Practice Paper 2 (Pure & Mechanics) (AQA A Level Maths: Statistics)

Practice Paper Questions

1
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1 mark

Four possible sketches of y space equals space a x squared space plus space b x space plus space c are shown below.

Given b squared space minus space 4 a c space less than space 0, which sketch is the only one that could possibly be correct?

Tick (✓) one box.

q1-aqa-a-level-maths-practise-paper-set-a-mechanics

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2
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1 mark

A curve has equation y space equals space g left parenthesis x right parenthesis.

The curve has a maximum point at x space equals space 2 which is concave at that point.

It is given that g apostrophe left parenthesis 2 right parenthesis space equals space a and g " left parenthesis 2 right parenthesis space equals space b where a and b are real numbers.

Identify which one of the statements below must be true.

Circle your answer.

g apostrophe open parentheses 2 close parentheses not equal to 0 g double apostrophe open parentheses 2 close parentheses less or equal than 0 g apostrophe open parentheses 2 close parentheses greater than 0 g apostrophe apostrophe open parentheses 2 close parentheses greater than 0

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3
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1 mark

A sequence is defined by

u subscript 1 equals k space and space u subscript n plus 1 end subscript equals k minus u subscript n


Find

sum from n equals 1 to 60 of u subscript n

Circle your answer.

30 k 0 60 k k

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4a
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3 marks

On the same axes, sketch the graphs of space y space equals space open vertical bar straight f open parentheses x close parentheses close vertical bar and y space equals open vertical bar straight g open parentheses x close parentheses close vertical bar where

straight f left parenthesis x right parenthesis space equals space 3 x space long dash space 1 space space space space space 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 2 x space plus space 2 space space space space space space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

4b
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3 marks

Solve the equation = open vertical bar straight f open parentheses x close parentheses close vertical bar space equals space open vertical bar straight g open parentheses x close parentheses close vertical bar.

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5
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3 marks

Write

fraction numerator 2 left parenthesis x minus 11 right parenthesis over denominator x squared plus 2 x minus 15 end fraction

in the form

fraction numerator A over denominator x plus 5 end fraction plus fraction numerator B over denominator x minus 3 end fraction

where A spaceand B are integers to be found.

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6a
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2 marks

Simplify

2 space ln space 3 to the power of 4 plus ln space 3 cubed minus ln space 9

giving your answer in the form a ln b, where a and b are integers to be found.

6b
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2 marks

Write

2 space log subscript a space x plus 3 space log subscript a open parentheses x plus 1 close parentheses minus log subscript a space 4 open parentheses x plus 2 close parentheses

as a single logarithm.

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7a
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2 marks

Show that x squared plus y squared plus 5 x minus 2 y minus 5 equals 0 can be written in the form open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared, where a comma space b and r are constants to be found.

7b
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2 marks

Hence write down the centre and radius of the circle with equation x squared plus y squared plus 5 x minus 2 y minus 5 equals 0.

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8
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4 marks

The line space y plus 2 x equals 11 spacemeets the circle with equation x squared plus y squared plus 6 x minus 14 y equals negative 38 .

(i)
Show that the line and circle meet at one point only.

(ii)
Find the coordinates of the point of intersection.

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9
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3 marks

Prove that the sum of any three consecutive even numbers is always a multiple of 2, but not always a multiple of 4.

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10
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3 marks

Prove that the (positive) difference between an integer and its cube is the product of three consecutive integers.

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11a
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3 marks

A Ferris wheel with 30 passenger “pods” is modelled as a circle with centre open parentheses 0 comma 0 close parentheses and radius 60 spacem.  A pod’s position can be determined by the angle straight theta radians, which is measured anticlockwise from the positive x-direction, as shown in the diagram below.

q5a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-hard

The coordinates of a pod,open parentheses x comma y close parentheses  are given by  open parentheses A space cos open parentheses straight theta close parentheses comma B space sin open parentheses straight theta close parentheses close parentheses  where A spaceand B are positive constants. Ground level is represented by the line with equation y=-62.

(i)
Write down the values of A spaceand B.

(ii)
The pods are evenly distributed around the wheel.
Find the angle between each pod.
11b
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3 marks

Find the height above the ground of a passenger pod when straight theta equals fraction numerator 7 straight pi over denominator 6 end fraction radians.

11c
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2 marks

Find the angle straight theta, to three significant figures, for a passenger pod located at the point open parentheses 48 comma negative 36 close parentheses.

11d
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1 mark

What would you be able to say about the Ferris wheel in the case where A not equal to B?

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12
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4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar tan space x
(ii)
straight f colon x rightwards arrow from bar open vertical bar 1 over x close vertical bar
(iii)
straight f colon x rightwards arrow from bar square root of x squared end root
(iv)
straight f colon x rightwards arrow from bar plus-or-minus square root of 25 minus x squared end root

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13a
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4 marks

The graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals space x (dotted line) are shown in the diagram below.
straight f left parenthesis x right parenthesis has rotational symmetry about the origin and for x space greater than space 0, there is a vertical line of symmetry at x space equals space 4.5.

q5a-2-8-very-hard-aqa-a-level-maths-pure

(i)
Use the graph to write down the domain and range of straight f left parenthesis x right parenthesis.
(ii)
On the diagram above sketch the reflection of straight f left parenthesis x right parenthesis in the line y space equals space x and explain why this cannot be the graph of straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.
13b
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3 marks
(i)
Given that the maximum solution to straight f left parenthesis x right parenthesis space equals space 6 is x space equals space 6, state the restriction on the domain of straight f left parenthesis x right parenthesis such that straight f to the power of negative 1 end exponent left parenthesis x right parenthesis exists.
(ii)
Hence, or otherwise, write down the domain and range of straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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14
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1 mark

A particle's displacement, r metres, with respect to time, t seconds, is defined by the equation

r space equals space 10 space sin space 4 t

Find an expression for the velocity, v ms-1, of the particle at time t seconds.

Circle your answer.

v space equals space minus 2.5 space cos space 4 t v space equals space 40 space sin space 4 t v space equals space minus 160 space sin space 4 t v space equals space 40 space cos space 4 t

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15
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1 mark

A particle has a speed of 6 ms-1 in a direction relative to unit vectors i and j as shown in the diagram below.

q14-aqa-a-level-maths-practise-paper-set-a-pure

The velocity of this particle can be expressed as a vector open square brackets table row cell v subscript 1 end cell row cell v subscript 2 end cell end table close square brackets ms to the power of negative 1 end exponent.

Find the correct expression for v subscript 1.

Circle your answer.

v subscript 1 space equals space 6 space sin space 30 degree v subscript 1 space equals space minus 6 space cos space 30 degree v subscript 1 space equals space 6 space cos space 30 degree v subscript 1 equals space minus 6 space sin space 30 degree

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16
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3 marks

Two particles A and B are connected by a light inextensible string. Particle A has a mass of 7 kg, particle B has a mass of 3 kg, and particle B hangs directly below particle A. A force of 120 N is applied vertically upwards on particle A, causing the particles to accelerate.

By considering particles A and B as a single object, use Newton's Second Law of Motion to find the magnitude of the acceleration.

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17
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4 marks

The diagram below shows the velocity-time graph for a train travelling between two stations, starting at station P and finishing at station Q. The graph indicates velocity in kilometres per hour and time in minutes.

edexcel-al-maths-mechanics-topic-2-1-h---q3

(i)
Find the distance between station P and station Q.

(ii)
Find the deceleration of the train in the last 20 minutes.

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18a
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3 marks

A train locomotive of mass 7000 kg and a carriage of mass 2000 kg are at rest on a section of horizontal track.  The connection between the locomotive and carriage may be modelled as a light rod parallel to the direction of their motion forward or backward along the track.  The resistances to motion of the locomotive and the carriage are modelled as constant forces of 2300 N and 1000 N respectively.

The locomotive begins to accelerate in the backwards direction, with its engine providing a constant driving force of 15000 N.

Find:

the magnitude of the acceleration of the locomotive and carriage

18b
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2 marks

the thrust in the connecting rod.

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19
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4 marks

A B is a non-uniform rod of mass 12 kg and length 4 m. A B is held horizontally in equilibrium by a support placed at point C and a vertical wire attached to point D such that A C space equals space 0.8 m and D B space equals space 1 m as shown in the diagram below:

q4a-4-1-easy-aqa-a-level-maths-mechanics

The distance from point A to the centre of mass of the rod is 1.75 m.

Find the ratio of the reaction force at C to the tension in the wire at D. Give your answer in the form p colon space q where p and q are integers with no common factors other than 1.

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20a
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1 mark

A high-speed train has a maximum acceleration of 0.6 space m space s to the power of negative 2 end exponent which, from rest, takes 20 seconds to reach.

One such train leaves a station at t equals 0 seconds and its displacement, s space straight m, from the station is modelled using the equation

 s space equals space 1 over m t cubed space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 20

 where m is a constant.

 

Find an expression for the velocity of the high-speed train for 0 space less or equal than t space less or equal than space 20.

20b
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3 marks
(i)
Find an expression for the acceleration of the high-speed train for 0 less or equal than t less or equal than 20.

(ii)
Thus find the value of the constant m, assuming that the train reaches its maximum acceleration in the quickest time possible.

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21a
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3 marks

A home-made rocket is launched from rest, at time t = 0 seconds, from ground level with an acceleration of 56 space straight m space straight s to the power of negative 2 end exponent. The rocket’s acceleration is then modelled by the equation.                          

a space equals space 56 space plus space t space minus space t squared                              t space greater or equal than space 0 

(i)
Find an expression for the velocity of the home-made rocket.

(ii)
Other than at launch, find the time when the velocity of the rocket is 0 space straight m space straight s to the power of negative 1 end exponent.
21b
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4 marks

Find the greatest height the rocket reaches, giving your answer in kilometres to three significant figures.

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22
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5 marks

For a particle modelled as a projectile with initial velocity U space straight m space straight s to the power of negative 1 end exponent at an angle of α° above the horizontal, show that the equation of the trajectory of the particle is given by

y equals left parenthesis t a n space alpha right parenthesis space x minus fraction numerator g x squared over denominator 2 U squared c o s squared space alpha end fraction

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23a
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2 marks

The flight of a particle projected with an initial velocity of U space straight m space straight s to the power of negative 1 end exponent at an angle α above the horizontal is modelled as a projectile moving under gravity only. The particle is projected from the point  (x0, y0) with the upward direction being taken as positive, and with the coordinates being expressed in metres. g space straight m space straight s to the power of negative 2 end exponent is the constant of acceleration due to gravity.

Write down expressions for

(i)
the x-coordinate of the projectile at time t seconds
(ii)
the y-coordinate of the projectile at time t seconds.
23b
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4 marks

For a particular projectile, space tan space alpha equals space 3 over 4, U space equals space 10 space straight m space straight s to the power of negative 1 end exponent and the particle is projected from the point left parenthesis 3 space comma space 8 right parenthesis.  Find an expression for the trajectory of the particle, giving your answer in the form

 y equals fraction numerator a x squared plus b x plus c over denominator 128 end fraction

where the constants a, b and c are expressed in terms of g.

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24
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3 marks

A particle moves with acceleration open parentheses table row cell negative 3 end cell row 4 end table close parentheses m s-2 and after 7 seconds of motion has velocity open parentheses table row 5 row 3 end table close parentheses m s-1. Find the displacement of the particle in this time.

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25a
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5 marks

Two stones are slid across a large icy pond. The first stone is released from rest at the origin with constant acceleration left parenthesis 2 bold i space plus space 3 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. The second stone is released from rest from the position with coordinates left parenthesis 50 comma space minus 100 right parenthesis with constant acceleration  left parenthesis bold i space plus space 5 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.

 

(i)
Find the position vectors of both stones after 5 seconds.

(ii)
Find the distance between the two stones after 5 seconds.
25b
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2 marks

Show that the two stones collide after 10 seconds.

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