CIE AS Maths: Pure 2

Topic Questions

1.1 Modulus Functions

1
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3 marks

Solve the equation vertical line 3 x space long dash space 2 vertical line space equals space 7.

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2
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6 marks

Solve the following:

i)
open vertical bar 4 x plus 3 close vertical bar less than 9.
ii)
open vertical bar 2 x close vertical bar plus 5 space greater or equal than 15.
iii)
open vertical bar x plus 8 close vertical bar equals open vertical bar 2 x plus 10 close vertical bar.

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3
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6 marks

The graph of y equals straight f left parenthesis x right parenthesis where straight f open parentheses x close parentheses equals open vertical bar 3 x plus 1 close vertical bar is shown below.

q3-easy-1-1-modulus-function-cie-maths-pure-

i)
On the diagram above sketch the graph of y equals open vertical bar x minus 5 close vertical bar comma state the coordinates of any points of intersection with y equals straight f left parenthesis x right parenthesis.

ii)
Hence, or otherwise, solve the inequality open vertical bar 3 x plus 1 close vertical bar less than open vertical bar x minus 5 close vertical bar.

 

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4a
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3 marks

Solve the equation open vertical bar 6 minus 2 x close vertical bar equals 4.

4b
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4 marks

On the same diagram, sketch the graphs of y equals vertical line 6 minus 2 x vertical line and y equals 4.
Label the coordinates of the points where the two graphs intersect each other and the coordinate axes.

4c
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3 marks

Consider the graphs of space y equals vertical line 6 minus 2 x vertical line spaceand y equals k, where k is a constant.
For which values of k ...

i)
... will the two graphs have no points of intersection?
ii)
... will the two graphs have one point of intersection?
iii)
... will the two graphs have two points of intersection?

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5a
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1 mark

The graph of space y equals straight f left parenthesis x right parenthesis spacewhere straight f open parentheses x close parentheses equals 2 x minus 3 is shown below.

q5-easy-1-1-modulus-function-cie-maths-pure-

Determine the coordinates of the points marked A and B.

5b
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5 marks
i)
On the diagram above sketch the graph of space y equals open vertical bar straight f open parentheses x minus 1 close parentheses close vertical bar. space

ii)
Determine the coordinates of the image of the points A and B under the transformation in part (i).

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1a
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3 marks

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


straight f open parentheses x close parentheses equals 3 x minus 1 space space space space space space space space space space space space space space space space x element of straight real numbers space
straight g open parentheses x close parentheses equals 2 x plus 2 space space space space space space space space space space space space space space space space x element of straight real numbers space


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

1b
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3 marks

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

1c
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1 mark

Which of the solutions to  vertical line straight f open parentheses x close parentheses vertical line equals vertical line straight g open parentheses x close parentheses vertical line  is also a solution to  straight f open parentheses x close parentheses equals straight g open parentheses x close parentheses?

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2a
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1 mark

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

straight f colon x rightwards arrow from bar vertical line 3 x minus 2 vertical line space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers

Explain why the inverse of  straight f left parenthesis x right parenthesis  does not exist.

2b
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1 mark

Suggest an adaption to the domain of space straight f left parenthesis x right parenthesis space space so its inverse does exist, but also produces the maximum possible range for space straight f left parenthesis x right parenthesis.

2c
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3 marks

Using your adaption from part (b), find an expression for  straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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3
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3 marks

Solve the equation  open vertical bar x squared minus 4 close vertical bar equals 3, giving your answers in exact form.

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4a
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4 marks

The functions space straight f left parenthesis x right parenthesis comma space space straight g left parenthesis x right parenthesis space spaceare defined as follows

straight f open parentheses x close parentheses equals open vertical bar x minus 2 close vertical bar minus 5 space space space space space space space space space space space x element of straight real numbers space space space space
straight g open parentheses x close parentheses equals vertical line x vertical line space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers space space

Sketch the graph of  y equals gf left parenthesis x right parenthesis, stating the coordinates of all points where the graph intercepts the coordinate axes.

4b
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2 marks
i)
How many solutions are there to the equation  gf open parentheses x close parentheses equals 1 ?

 

ii)
How many solutions are there to the equation  gf open parentheses x close parentheses equals 10 ?
4c
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3 marks

Solve the equation  gf open parentheses x close parentheses equals 2.

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5a
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3 marks

The minimum point on the graph of y equals straight f left parenthesis x right parenthesis has coordinates space left parenthesis 4 space comma space minus 8 right parenthesis spaceas shown on the diagram below.

q5-hard-1-1-modulus-function-cie-maths-pure-

Sketch the graph of y equals open vertical bar straight f open parentheses 2 straight x close parentheses close vertical bar minus 3 and state the coordinates of the maximum point.

5b
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2 marks

Find the exact distance between the minimum point on the graph of space y equals straight f open parentheses x close parentheses and the maximum point on the graph of y equals open vertical bar straight f open parentheses 2 straight x close parentheses close vertical bar minus 3.

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6a
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3 marks

On the same axes sketch the graphs of space y equals straight p left parenthesis x right parenthesis spaceand y equals straight p to the power of negative 1 end exponent left parenthesis x right parenthesis, where straight p open parentheses x close parentheses equals open vertical bar 2 x close vertical bar comma space space space x less or equal than 0.

6b
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3 marks

Find an expression for straight p to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain.

6c
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3 marks

Show that straight p to the power of negative 1 end exponent open parentheses x close parentheses equals negative 1 half straight p left parenthesis negative 1 half x right parenthesis.

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7a
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2 marks

The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius 10 m, as shown in the diagram below.

q7-hard-1-1-modulus-function-cie-maths-pure-

At time t seconds, the x-coordinate of the boat is modelled by the function

x open parentheses t close parentheses equals 10 space sin space space open parentheses straight pi over 5 t close parentheses comma space space t greater or equal than 0 space comma

and the height, h m, of the boat above the ground, at time space t spaceseconds, is modelled by

h open parentheses t close parentheses equals 12 minus 10 open vertical bar cos space open parentheses straight pi over 5 t close parentheses space close vertical bar comma space space t greater or equal than 0.

Verify that the initial position of the boat is left parenthesis 0 space comma space 2 right parenthesis.

7b
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3 marks
i)
Write down the coordinates of the boat when it is at its maximum height.

ii)
Find the time it takes the boat to swing between these two points.
7c
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2 marks

Find the position of the boat when it has swung through an angle of  straight pi over 6 anticlockwise from the y-axis, as shown in the diagram above. 
Find the time at which the boat first reaches this position.

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1a
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3 marks

On the same axes, sketch the graphs of  y equals straight f left parenthesis x right parenthesis  and  y equals vertical line straight g open parentheses x close parentheses vertical line  where


straight f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses to the power of 2 space space space space space space space space space space space end exponent space x element of straight real numbers
straight g open parentheses x close parentheses equals 2 x plus 4 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.

1b
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2 marks

Solve the equation straight f open parentheses x close parentheses equals vertical line straight g open parentheses x close parentheses vertical line.

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2
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3 marks

Solve the equation  open vertical bar 4 x plus 2 close vertical bar equals 5.

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3a
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4 marks

The functions space straight f left parenthesis x right parenthesis comma space space straight g left parenthesis x right parenthesis space space are defined as follows

straight f open parentheses x close parentheses equals open vertical bar x minus 9 close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers space space space space
straight g open parentheses x close parentheses equals x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers

Sketch the graph of space y equals fg left parenthesis x right parenthesis, stating the coordinates of all points where the graph intercepts the coordinate axes.

3b
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2 marks
i)
How many solutions are there to the equation  fg open parentheses x close parentheses equals 5 ?

ii)
How many solutions are there to the equation  fg open parentheses x close parentheses equals 9 ?
3c
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2 marks

Write down the solutions to the equation space fg open parentheses x close parentheses equals 0.

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4
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4 marks

The turning point on the graph of space y equals straight f left parenthesis x right parenthesis space has coordinates left parenthesis 2 space comma space minus 5 right parenthesis as shown on the diagram below.

q4-medium-1-1-modulus-function-cie-maths-pure-

i)
On the diagram above sketch the graph of y equals open vertical bar straight f open parentheses straight x close parentheses close vertical bar plus 1 and state the coordinates of the turning point.

 

ii)
State the distance between the turning points on the graphs of space y equals straight f open parentheses x close parentheses and y equals open vertical bar straight f open parentheses straight x close parentheses close vertical bar plus 1.

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5a
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2 marks

The diagram below shows the graph of y equals straight g left parenthesis x right parenthesis where

straight g open parentheses x close parentheses equals fraction numerator 2 x plus 1 over denominator x minus 1 end fraction comma space space x not equal to 1

q5-medium-1-1-modulus-function-cie-maths-pure-

Write down the equations of the two asymptotes.

5b
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2 marks

Determine the equations of the two asymptotes on the graph of y equals straight g open parentheses 2 x close parentheses minus 3.

5c
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2 marks

Determine the range of open vertical bar straight g open parentheses 3 x close parentheses minus 2 close vertical bar.

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1
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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 space
ii)
straight f colon x rightwards arrow from bar vertical line 1 over x vertical line
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 space end root

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2
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3 marks

Solve the equation  open vertical bar x squared minus 9 close vertical bar equals 6 minus 0.25 x squared, giving your answers in exact form.

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3a
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3 marks

The functions  straight f left parenthesis x right parenthesis comma space space straight g left parenthesis x right parenthesis  are defined as follows

straight f open parentheses x close parentheses equals open vertical bar x cubed minus 8 close vertical bar space space space space space space space space space space space space space space space space space x element of straight real numbers space space space
straight g open parentheses x close parentheses equals vertical line x vertical line space space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers space space space

Sketch the graph of  y equals fg left parenthesis x right parenthesis, stating the coordinates of all points where the graph intercepts the coordinate axes.

3b
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3 marks

There are between 0 and 4 solutions to the equation  fg open parentheses x close parentheses equals c, where c is a real number.  Determine the values of c that produce each number of solutions.

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4a
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3 marks

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


straight f open parentheses x close parentheses equals square root of x space space space space space space space space space space space space space space space space space space space x greater or equal than 0 space
straight g open parentheses x close parentheses equals 2 x minus 3 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  straight f open parentheses x close parentheses equals vertical line straight g open parentheses x close parentheses vertical line.

4c
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1 mark

Which of the solutions to  straight f open parentheses x close parentheses equals vertical line straight g open parentheses x close parentheses vertical line spaceis not a solution to  straight f open parentheses x close parentheses equals straight g open parentheses x close parentheses?

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5a
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3 marks

A sketch of the graph with equation y equals f open parentheses x close parentheses where f open parentheses x close parentheses equals 10 x minus x squared minus 16 is shown below.
Points A and B are the x-axis intercepts and point C is the maximum point on the graph.

q5-vhard-1-1-modulus-function-cie-maths-pure-

On the diagram above, sketch the graph of y equals negative open vertical bar begin inline style 1 fourth end style straight f open parentheses begin inline style 1 half end style x close parentheses close vertical bar labelling the image of the points A comma space B and C with A apostrophe comma space B apostrophe and space C apostrophe.

5b
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4 marks

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

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6
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4 marks

The function straight f open parentheses x close parentheses equals e to the power of 3 x end exponent minus x minus 6 is transformed by a sequence of transformations as described below.

  1. Horizontal stretch by scale factor 3,
  2. The modulus of the function is then taken,
  3. Reflection in the y-axis.

Write down the resulting transformation in terms of straight f left parenthesis x right parenthesis as well as an expression in terms of x.

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7a
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3 marks

A swing boat fairground ride is modelled as moving forwards and backwards along the path of a semi-circle, radius 18 m, as shown in the diagram below.

q7-vhard-1-1-modulus-function-cie-maths-pure-

Show that, for 0 less or equal than theta less or equal than straight pi over 2 comma

i)
the x-coordinate of the boat is given by  x equals 18 space sin space theta ,

ii)
the y-coordinate is given by  y equals 20 minus 18 space cos space space theta
7b
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3 marks

The model is refined so that the coordinates of the boat can be calculated from the time, space t seconds, after the boat is set in motion.   The x and y coordinates are now given by 

x equals 18 space sin space B t space space space space space space space space space space space space space space space space space space space space space space space space space space space y equals 20 minus 18 space open vertical bar cos space B t close vertical bar

where B is a constant.

i)
Briefly explain why the modulus of cos space theta is required for the y- coordinate.

ii)
Given that the time between the boat reaching its maximum height at either end of the ride is 8 seconds, find the value of B.
7c
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3 marks

For 0 less or equal than space t space less or equal than 4, find the times when the boat is equidistant from the ground and horizontally from the origin.

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