b), show that the total shaded area enclosed by the two curves in the diagram is equal to

fraction numerator 25 pi over denominator 4 end fraction minus 4 minus 25 over 2 open parentheses 2 arcsin space open parentheses 4 over 5 close parentheses minus arcsin space open parentheses 3 over 5 close parentheses close parentheses space units squared

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16
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5 marks

The curve C is described by the equation

ln space y space plus x squared y squared equals 9.

Show that the tangents of the two points on C wherespace y equals 1 spacemeet at the point open parentheses 0 space comma 37 over 19 close parentheses.

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17a
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3 marks

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.
At a certain point along the tank the height of water is measured relative to the calm water level which has a height of 0 space c m.
The height of water in the tank is modelled by the function

h left parenthesis t right parenthesis equals 12 space cos left parenthesis 20 t right parenthesis degree space space space space space space space space space t greater or equal than 0

where h cm is the height of water and t seconds is the time after the peak of the first wave passes the measuring point.

Sketch a graph of h spaceagainst t spacefor 0 less or equal than t less or equal than 54.

17b
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4 marks
(i)
What is the maximum height the water reaches according to the model?

(ii)
How frequent are the waves generated by the tank?

(iii)
How often is the water at its calm level?

(iv)
When will the peak of the 12th wave pass the measuring point?

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