QCE Mathematical Methods: Exam practice Q14 (2025, Paper 2)

Mathematical Methods — Question 14

QCAA 2025, Paper 2 · 6 marks

Q14 · 2025 · Technology-activeSimple familiar6 marks

QUESTION 14 (6 marks)

The number of tourists visiting a country at any given time is modelled by

N(t)=18000\sin\left(\frac{\pi}{6}t+6\right)+22000,\qquad0\le t\le12,

where

t

is the time (months) from the start of the year.

Determine when the number of tourists first reaches 36 000. Provide your response as a decimal number.

[1 mark]

Determine the equation of the second derivative of

N(t)

[2 marks]

Determine the value of the second derivative when the number of tourists is a minimum.

[2 marks]

Use your result from Question 14c) to explain how the value of the second derivative is consistent with the number of tourists being a minimum.

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

t=2.2428

months; b)

N''(t)=-500\pi^2\sin(\frac{\pi}{6}t+6)

; c)

4934.80

; d) it is positive, consistent with a local minimum.

Worked solution

a) Solving

36000=18000\sin\left(\frac\pi6t+6\right)+22000

with a GDC gives the first solution

t=2.2428\text{ months}.

N'(t)=3000\pi\cos\left(\frac\pi6t+6\right),

N''(t)=-500\pi^2\sin\left(\frac\pi6t+6\right).

c) Using a GDC, the minimum occurs at

t=9.5408\text{ months}.

Then

N''(9.5408)=4934.80.

d) The second derivative is positive, which supports the number of tourists being at a local minimum.

correctly determines when tourists first reach 36 000 as a decimal

[1 mark]

correctly determines the equation of the first derivative

[1 mark]

determines the equation of the second derivative

[1 mark]

correctly determines the time when the number of tourists is a minimum

[1 mark]

determines the value of the second derivative when the number of tourists is a minimum

[1 mark]

explains how a positive value of the second derivative is consistent with a local minimum

[1 mark]

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Mathematical Methods — Question 14

QUESTION 14 (6 marks)

QCAA guide · typeset solution

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