Modelling (Cambridge (CIE) IGCSE International Maths: Extended)

Exam Questions

1 hour12 questions
1a
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1 mark

This task looks at how to model the path of a bending river, using approximations to predict the speed of the river.

When viewed from above or drawn on a map, the path of a river is not a straight line but instead has many bends, especially as the river approaches the sea.

An S-bend is a bend in one direction followed immediately by a bend in the opposite direction. The map below shows a river path completing two S-bends before it reaches the sea (the water in the river flows from right to left).

The x-axis is the centre line of the river path, about which the river bends, measured inland from the sea. The y-axis is perpendicular to the x-axis. The units on both axes are kilometres.

The flat land either side of the river path is called the flood plain. The lines y equals 2.4 and y equals negative 2.4 indicate the boundaries of the flood plain.

Graph showing a sine wave river path on a grid. Axes labelled x (0 to 8 km) and y (-2.4 to 2.4 km), with sea and flood plain areas marked. The direction of flow of the river is from right to left.

The amplitude, A km, of an S-bend is the maximum distance (in the y-direction) from the centre line to the river path. From the diagram above:

A equals 1.5

The wavelength, L km, of an S-bend is the distance along the x-axis that it takes to complete one S-bend.

Find the value of L from the diagram.

1b
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1 mark

A geographer wants to model the river path shown in part (a).

They suggest two possible models, where angles are measured in degrees:

y equals A space sin open parentheses fraction numerator 360 x over denominator L end fraction close parentheses

y equals A space cos open parentheses fraction numerator 360 x over denominator L end fraction close parentheses

Decide which of the models is not suitable for the river shown in the diagram.

Give a reason for your answer.

1c
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1 mark

Substitute the values of A and L into the correct model from part (b).

Simplify your answer.

1d
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2 marks

Use the model in part (c) to find the perpendicular distance from the point on the centre line that is 500 metres inland from the sea, to the path of the river.

Give your answer correct to 3 significant figures.

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2a
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2 marks

The geographer wants to find the length of the river in question 1, but does not know how to measure the length of a trigonometric curve.

Instead, they decide to approximate the length of the river by dividing it into straight-line segments, as shown.

Graph with zigzag straight lines on x-axis (0 to 8 km) and y-axis (-1.5 to 1.5 km). Peaks at 1.5 km and troughs at -1.5 km.

Each straight line segment passes through the correct maximum point, minimum point and x-intercept.

Show that the river is approximately 14.4 km in length.

Explain your method.

2b
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2 marks

The geographer wants to estimate the speed of the river.

They place a paper boat into the river at x equals 8 and time how long it takes for the paper boat to reach the sea, at x equals 0.

The total time taken is 2 hours and 24 minutes.

Use this information, and the rounded value of the length in part (a), to estimate the speed of the river.

Give your answer in km/h.

2c
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2 marks

Based on the approximation made in part (a), explain whether the true speed of the river is likely to be higher or lower than the value calculated in part (b).

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3a
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1 mark

In reality, the speed of the river increases by 2 km/h as it approaches the sea.

The geographer creates a simple model to show this change:

  • The S-bend from x equals 8 to x equals 4 has a velocity of v km/h

  • The S-bend from x equals 4 to x equals 0 has a velocity of open parentheses v plus 2 close parentheses km/h

The approximate length of the river in question 2(a) is rounded to the nearest integer.

Find, in terms of v, a formula for the approximate time taken (in hours) to travel along the S-bend from x equals 8 to x equals 4.

3b
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1 mark

Find, in terms of v, a formula for the approximate time taken (in hours) to travel along the S-bend from x equals 4 to x equals 0.

3c
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2 marks

Show that the approximate total time (in hours) to travel along both S-bends is given by the formula

fraction numerator 14 open parentheses v plus 1 close parentheses over denominator v open parentheses v plus 2 close parentheses end fraction

3d
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2 marks

Use the exact total time, given by the geographer in question 2(b), to show that v satisfies the equation

6 v squared minus 23 v minus 35 equals 0

3e
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2 marks

Solve the equation in part (d) to find v.

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4a
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2 marks

A different model for a river path approaching the sea has the form

y equals A space sin open parentheses 90 x close parentheses

In this model, the amplitude of an S-bend, A, at a distance x km inland from the sea is inversely proportional to f, the distance to where the flood plain boundary meets the sea, shown below:

Diagram of a flood plain next to the sea with marked axes in kilometres. Line f connects sea at (0, 2.4) to a point a distance of x km along the centre line.

Show that this model has the equation

y equals fraction numerator k over denominator square root of 2.4 squared plus x squared end root end fraction sin open parentheses 90 x close parentheses

where k is an unknown constant of proportionality.

4b
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1 mark

In this model, it is known that y equals 1.5 when x equals 1.

Use this information to show clearly that

k equals 3.9

4c
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2 marks

Sketch the model on the axes provided.

Label any x-intercepts clearly.

Axes with x-axis labelled 0 to 8 km, y-axis from -2 to 2 km.
4d
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1 mark

Describe how the S-bends change as the river approaches the sea.

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