Practice Paper 2 (Pure) (Edexcel A Level Maths: Mechanics)

Practice Paper Questions

1a
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1 mark

Given that space y equals 81 over 16 x to the power of negative 12 end exponent, express each of the following in the form a x to the power of n, where a and n are constants.

space y to the power of 3 over 4 end exponent

1b
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1 mark

space y to the power of negative 1 half end exponent

1c
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2 marks

open parentheses y to the power of 1 half end exponent close parentheses to the power of negative 3 end exponent

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2
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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.

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3
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4 marks

A rainbow-shaped logo is formed from two semicircles as shown below.
The radius of the smaller semicircle is 1 cm less than the radius of the larger semicircle.

q5-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-medium

Find the area of the logo.

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4a
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3 marks

The graph of the ellipse E shown below is defined by the parametric equations

space space x equals 2 cos open parentheses space theta plus pi over 3 space space close parentheses    space space y equals 4 sin space theta    negative pi less or equal than theta less or equal than pi

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Find an expression for  fraction numerator straight d y over denominator straight d x end fraction  in terms of θ.

4b
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4 marks

Find the equation of the tangent to E, at the point where  theta equals negative space pi over 6 , giving your answer in the form y equals a minus b x, where a and b are real numbers that should be given in exact form.

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5
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3 marks

The diagram below shows a sketch of the curve with equation  y equals 1 plus 2 x minus 1 fourth x squared.

q1-8-2-further-integration-hard-a-level-maths-pure-screenshot

The point  P left parenthesis x comma space y right parenthesis lies on the curve.

The shaded rectangle shown has width  δx and height y.

Calculate

limit as delta x rightwards arrow 0 of space sum from x equals 1 to 8 of open parentheses space 1 plus 2 x minus 1 fourth x squared close parentheses space delta x space

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6a
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1 mark

The function f(x) is defined as

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

Explain why the inverse of  f(x) does not exist.

6b
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2 marks

Suggest an adaption to the domain of f(x) so the following conditions are met:

  • the inverse of f(x) exists,
  • the graph of y = f(x) lies in the first quadrant only, and,
  • the domain of f(x) is as large as possible.

State the range for your adapted f(x).

6c
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3 marks

The domain of f(x) is changed to negative 5 less or equal than 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.

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7a
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2 marks

The leakage rate of water from a pipe, L space 1 space straight s to the power of negative 1 end exponent (litres per second), is directly proportional to flow rate, straight s space straight m space straight s to the power of negative 1 end exponent (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was 0.31 space straight s to the power of negative 1 end exponent when the flow rate was 0.6 space straight m space straight s to the power of negative 1 end exponent.

Show that the constant of proportionality is 0.5 and hence write down an equation connecting L spaceand straight s.

7b
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2 marks

Find the leakage rate when the flow rate is 1.8 space straight m space straight s to the power of negative 1 end exponent.

7c
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2 marks

The flow rate is reduced should the leakage rate exceed 0.8 space l space s to the power of negative 1 end exponent.
Find the maximum possible flow rate before it is reduced.

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8a
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4 marks

Given that sum from r equals 1 to k of space 7 cross times 3 to the power of r space space equals 620004,

Show that k equals fraction numerator log space 59049 over denominator log space 3 end fraction

8b
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3 marks

For this value of k, calculate sum from r equals 0 to k plus 3 of space 7 cross times 3 to the power of r.

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9a
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2 marks

An exponential model of the form  D equals A e to the power of negative k t end exponent  is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, t hours after the drug was administered by injection. A  and k are constants.

The graph below shows values of In D plotted against t with a line of best fit drawn.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-medium

(i)        Use the graph and line of best fit to estimate ln space D at time t equals 0.

(ii)       Work out the gradient of the line of best fit.

9b
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1 mark

Use your answers to part (a) to write down an equation for the line of best fit in the form ln space D equals m t plus ln space c,  where m and c are constants.

9c
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1 mark

Show that D equals A e to the power of negative k t end exponent can be rearranged to give ln space D equals negative k t plus ln space A

9d
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2 marks

Hence find estimates for the constants A spaceand k.

9e
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2 marks

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

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10a
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1 mark

In triangle ABC, point F lies on AB and point G lies on BC.  

F divides AB in the ratio m:n.  

The line segment FG is parallel to AC.

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Explain why stack B G with rightwards arrow on top equals lambda stack B C with rightwards arrow on top for some constant lambda, where 0 less or equal than lambda less or equal than 1.

10b
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4 marks

Given that space stack A B with rightwards arrow on top equals bold a  and space stack A C with rightwards arrow on top equals bold b,   show that

stack F G with rightwards arrow on top equals open parentheses fraction numerator n over denominator m plus n end fraction minus lambda close parentheses bold a plus lambda bold b

10c
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3 marks

Using your result from (b), prove that G divides BC in the ratio n:m.

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11a
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2 marks

The function,space straight f left parenthesis x right parenthesis space is defined by straight f open parentheses x close parentheses equals 1 over e to the power of x minus x plus 1 space space space space space space space space space space space space space space x element of straight real numbers

Show that the equation straight f open parentheses x close parentheses equals 0 can be written in the form

space space x equals e to the power of negative x end exponent plus 1

11b
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2 marks

On the same diagram sketch the graphs of y equals x and y equals e to the power of negative x end exponent plus 1.

11c
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2 marks

The equation straight f open parentheses x close parentheses equals 0 has a root, α, close to x equals 1.
The iterative formula     xn+1=e-xn+1      with x0=2 is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root .

11d
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3 marks
(i)
Find the values ofspace x subscript 1 comma space x subscript 2 spaceand x subscript 3, giving each correct to three significant figures.
(ii)
How many iterations are required before x subscript n and x subscript n minus 1 end subscript agree to two decimal places?

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12a
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4 marks

Use an identity for cos 2 A spaceto derive an identity for cos 4 A, in terms of cos A.

12b
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5 marks

Hence, or otherwise, solve the equation

         2 cos space 4 x equals 7 sin squared space x minus 2 space space space space space space space space 0 less or equal than space x less or equal than straight pi

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13a
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1 mark

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

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The base of each triangle is 2x metres, and the equal sides are each y metres in length.

Although x and y can vary, the total amount of fencing to be used is fixed at P metres.

Explain why 0 less than x less than space space P over 6.

13b
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4 marks

Show that

    space space A squared equals 4 over 9 P squared x squared minus 16 over 3 P x cubed

where A is the total area of the garden bed.

13c
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4 marks

Using your answer to (b) find, in terms of P, the maximum possible area of the garden bed.

13d
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1 mark

Describe the shape of the bed when the area has its maximum value.

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14
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5 marks

Find

integral open parentheses 2 tan space x space plus 3 sec space x close parentheses squared space straight d x

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15a
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5 marks

Show that the general solution to the differential equation

y cot space x space space fraction numerator d y over denominator d x end fraction equals y squared plus 3

can be written in the form

y squared plus 3 equals A space sec to the power of 2 space end exponent x

where A is a constant.

15b
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6 marks

Find the particular solution to the following differential equation, using the given boundary condition

e to the power of x squared end exponent space fraction numerator d y over denominator d x end fraction equals 2 x space cosec space 3 y space           space x equals 0 comma space space space y equals pi over 3

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