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Investigations (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

Exam code: 0607

1 hour18 questions

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Investigations

1a
Sme Calculator2 marks

This investigation looks at the geometry and properties of an ellipse, including its area and its circumference.

An ellipse can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

When a equals b, the ellipse is a circle with radius a.

Three shapes: a horizontal ellipse with a greater width than height labelled 'a > b,' a circle labeled 'a = b,' and a vertical ellipse labelled 'a < b.'

This investigation will only consider ellipses in the form a greater or equal than b.

Sketch an ellipse with a equals 5 and b equals 2.

Label the two lengths on your diagram.

How did you do?

1b
Sme Calculator1 mark

The formula for the area of an ellipse, A, is given by

A equals pi a b

Complete the table below, leaving all answers in terms of pi.

The first example has been done for you:

A equals pi cross times 5 cross times 2 equals 10 pi

a

b

A

5

2

10 pi

8

3

3

36 pi

10

50 pi

How did you do?

2
Sme Calculator3 marks

An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

The formula for the area of an ellipse, A, is given by

A equals pi a b

A minor circle refers to the biggest circle that fits inside an ellipse, with the same centre, as shown.

Diagram of an ellipse with a dashed inner circle, labelled "minor circle."

A major circle refers to smallest circle that fits outside an ellipse, with the same centre, as shown.

Diagram showing an ellipse inside a dashed major circle, labelled "major circle."

The following notation will be used for the different areas:

  • A subscript E is the area of an ellipse

  • A subscript C is the area of its minor circle

  • A subscript D is the area of its major circle

Complete the table below, leaving answers fully simplified and in terms of pi (where necessary).

Parts of the table have been done for you.

a

b

a over b

A subscript E

A subscript C

A subscript E over A subscript C

A subscript D

A subscript E over A subscript D

8

2

4

16 pi

4 pi

4

64 pi

1 fourth

10

5

2

50 pi

25 pi

2

12

4

How did you do?

3
Sme Calculator3 marks

An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

The formula for the area of an ellipse, A, is given by

A equals pi a b

A minor circle refers to the biggest circle that fits inside an ellipse, with the same centre, as shown.

Diagram of an ellipse with a dashed inner circle, labelled "minor circle."

A major circle refers to smallest circle that fits outside an ellipse, with the same centre, as shown.

Diagram showing an ellipse inside a dashed major circle, labelled "major circle."

The following notation is used for the different areas:

  • A subscript E is the area of an ellipse

  • A subscript C is the area of its minor circle

  • A subscript D is the area of its major circle

Use the area formulas for A subscript E, A subscript C and A subscript D to prove algebraically that

A subscript E over A subscript C

is the reciprocal of

A subscript E over A subscript D

How did you do?

4a
Sme Calculator2 marks

An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

The formula for the area of an ellipse, A, is given by

A equals pi a b

In the diagram below, the area of the circle, radius R, is equal to the area of the ellipse, pi a b.

A circle, radius R, with the same area as the ellipse.
A circle, radius R, with the same area as the ellipse

Use algebra to show that

R equals square root of a b end root

How did you do?

4b
Sme Calculator2 marks

There is no algebraic formula for the circumference of an ellipse, C.

However, there are many formulas that give approximations to the circumference of an ellipse in the form

2 pi r

One approximate formula uses r equals square root of a b end root from part (a), giving:

C subscript 1 equals 2 pi square root of a b end root

A second approximate formula uses r as the mean of a and b, giving:

C subscript 2 equals 2 pi open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses

The table below compares the approximations C subscript 1 and C subscript 2 to the true circumferences, C.

Complete the table, giving answers correct to 1 decimal place.

The first two rows have been done for you.

a

b

C

C subscript 1

C subscript 2

10

9

59.7

59.6

59.7

10

1

40.6

19.9

34.6

15

13

88.1

15

2

61.6

How did you do?

4c
Sme Calculator2 marks

Explain, with evidence from the table in part (b), which approximation out of C subscript 1 and C subscript 2 you would not recommend for ellipses that are very flat.

How did you do?

5a
Sme Calculator1 mark

An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

The point F on the diagram below is called the focus of an ellipse.

The focus is:

  • a point on the line O A

  • that is a distance c away from O

  • and that forms a right-angled triangle, B O F

  • with a hypotenuse of length a, as shown.

The length O A equals a is not shown.

Diagram of an ellipse with centre O and the points A and B on the ellipse. F is the focus of the ellipse.

The eccentricity of an ellipse, e, is the ratio of the length c to the length a, given by

e equals c over a

By finding c in terms of a and b from the diagram, show that the eccentricity can be written in the form

e equals fraction numerator square root of a squared minus b squared end root over denominator a end fraction

How did you do?

5b
Sme Calculator3 marks

Two formulas for approximating the circumference of an ellipse are space C subscript 1 equals 2 pi square root of a b end root space and space C subscript 2 equals 2 pi open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses.

A third formula to approximate the circumference of an ellipse is

C subscript 3 equals 2 pi a open parentheses 1 minus 1 fourth e squared minus 3 over 64 e to the power of 4 close parentheses

where e is the eccentricity of the ellipse, as given in part (a).

Use this formula for C subscript 3 to work out an estimate for the circumference of an ellipse with a equals 26 and b equals 24.

You must show your working clearly.

Give your answer correct to 2 decimal places.

How did you do?

5c
Sme Calculator1 mark

The true circumference of an ellipse with a equals 26 and b equals 24 is 157.14, rounded to 2 decimal places.

Explain whether the estimate in part (b) is an overestimate or an underestimate.

How did you do?

6
Sme Calculator5 marks

An ellipse is a shape that can be thought of as a squashed or stretched circle.

It can be described by a horizontal length, a, and a vertical length, b, measured from its centre, as shown below.

A horizontal ellipse with width from centre a and height from centre b, with a > b.

An ellipse can be rotated 360degree about the x-axis to form a 3D shape called a prolate spheroid, as shown, where a greater than b.

Diagram of a prolate spheroid with horizontal axis labelled 'a' and cross-sectional axes labelled 'b'.

The total surface area of a prolate spheroid, A, is given by the formula

A equals 2 pi a squared open square brackets 1 minus e squared plus fraction numerator square root of 1 minus e squared end root over denominator e end fraction open parentheses fraction numerator theta pi over denominator 180 end fraction close parentheses close square brackets

where e equals fraction numerator square root of a squared minus b squared end root over denominator a end fraction is the eccentricity of the ellipse, and theta is the acute angle in degrees that satisfies the equation

sin theta equals e

Use the information above to find the total surface area of a prolate spheroid with a equals 2 and b equals square root of 3.

Leave your answer in the exact form

A equals p pi plus q square root of 3 pi squared

where p is an integer and q is a fraction, both of which you should find.

How did you do?