Mathematical Methods — Question 11

QCAA 2025, Paper 2 · 6 marks

Q11 · 2025 · Technology-activeSimple familiar6 marks

QUESTION 11 (6 marks)

Determine the equation of

f(x)

f'(x)=\cos(x)-4\sin(2x+\pi)

and

f(\pi)=2

[4 marks]

Use your result from Question 11a) to determine the value of

x

where

f(x)=4.8

, for

2\le x\le4

[1 mark]

Use your result from Question 11a) to determine the minimum value of

f(x)

, for

2\le x\le4

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

f(x)=\sin x+2\cos(2x+\pi)+4

; b)

x=2.3364

; c) minimum

f(x)=1.9375

Worked solution

a) Integrating,

f(x)=\sin x+2\cos(2x+\pi)+c.

Using

f(\pi)=2

2=\sin\pi+2\cos(3\pi)+c=0-2+c,

c=4

. Therefore

f(x)=\sin x+2\cos(2x+\pi)+4.

b) Using a GDC to solve

f(x)=4.8

for

2\le x\le4

gives

x=2.3364.

c) Using a GDC on the stated domain gives a minimum value

f(x)=1.9375.

correctly integrates the cosine term in

f'(x)

[1 mark]

correctly integrates the sine term in

f'(x)

[1 mark]

substitutes

\pi

and 2 appropriately into the equation for

f(x)

[1 mark]

determines the constant of integration in the function

f(x)

[1 mark]

determines the solution in the stated domain

[1 mark]

determines the minimum

f(x)

value

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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Mathematical Methods — Question 11

QUESTION 11 (6 marks)

QCAA guide · typeset solution

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