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

Practice Paper Questions

1
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3 marks

The 4th and 8th terms of an arithmetic sequence are 20 and 64 respectively.
Find the first term and the common difference.

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2a
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3 marks

The functions f(x) and  g(x)  are defined as follows

straight f left parenthesis x right parenthesis space equals space 1 half space left parenthesis 4 x space minus space 3 right parenthesis space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space 0.5 x space plus space 0.75 space space space space space space space space space x element of straight real numbers

Find

(i)
straight f g left parenthesis x right parenthesis
(ii)
g straight f left parenthesis x right parenthesis
2b
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3 marks

Write down f−1(x) and state its domain and range.

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3
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3 marks

Solve the equation

log subscript 3 open parentheses x plus 4 close parentheses equals 4 plus 2 space log subscript 3 space x

giving your answers correct to 3 significant figures.

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4
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3 marks

Given that theta is small, write an approximation in terms of theta for

                                             2sin thetapluscos theta minustan2 theta

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5a
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3 marks

A curve has the equation y equals x cubed minus 12 x plus 7.

Find expressions forfraction numerator d y over denominator d x end fractionand fraction numerator d squared y over denominator d x squared end fraction.

5b
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3 marks

Determine the coordinates of the local minimum of the curve.

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6
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6 marks

The diagram below shows the sector of a circle with centre O.  The radii O A and O B are each equal to r cm, and the angle at the centre,A O B , is equal to straight theta radians.  The line D C is perpendicular to the line O B.

q9-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-veryhard

Given that  B Ccolon C O equals 2 colon 3, show that the area of the shaded shape A B C D is given by

1 over 50 r squared open parentheses 25 straight theta minus 9 space tan space theta close parenthesescm2.
                         

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7a
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2 marks

The curve C has equation y equals 3 x squared minus 6 plus 4 over x.  The point Popen parentheses 1 comma space 1 close parentheses lies on C.

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

7b
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3 marks

Show that an equation of the normal to C at point P is x plus 2 y equals 3.

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8
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4 marks

The diagram below shows part of the graph of y equals 2 x plus 3 x squared minus 2 x cubed.

q5-8-1-integration-hard-a-level-maths-pure-screenshot

Find the area of the shaded region labelled R.

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9a
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5 marks

Show that the derivative function of the curve given by 

ln space y minus 2 x y cubed equals 8

is given by

fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y to the power of 4 over denominator 1 minus 6 x y cubed end fraction.

9b
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3 marks

Find the equation of the normal to the curve given in part (a) at the point where y equals 1, giving your answer in the form a x plus b y plus c equals 0 comma where a comma space b and c are integers to be found.

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10
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4 marks

A geometric series is given by 1 plus 2 x plus 4 x squared plus horizontal ellipsis

(i)
Write down the common ratio, r, of the series.
(ii)
Given that the series is convergent, and that sum from n equals 1 to infinity of space open parentheses 2 x close parentheses to the power of n minus 1 end exponent equals 19, calculate the value of x.

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11a
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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 D plotted against t

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-hard

Using the points marked P and Q, find an equation for the line of best fit, giving your answer in the form ln space D equals m t plus ln space c, where m and c are constants to be found.

11b
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2 marks

Hence find estimates for the constants A and space k

11c
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2 marks

The patient is allowed a second injection of the drug once the amount of drug in the bloodstream falls below 1% of the initial dose.
Find, to the nearest minute, how long until the patient is allowed a second injection of the drug.

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12a
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3 marks

A sketch of the graph with equation y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis equals 10 x minus x squared minus 16 is shown below.

Points A spaceand B are the x-axis intercepts and point C is the maximum point on the graph.

q9a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-veryhard

On the diagram above, sketch the graph of y equals negative vertical line 1 fourth space straight f left parenthesis 1 half space x right parenthesis vertical linelabelling the image of the points A comma B and C with A to the power of apostrophe comma B to the power of apostrophe spaceand C apostrophe.

12b
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4 marks

Show that the area of A B C is twice the area of triangle A apostrophe B apostrophe C apostrophe.

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13
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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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14a
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4 marks

The curvespace C space has parametric equations

x equals t squared minus 4   space y equals 3 t

Show that at the point (0 , 6),space t equals 2 spaceand find the value of  fraction numerator straight d y over denominator straight d x end fraction  at this point.

14b
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3 marks

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

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15a
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5 marks

A large block of ice used by sculptors is in the shape of a cuboid with dimensions x m by 2x m by 5x m.  The block melts uniformly with its surface area decreasing at a constant rate of k m2 s-1. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

  fraction numerator 15 k x over denominator 34 end fraction space straight m cubed space straight s to the power of negative 1 end exponent.

15b
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3 marks

In the case whenspace k equals 0.2, the block of ice remains solid enough to be sculpted as long as the rate of melting, by volume, does not exceed 0.05 space straight m cubed space straight s to the power of negative 1 end exponent.

Find the value of x for the largest block of ice that can be used for ice sculpting under such conditions, giving your answer as a fraction in its lowest terms.

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16a
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2 marks

The number of daylight hours, h, in the UK, d days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function

h equals A plus B spacesin open parentheses fraction numerator 2 straight pi over denominator 365 end fraction d close parentheses

where A are B constants.

(i)
Write down the value of A.

(ii)
Given that the maximum number of daylight hours is 16.5, write down the value of B.
16b
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2 marks

For how many days of the year does the number of daylight hours remain below 10? Give your answer as a whole number of days.

16c
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3 marks

If the spring equinox falls on the 21st March, find the dates throughout the year when there are 16 hours of daylight.

16d
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1 mark

The model needs to be adjusted every four years. Suggest a reason why.

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17a
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3 marks

A particle of mass 0.2 kg is acted upon by a force of open parentheses negative 3 bold i plus p bold j plus 4 bold k close parentheses space straight N

Given that the magnitude of the acceleration experienced by the particle under the influence of this force is 65 m s-2, find the possible values of p.

17b
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5 marks

An additional second force with a magnitude of qj N now acts on the particle, where  q greater than 0.   Under the influence of the resultant of the two forces, the particle now experiences an acceleration of magnitude fraction numerator 5 square root of 221 over denominator 2 end fraction space straight m space straight s to the power of negative 2 end exponent.

Find the possible values of q.

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18
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3 marks

Prove that the square of an odd number is always odd.

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