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

Practice Paper Questions

1
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1 mark

State the set of values of x which satisfies the inequality

open parentheses x space plus space 2 close parentheses open parentheses 3 x minus 1 close parentheses less than 0

Tick (✓) one box.

open curly brackets x space colon space x space less than space minus 1 third space or space x space greater than 2 close curly brackets square
open curly brackets x space colon space minus 2 space less than space x less than 1 third close curly brackets square
open curly brackets x space colon space x space less than space minus 2 space or space x greater than 1 third close curly brackets square
open curly brackets x space colon space minus 1 third space less than space x space less than space 2 close curly brackets square

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2
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1 mark

Given that y space equals space e to the power of 5 x end exponent find fraction numerator straight d y over denominator straight d x end fraction.

Circle your answer.

fraction numerator straight d y over denominator straight d x end fraction equals e to the power of 5 x end exponent fraction numerator straight d y over denominator straight d x end fraction equals 5 x e to the power of 5 x end exponent fraction numerator straight d y over denominator straight d x end fraction equals 5 e to the power of 5 x end exponent fraction numerator straight d y over denominator straight d x end fraction equals 1 fifth e to the power of 5 x end exponent

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3
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1 mark

A geometric sequence has a sum to infinity of negative 1 half.

A second sequence is formed by multiplying each term of the original sequence by negative 10.

What is the sum to infinity of the new sequence?

Circle your answer.

5 –5 the sum to infinity does not exist negative 1 half

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4
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1 mark

Milo is attempting to use proof by contradiction to show that the result of multiplying a prime number by a multiple of 3 is never a prime number.

Select the assumption he should make to start his proof.

Tick (✓) one box.

Every prime multiplied by a multiple of 3 is never prime. square
Every prime multiplied by a multiple of 3 is prime. square
There exists a prime and a multiple of 3 whose product is prime. square
There exists a prime and a multiple of 3 whose product is not prime. square



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5a
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2 marks

The line l passes through the points (3, 4) and (9, 2).

Find the equation of the line l, giving your answer in the form y equals m x plus c.

5b
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1 mark

Write down the gradient of a line perpendicular to l.

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6
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3 marks

The line segment AB is the diameter of a circle.

A has coordinates (-7,-9) and B has coordinates (9, 3).

Find the coordinates of the centre of the circle and the length of the diameter.

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7a
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2 marks

An arithmetic series is defined by

          S subscript n equals left parenthesis k plus 13 right parenthesis plus left parenthesis 2 k plus 9 right parenthesis plus left parenthesis 3 k plus 5 right parenthesis plus horizontal ellipsis plus u subscript n plus horizontal ellipsis

Find an expression for n in terms of u subscript n and k.

7b
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5 marks

For a particular value of nu subscript n equals negative 16  and S subscript n equals negative 11.

Find the value of k.

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8a
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1 mark

The diagram below shows the graphs of y equals x subscript blank and space y equals ln space open parentheses x minus 1 close parentheses plus 3 space.

q6-10-1-solving-equations-medium-a-level-maths-pure

The iterative formula

x subscript n plus 1 end subscript equals ln open parentheses space x subscript n minus 1 close parentheses plus 3

is to be used to find an estimate for a root, alpha, of the functionspace straight f left parenthesis x right parenthesis.

Write down an expression for straight f left parenthesis x right parenthesis.

8b
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2 marks

Using an initial estimate, x subscript 0 space equals space 2, show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates x subscript 1 comma x subscript 2 comma x subscript 3 comma horizontal ellipsis will converge to.
Hence deduce whether alpha  is the x-coordinate of point S or point T.

8c
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2 marks

Find the estimates x subscript 1 comma x subscript 2 comma space x subscript 3 and x subscript 4, giving each to three decimal places.

8d
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2 marks

Confirm that α = 4.146 correct to three decimal places.

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9
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4 marks

Show that

sin theta open parentheses cosec squared space theta minus 2 close parentheses identical to fraction numerator cos space 2 theta over denominator sin space theta end fraction

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10a
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2 marks

Show that the equation  cosec2 x equals 2 cosec x minus 1  can be written as

open parentheses cosec space x minus 1 close parentheses squared equals 0

10b
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3 marks

Hence, or otherwise, solve the equation

cosec2 x equals 2 cosec x minus 1 comma space space space space space space space space space space space space space space space space space space space space space space space space minus 2 straight pi less or equal than space x less or equal than 2 straight pi

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11a
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1 mark

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, m g, in an object of age t years is

m equals m subscript 0 e to the power of negative k t end exponent

where m subscript 0 and k are constants.

For an object initially containing 100g of carbon-14, write down the value of m subscript 0.

11b
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2 marks

Briefly explain why, if m subscript 0 equals 100,m  will equal 50g  when t equals 5700 years.

11c
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2 marks

Using the values from part (b), show that the value of k spaceis 1.22 cross times 10 to the power of negative 4 end exponent to three significant figures.

11d
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2 marks

A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

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12a
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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.

12b
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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.

12c
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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

12d
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2 marks

Hence find estimates for the constants A spaceand k.

12e
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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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13
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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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14
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4 marks

Use the substitution space u equals 4 x plus 1 spaceto find

integral subscript 2 superscript 6 4 open parentheses 4 x plus 1 close parentheses to the power of 1 half end exponent d x

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15a
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5 marks

Show that the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 3 x squared y comma space space space space space space space space space space space space space space space space space y not equal to 0


is

y equals A e to the power of x cubed end exponent


where A is a constant.

15b
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3 marks

On the same set of axes sketch the graphs of the solution for the instances where

(i)
the constant A is greater than 0
(ii)
the constant A is less than 0

In each case be sure to state where the graph intercepts the y-axis.

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16a
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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.

16b
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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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17a
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2 marks

straight f left parenthesis x right parenthesis equals 6 x cubed minus 19 x squared plus 11 x plus 6

Show that straight f left parenthesis x right parenthesis equals left parenthesis 2 x minus 3 right parenthesis left parenthesis a x squared plus b x plus c right parenthesis where a comma space b and c are constants to be found.

17b
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4 marks

Hence factorise straight f left parenthesis x right parenthesis completely.

17c
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2 marks

Write down all the real roots of the equation straight f left parenthesis x right parenthesis equals 0.

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18a
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2 marks

The diagram below shows a sketch of the curve defined by the parametric equations

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

q6-9-2-further-parametric-equations-very-hard-a-level-maths-pure

(i)
Write down the equations of the two horizontal tangents to the curve.

(ii)

Write down the equations of the two vertical tangents to the curve.

18b
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8 marks

The four tangents from part (a) create a rectangle around the curve as shown below.

Find the percentage of the area of the rectangle enclosed by the curve

(the shaded area on the diagram).

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19a
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4 marks

The circle sector O A B is shown in the diagram below.

The angle at the centre is theta radians , and the radii O A and O B are each equal to r cm

Additionally,C D is parallel to A B, so that  A D equals B C and O D equals O C.

q7a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-hard

In the case when A D equals B C equals 1 cm, show that the area of the shaded shape A B C D is given by 1 half straight theta r squared minus 1 half open parentheses straight r minus 1 close parentheses squared sin theta.

.

19b
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3 marks

Show that for small values of theta, the area of A B C D is approximately 1 half theta open parentheses 2 straight r minus 1 close parentheses.

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20
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3 marks

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

fraction numerator sin squared space theta plus cos space theta over denominator tan space theta plus sin space theta end fraction

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