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IGCSEMathsCambridge (CIE)International MathsExtendedExam QuestionsCoordinate Geometry & GraphsUsing Calculators for Graphs, Equations & Inequalities

Using Calculators for Graphs, Equations & Inequalities (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

2 hours12 questions
Medium
Hard
Very Hard
1a
Sme Calculator2 marks
A coordinate plane with the x-axis labelled from 0 to 4 and the y-axis labelled from -1 to 5. Both axes have arrows indicating positive directions.

straight f open parentheses x close parentheses equals 5 minus 1.25 x 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 straight g open parentheses x close parentheses space equals 1.5 x minus 1

On the diagram, sketch the graph of y equals straight f open parentheses x close parentheses for 0 less or equal than x less or equal than 4.

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1b
Sme Calculator2 marks

On the diagram, sketch the graph of y equals straight g open parentheses x close parentheses for 0 less or equal than x less or equal than 4.

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1c
Sme Calculator1 mark

Find x when straight f left parenthesis x right parenthesis equals straight g left parenthesis x right parenthesis.

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2
Sme Calculator3 marks

2 to the power of 3 x plus 1 end exponent equals 8 to the power of 2 minus x end exponent

Find the value of x

x equals.......................

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3a
Sme Calculator2 marks
Cartesian graph with x-axis from -3 to 3 and y-axis from -10 to 20. Axes are labelled with arrows, x pointing right, y pointing up.

straight f left parenthesis x right parenthesis equals x cubed minus 5 x plus 3 for negative 3 less or equal than x less or equal than 3

On the diagram, sketch the graph of y equals straight f open parentheses x close parentheses.

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3b
Sme Calculator2 marks

Find the coordinates of the local maximum.

( ...................... , ......................)

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3c
Sme Calculator3 marks

Find the zeros of the graph of y equals straight f left parenthesis x right parenthesis.

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4a
Sme Calculator3 marks
Graph with horizontal x-axis and vertical y-axis intersecting at the origin. 2 and -2 are labelled at each end of the x axis. Arrows indicate positive direction for each axis.

straight f left parenthesis x right parenthesis equals x cubed minus 2 x straight g left parenthesis x right parenthesis equals 1

On the diagram, sketch the graphs of space y equals straight f left parenthesis x right parenthesis and space y equals straight g left parenthesis x right parenthesis for negative 2 less or equal than x less or equal than 2.

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4b
Sme Calculator3 marks

Use your graphs to solve space x cubed minus 2 x minus 1 equals 0.

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4c
Sme Calculator2 marks

Use your graphs to solve space x cubed minus 2 x greater than 1.

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5
Sme Calculator3 marks

Use a graphical method to solve the inequality.

Show a sketch of the graph.

2 to the power of x plus x greater than 5

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1a
Sme Calculator3 marks
Graph with x-axis labeled from 0 to 360 and y-axis labeled from -10 to 10. Both axes have arrows indicating positive direction.

straight f open parentheses x close parentheses equals fraction numerator 1 over denominator sin x degree end fraction for 0 less or equal than x less or equal than 360

On the diagram, sketch the graph of y equals straight f open parentheses x close parentheses.

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1b
Sme Calculator1 mark

Find the coordinates of the local minimum point.

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1c
Sme Calculator2 marks

Write down the equations of the three asymptotes of the graph of y equals straight f left parenthesis x space right parenthesis.

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1d
Sme Calculator2 marks

The equation straight f open parentheses x close parentheses equals k has no solutions.

Write down the range of values of k.

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1e
Sme Calculator3 marks

By sketching another graph on the diagram, solve the equation fraction numerator 1 over denominator sin space x degree end fraction equals 5 sin open parentheses x over 2 close parentheses degree for 0 less or equal than x less or equal than 360 degree.

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2a
Sme Calculator4 marks

A medicine delivery company uses drones to deliver medications to remote mountain villages.

A radar system is used to control the drones, as long as the controller is within a certain distance from the drone, called the radar visibility, V metres,

A model for the radar visibility, V metres, at different altitudes, x metres up a particular mountain is

V equals fraction numerator 600 minus x over denominator 4 end fraction plus square root of 70 space 000 plus x squared end root

On the diagram, sketch the graph of V against x.

Graph with horizontal axis labelled 'Altitude (x metres)' from 0 to 400, and vertical axis labelled 'Visibility (V m)' from 400 to 600. No data is plotted.

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2b
Sme Calculator2 marks

Find, correct to the nearest metre,

(i) the minimum visibility,

[1]

(i) the altitude for which the visibility is at its minimum.

[1]

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2c
Sme Calculator1 mark

Write down a reason why the visibility given by the model might be different to the actual visibility.

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3a
Sme Calculator2 marks
A Cartesian coordinate plane with x-axis ranging from -1 to 4.7 and y-axis ranging from -10 to 9.

straight f left parenthesis x right parenthesis equals open parentheses x minus 2 close parentheses cubed minus 5 x plus 12 for negative 1 less or equal than x less or equal than 4.7

On the diagram, sketch the graph of y equals straight f open parentheses x close parentheses.

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3b
Sme Calculator2 marks

Write down the coordinates of the local maximum.

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3c
Sme Calculator2 marks

The equation open parentheses x minus 2 close parentheses cubed minus 5 x plus 12 equals k has exactly 2 solutions.

Find the values of k.

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3d2 marks

straight g left parenthesis x right parenthesis equals negative open parentheses x minus 1 close parentheses squared for negative 1 less or equal than x less or equal than 4.7

On the diagram, sketch the graph of y equals straight g open parentheses x close parentheses.

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3e
Sme Calculator1 mark

Solve straight f left parenthesis x right parenthesis equals straight g open parentheses x close parentheses.

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4
Sme Calculator1 mark

The curve  y equals x squared minus 2 x plus 1  is drawn on a grid.

A line is drawn on the same grid.

The points of intersection of the line and the curve are used to solve the equation  x squared minus 7 x plus 5 equals 0.

Find the equation of the line in the form  y equals m x plus c.

y equals ..................................................

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1a
Sme Calculator3 marks
Cartesian coordinate plane with x-axis ranging from -6 to 6 and y-axis from -9 to 9, intersecting at the origin (0, 0).

straight f open parentheses x close parentheses equals fraction numerator x squared plus 3 x over denominator open parentheses x minus 2 close parentheses open parentheses x plus 1 close parentheses end fraction

On the diagram sketch the graph of y equals straight f open parentheses x close parentheses for values of x between negative 6 and 6.

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1b
Sme Calculator2 marks

Write down the equations of the asymptotes parallel to the y‑axis.

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1c
Sme Calculator2 marks

Find the zeros of the graph of y equals straight f open parentheses x close parentheses.

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1d
Sme Calculator7 marks

straight g open parentheses x close parentheses equals x minus 3

(i) On the diagram, sketch the graph of y equals straight g open parentheses x close parentheses for negative 6 less or equal than x less or equal than 6.

[1]

(ii) Use your graphs to solve straight f open parentheses x close parentheses equals straight g open parentheses x close parentheses.

[3]

(iii) Solve straight g open parentheses x close parentheses greater than straight f open parentheses x close parentheses.

[3]

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2a
Sme Calculator2 marks
Graph with a horizontal x-axis from 0 to 360 and a vertical y-axis from -1 to 1. Both axes are labelled x and y respectively.

straight f open parentheses x close parentheses equals open parentheses cos space x degree close parentheses squared straight g open parentheses x close parentheses equals 0.5 minus 0.001 x

On the diagram, sketch the graph of space y equals straight f open parentheses x close parentheses for 0 less or equal than x less or equal than 360.

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2b
Sme Calculator2 marks

On the same diagram, sketch the graph of y equals straight g left parenthesis x right parenthesis for 0 less or equal than x less or equal than 360.

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2c
Sme Calculator4 marks

Solve straight f open parentheses x close parentheses equals straight g open parentheses x close parentheses for 0 less or equal than x less or equal than 360.

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2d3 marks

Solve straight f open parentheses x close parentheses less than straight g open parentheses x close parentheses for 0 less or equal than x less or equal than 360.

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3a
Sme Calculator4 marks
Graph with x and y axes, centred at zero. x-axis ranges from -5 to 5, y-axis from -9 to 8, both with arrows indicating positive direction.

space straight f open parentheses x close parentheses equals x plus fraction numerator 5 over denominator open parentheses x minus 2 close parentheses open parentheses x plus 3 close parentheses end fraction

Sketch the graph of space y equals straight f open parentheses x close parentheses for values of x between negative 5 and 5.

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3b2 marks

Write down the equations of the asymptotes parallel to the y-axis.

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3c
Sme Calculator6 marks

(i) Find the coordinates of the local maximum.

(............ , ............) [2]

(ii) Find the coordinates of the local minimum.

(............ , ............) [2]

(iii) Write down the range of values of k for which straight f open parentheses x close parentheses equals k has exactly one solution.

[2]

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3d
Sme Calculator6 marks

straight g open parentheses x close parentheses equals negative 4 minus x

(i) Solve the equation straight f open parentheses x close parentheses equals straight g open parentheses x close parentheses.

[3]

(ii) Find the solutions to the inequality straight f open parentheses x close parentheses greater than straight g open parentheses x close parentheses.

[3]

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