Mathematical Methods — Question 13

QCAA 2023, Paper 2 · 4 marks

Q13 · 2023 · Technology-activeSimple familiar4 marks

QUESTION 13 (4 marks)

The curved lines represent graphs of the equations

y=x^2-4x+8

and

y=10\cos(x+10)

Determine the coordinates of the points of intersection A and B.

[1 mark]

State an integral expression representing the area enclosed by the two graphs.

[2 marks]

Determine the area enclosed by the two graphs.

[1 mark]

WORKED SOLUTION

4 marks

ANSWER

A(1.44140,4.31203)

B(3.47247,6.16818)

; b)

\displaystyle\int_{1.44140}^{3.47247}[10\cos(x+10)-(x^2-4x+8)]\,dx

; c)

7.64702

units

^2

Worked solution

a) Using GDC:

A(1.44140,4.31203),\qquad B(3.47247,6.16818).

b) The enclosed area is represented by

\int_{1.44140}^{3.47247}\left(10\cos(x+10)-(x^2-4x+8)\right)\,dx.

c) Using GDC,

\text{Area}=7.64702\text{ units}^2.

correctly determines A and B

[1 mark]

states the difference in functions required for area, in the correct order (to obtain a positive value)

[1 mark]

uses the x-coordinates from Q13a) in the definite integral

[1 mark]

determines the area value

[1 mark]

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

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