Specialist Maths — Question 13

QCAA 2025, Paper 2 · 7 marks

Q13 · 2025 · Technology-activeSimple familiar7 marks

QUESTION 13 (7 marks)

The sketch shows sections of the functions

f(x)=-0.5x^2+7.5x-18

and

g(x)=4\operatorname{cosec}\left(\frac{\pi x}{24}\right)-4.

Two points of intersection at

(12,0)

and point

A

are shown.

Graph of f(x) and g(x), intersecting at A and (12,0), with the bounded region between them shaded.

Determine the coordinates of point

A

[1 mark]

Consider the shaded bounded region between the functions. Determine an approximate area of this region using Simpson’s rule with four intervals. Show evidence of the values substituted into this rule in your solution.

[3 marks]

State a definite integral that represents the area of the shaded bounded region.

[1 mark]

Determine the value of your result from Question 13c). Give your answer to two decimal places.

[1 mark]

Other than working to more decimal places, state a strategy involving Simpson’s rule that could be used to improve the accuracy of your result from Question 13b).

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

A=(4,4)

; Simpson approximation

\approx50.37

units

^2

; exact-area value

\approx50.42

units

^2

Worked solution

a) Using a GDC,

A=(4,4)

b) Method 1: using four intervals from

x=4

x=12

gives

w=2

. For

f(x)-g(x)

x

4,\ 6,\ 8,\ 10,\ 12

f(x)-g(x)

0,\ 7.34,\ 9.38,\ 6.86,\ 0

Using Simpson’s rule,

A\approx\frac23\left[0+4(7.34+6.86)+2(9.38)+0\right]\approx50.37\text{ units}^2.

Alternative method 2: apply Simpson’s rule separately to

f

and

g

using

x

4,\ 6,\ 8,\ 10,\ 12

f(x)

4,\ 9,\ 10,\ 7,\ 0

g(x)

4,\ 1.66,\ 0.62,\ 0.14,\ 0

For

f

A_f\approx\frac23\left[4+4(9+7)+2(10)+0\right]\approx58.667.

For

g

A_g\approx\frac23\left[4+4(1.66+0.14)+2(0.62)+0\right]\approx8.293.

Therefore the required area is approximately

58.667-8.293=50.37\text{ units}^2.

c) The exact area is represented by

\int_4^{12}\left[-0.5x^2+7.5x-18-\left(4\operatorname{cosec}\left(\frac{\pi x}{24}\right)-4\right)\right]dx.

A\approx50.42\text{ units}^2

e) The accuracy could be improved by using more intervals (an even number).

correctly determines point

A

[1 mark]

correctly recognises the requirement to determine the relevant function values for Simpson’s rule

[1 mark]

shows evidence of use of Simpson’s rule

[1 mark]

determines the required approximate area

[1 mark]

correctly determines a definite integral, including the

dx

term, that represents the exact area

[1 mark]

determines the rounded value of the exact area

[1 mark]

correctly states a suitable strategy

[1 mark]

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Specialist Maths — Question 13

QUESTION 13 (7 marks)

QCAA guide · typeset solution

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