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

Practice Paper Questions

1
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1 mark

Identify the graph of the following equation from the options below.

y space equals space minus open vertical bar x plus 2 close vertical bar minus 1

Tick (✓) one box.

q1-aqa-a-level-maths-practice-paper-mechanics

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2
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1 mark

Simplify the following expression.

cube root of a to the power of 3 over 2 end exponent cross times a to the power of 1 third end exponent end root

Circle your answer.

a to the power of 11 over 2 end exponent a to the power of begin inline style 1 over 6 end style end exponent a to the power of begin inline style 11 over 18 end style end exponent a to the power of begin inline style 1 over 18 end style end exponent

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3
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1 mark

Each of these functions has domain x space element of space straight real numbers.

Which function has an inverse?

Circle your answer.

straight f left parenthesis x right parenthesis space equals space x squared straight f open parentheses x close parentheses equals x cubed straight f open parentheses x close parentheses equals sin space x straight f open parentheses x close parentheses equals open vertical bar x close vertical bar

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4
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4 marks

Given that x equals 1 half is a root of the function straight f left parenthesis x right parenthesis equals 2 x cubed plus left parenthesis p squared plus 1 right parenthesis x squared minus 11 x plus 4, find the possible values of space p.

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5a
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4 marks

Given that y greater than 2, find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals x squared left parenthesis y minus 2 right parenthesis

5b
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3 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals sin to the power of 2 space end exponent 2 y

giving your answer in the form x equals straight f left parenthesis y right parenthesis.

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6a
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4 marks
(i)

Show that R space cos open parentheses x plus alpha close parentheses identical to R space cos space alpha space space cos space x minus R space sin space alpha space sin space x, where R and alpha are constants.

(ii)
Use your result from part (i) to show that  cos space x minus square root of 3 sin space x identical to 2 space cos open parentheses x plus straight pi over 3 close parentheses.
6b
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3 marks

Hence solve the equation cos x minus square root of 3sin x equals 1 for 0 less or equal than space x less or equal than 2 straight pi.

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7a
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2 marks

Express 2 x cubed plus 2 x squared minus 12 x in the form a x left parenthesis x plus b right parenthesis left parenthesis x plus c right parenthesis, where a comma space b space and space c are integers to be found.

7b
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3 marks

Hence sketch the graph of space y equals 2 x cubed plus 2 x squared minus 12 x, labelling any points where the graph intersects the coordinate axes.

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8a
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3 marks

The diagram below shows a part of the curve with equation y equals f open parentheses x close parentheses, where

straight f open parentheses x close parentheses equals 460 minus x cubed over 300 minus 8100 over x comma space space space space space space space space space space space space space space x greater than 0

Point A is the maximum point of the curve.

KTI0dIN4_q7a-7-2-applications-of-differentiation-medium-a-level-maths-pure

Find straight f apostrophe open parentheses x close parentheses.

8b
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4 marks

Use your answer to part (a) to find the coordinates of point A.

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9a
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2 marks

Show that the equation

x equals 7 e to the power of negative 0.2 t end exponent

can be written as

ln space x equals ln space 7 minus 0.2 straight t

9b
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2 marks

Rewrite the equation ln space y equals 4.1 x plus ln space 8 in the form  y equals A e to the power of k x end exponent.

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10a
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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form N equals N subscript 0 space a to the power of k t end exponent. N is the number of bacteria and t is the time in hours since the experiment began.N subscript 0 comma a and k are constants.  A scientist records the number of bacteria at various points over a six-hour period.  The results are shown in the table below.

  t, hours 0 2 4 6

  N, no. of bacteria

100 180 340 620

  log subscript 3 space N (3SF)

4.19

4.73

5.31 5.85

Plot the observations on the graph below - plotting log subscript 3 space N against t

q9a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy.

10b
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2 marks

Using the points (0, 4.19) and (6, 5.85), find an equation for a line of best fit in the form log subscript 3 space N equals m t plus log subscript 3 space c, where m spaceand c are constants to be found.

10c
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2 marks

The equation N equals N subscript 0 space a to the power of k t end exponent  can be written in the form  log subscript a space N equals k t plus log subscript a space N subscript 0.
Use your answer to part (b) to estimate the values of , N subscript 0 comma a space, and k.

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11a
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2 marks

The functions straight f left parenthesis x right parenthesis and  straight g left parenthesis x right parenthesis are given as follows

               straight f left parenthesis x right parenthesis equals open parentheses 4 plus 3 x close parentheses to the power of 1 half end exponent space space space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals open parentheses 9 minus 2 x close parentheses to the power of negative 1 half end exponent

Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.

11b
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2 marks

Expand straight g open parentheses x close parentheses, in ascending powers of x up to and including the term in x squared.

11c
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2 marks

Find the expansion of square root of fraction numerator 4 plus 3 x over denominator 9 minus 2 x end fraction end root  in ascending powers of x, up to and including the term in x squared.

11d
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2 marks

Find the values of x for which your expansion in part (c) is valid.

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12
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1 mark

The diagram below shows a velocity-time graph for a particle moving with velocity v space ms to the power of negative 1 end exponent at time t seconds.

q5-aqa-a-level-maths-practice-paper-pure

Which statement is incorrect?

Tick (✓) one box.

The particle had a positive acceleration when t equals 6 square
The particle's speed when t space equals space 3 was negative 12 ms-1 square
The particle was at instantaneous rest when t equals 6 square
The particle had a constant speed for 9 space less or equal than space t space less or equal than space 12 square

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13
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1 mark

An object rests on a rough horizontal surface.

The coefficient of friction between the crate and the surface is 0.4.

A forward force acts on the object, parallel to the surface.

When this force is 588 N, the object is on the point of moving.

Find the weight of the crate.

Circle your answer.

1470 N 150 kg 235.2 N 24 kg

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14
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1 mark

A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of 24 ms-1.

The first force acting on the particle is (280i + 210j) N.

The second force acting on the particle is left parenthesis p bold i minus 210 bold j right parenthesis N.

Find the value of p.

Circle your answer.

280 –256 256 –280

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15
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5 marks

A car travelling along a horizontal road passes a point A with velocity 21 space straight m space straight s to the power of negative 1 end exponent and constant acceleration 0.2 space straight m space straight s to the power of negative 2 end exponent. Point B is 1.5 km from point A. When the car reaches point B it decelerates uniformly at 3.1 space straight m space straight s to the power of negative 2 end exponent until it comes to rest.

Find the distance the car travels from the moment it starts to decelerate until it comes to rest. Give your answer to an appropriate degree of accuracy.

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16
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6 marks

In the diagram below A B is a uniform beam of length 4 m. It rests horizontally on two supports placed at points C and D, such that A C space equals space 1.5 space m spaceand D B space equals space 1.2 m as shown:

q3-4-1-easy-aqa-a-level-maths-mechanics


A stone of mass 10 kg is placed at point B and the beam is on the point of tilting. That stone is removed, and another stone of mass m subscript A kg is placed at point A which causes the beam to begin tilting.

Given that the stones may be modelled as particles, show that m subscript A greater than space k, where k is the largest possible constant for which that inequality must be true.

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17a
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2 marks

A horse running across a large area of open countryside starts to gallop with constant acceleration (0.6i + 0.4j) m s−2. After 12 seconds of galloping the horse has velocity (8i + 10j) m s−1.

Find the displacement of the horse at the end of its 12 second gallop.

17b
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4 marks

Find the change in speed of the horse between the start and end of its 12 second gallop.

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18
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6 marks

It takes 6 minutes for a particle to travel  open parentheses fraction numerator 5.94 over denominator 13.86 end fraction close parentheses km with constant acceleration. The particle’s velocity at the start of the 6 minutes is one-tenth of its velocity at the end. Find the acceleration of the particle.

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19a
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4 marks

A bullet train has a maximum acceleration of 0.72 space straight m space straight s to the power of negative 2 end exponent.

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

s space equals space 3 over 200 t cubed space space space space space space space space space space space space space space space 0 space less or equal than space t space less or equal than space 8

Show that it takes 8 seconds for the bullet train to reach its maximum acceleration.

19b
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3 marks

After reaching its maximum acceleration the bullet train continues to accelerate at that rate until its velocity reaches its maximum of 75 space straight m space straight s to the power of negative 1 end exponent.

How long does it take for this increase in velocity to happen?

19c
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3 marks

Once reaching its maximum velocity, the bullet train continues at this velocity for 10 minutes. Find the displacement of the train from the station at this time, giving your answer in kilometres to 3 significant figures.

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20a
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3 marks

Once an aircraft reaches its cruising height (at time t = 0 hours) its acceleration is modelled by

 bold a bold space equals space open parentheses 4 t cubed space minus space 6 t squared close parentheses bold i space plus space open parentheses 0.9 t squared space minus space 1 close parentheses bold j bold space km space straight h to the power of negative 2 end exponent

Given that the velocity of the aircraft at t space equals space 5 hours is bold v space equals space 400 bold i space plus space 40 bold j bold space km space straight h to the power of negative 1 end exponent, find the velocity of the aircraft in terms of t.

20b
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2 marks

Find the speed of the aircraft when it first reaches its cruising height.

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21
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9 marks

A particle of mass 0.5 kg is at rest on a rough plane which is inclined at 35 degree to the horizontal.  The particle is being acted upon by a force of 6 N, directed at an angle of theta degree to the plane.  The line of action of the force is in the same vertical plane as the line of greatest slope of the inclined plane.

3-3-v-h-q-6-a-level-maths-mechnics

Given that the coefficient of friction between the particle and the plane is 0.4, and that the particle is on the point of slipping up the plane, find the value of theta.

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