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

Practice Paper Questions

1
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3 marks

Given that left parenthesis x minus 4 right parenthesis space is a factor of x cubed minus k x squared minus 4 x plus 16, find the value of k.

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2a
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2 marks

Write the quadratic function y equals x squared plus 8 x minus 9 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 x squared plus 8 x minus 9.

2c
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3 marks

Sketch the graph of y equals x squared plus 8 x minus 9, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

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3a
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5 marks

A sequence is defined for k greater or equal than 1 by the recurrence relation

 u subscript k plus 1 end subscript equals left parenthesis p minus 2 right parenthesis u subscript k minus 2 comma space space space space space space space space space space u subscript 1 equals 3

where p is a constant.

Given that the sequence is periodic with order 2, and given as well that u subscript 1 not equal to u subscript 2,

Find the value of  p.

3b
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2 marks

For the value of p found in part (a),

Calculate  sum from n equals 50 to 900 of space u subscript n

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4a
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2 marks

The function straight f left parenthesis x right parenthesis is defined as

 space straight f open parentheses x close parentheses equals x squared minus ln space left parenthesis x plus 2 right parenthesis space   space x greater than 0

Use the sign change rule to show there is a root to the equationspace straight f open parentheses x close parentheses equals 0 space in the interval 1 less than x less than 1.2.

4b
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2 marks

Find straight f apostrophe open parentheses x close parentheses.

4c
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4 marks

Use the Newton-Raphson method with x subscript 0 equals 1 to find the root in the interval 1 less than x less than 1.2 correct to three decimal places.

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5a
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2 marks

Stephen opens a savings account with £600.

Compound interest is paid annually at a rate of 1.2%.

At the start of each new year Stephen pays another £600 into his account.

Show that at the end of two years Stephen has £1221.69 in the account

5b
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2 marks

Show that at the end of year n, the amount of money, in pounds, Stephen will have in his account is given by

         600 left parenthesis 1.012 plus 1.012 squared plus 1.012 cubed plus midline horizontal ellipsis plus 1.012 to the power of n right parenthesis

5c
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2 marks

Hence show that the total amount, in pounds, in Stephen’s account after n years is

         50 space 600 left parenthesis 1.012 to the power of n minus 1 right parenthesis.

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6
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5 marks

In the triangle ABCstack A B with rightwards arrow on top equals negative 2 bold i plus 3 bold j minus bold k space and  stack A C with rightwards arrow on top equals negative 5 bold i minus 4 bold j minus 7 bold k.

q1-11-1-vectors-in-2-dimensions-hard-a-level-maths-pureShow that angle B A C equals 81.9 degree to 1 d.p.

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7
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7 marks

The line x plus y equals c intersects the circle x squared plus y squared minus 6 x plus 10 y minus 16 equals 0 at exactly two points.  Find the range of possible values of c.

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8a
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2 marks

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years

A model for the mass of carbon-14, y g, in an object originally containing 100 g,
at time t years is

y equals 100 e to the power of negative k t end exponent

where k is a constant.

Find the value of k, giving your answer to three significant figures.

8b
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2 marks

The object is considered as having no radioactivity once the mass of carbon-14 it contains falls below 0.5 g. Find out how old the object would have to be considered non-radioactive.

8c
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2 marks

A different object currently contains 25g of carbon-14.
In 500 years’ time how much carbon-14 will remain in the object?

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9
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4 marks

Express  fraction numerator x squared minus 3 over denominator open parentheses x minus 3 close parentheses open parentheses x plus 2 close parentheses squared end fraction  as partial fractions.

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10a
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

         left parenthesis 2 plus 3 x right parenthesis to the power of negative 1 end exponent left parenthesis 3 minus 2 x right parenthesis to the power of negative 2 end exponent almost equal to 1 over 18 minus 1 over 108 x plus 19 over 216 x squared

10b
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3 marks

Find the percentage error between your calculator answer and the approximation in part (a) when x equals 0.1, giving your answer to one decimal place.

10c
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2 marks

For which values of x is the approximation in part (a) valid?

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11a
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2 marks

Show that the equation  3 tan2 x equals 18 minus 2 sec x  can be written as

3 sec2 x plus 2sec x minus 21 equals 0

11b
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4 marks

Hence, or otherwise, solve the equation

3 tan2 x equals 18 minus 2 sec x comma space space space space space space space space space space space space space space space space space space space space space space minus straight pi less or equal than space x less or equal than straight pi

Give your answers to three significant figures.

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12a
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3 marks

A stretch of road along Equality Street has a speed limit of 70 mph and traffic is monitored by six average speed cameras.

Ricky is driving his car along Equality Street and passes the first camera at time zero.  Ricky’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

Camera

1

2

3

4

5

6

Time (hours)

0

0.05

0.1

0.15

0.2

0.25

Speed (mph)

68

72

69

71

70

70

Use all the results above to estimate the distance between the first and last camera.

12b
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2 marks

A driver will receive a speeding ticket if their average speed between the first and last camera exceeds the speed limit.
Should Ricky receive a speeding ticket or not?  Justify your answer.

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13a
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2 marks

The diagram below shows a sketch of the curves with equations

  space y equals x cubed plus 3 space     and       y equals negative x cubed plus 2 x plus 3

q11-8-2-further-integration-hard-a-level-maths-pure-screenshot

Find the x-coordinates of the points of intersection of the two graphs.

13b
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5 marks

Use calculus to find the total shaded area enclosed by the two graphs.

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14a
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1 mark

A soft ball is thrown upwards from the top of a building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals H plus 9.8 t minus 4.9 t squared space space space space space space space space t greater than 0

What does the constant H indicate in the function?

14b
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2 marks

At what time is the ball at the same height as when it was thrown?

14c
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2 marks

Find in terms of H comma space how long it takes for the ball to first hit the ground.

14d
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2 marks

How much longer does a ball launched from a 25 m tall building stay in the air for compared to a ball launched from a 15 m tall building?

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15a
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3 marks

The graph of the curve C shown below is defined by the parametric equations

x equals 2 cos space 3 theta space    y equals 5 sin theta    space 0 less or equal than theta less or equal than 2 pi

usbvpK9r_q4-9-2-further-parametric-equations-hard-a-level-maths-pure

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

15b
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4 marks
(i)
Show that the gradient of the tangent to C, at the point where  theta equals space pi over 4,  is  negative 5 over 6 .

(ii)
Hence find the equation of the tangent to C at the point where  theta equals space pi over 4 .

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16
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4 marks

Differentiate  fraction numerator 5 x to the power of 7 over denominator sin space 2 x space space end fraction with respect to x.

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17
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4 marks

Prove by contradiction that if x cubed is odd, then x must be odd.

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