Mathematical Methods — Question 13

QCAA 2024, Paper 1 · 6 marks

Q13 · 2024 · Technology-freeSimple familiar6 marks

QUESTION 13 (6 marks)

F(x)=\displaystyle\int(4e^{2x}+\sin(2x))\,dx

. Use integration to determine

F(x)

, if

F(0)=5

[3 marks]

\displaystyle\frac{dy}{dx}=\left(\frac{3x^7-2x}{x^4}\right)^2

, determine

y

[3 marks]

WORKED SOLUTION

6 marks

ANSWER

F(x)=2e^{2x}-\frac12\cos(2x)+\frac72

; b)

y=\frac97x^7-12x-\frac45x^{-5}+c

Worked solution

F(x)=2e^{2x}-\frac12\cos(2x)+c.

Since

F(0)=5

5=2-\frac12+c,

c=\frac72

. Therefore

F(x)=2e^{2x}-\frac12\cos(2x)+\frac72.

\frac{dy}{dx}=\left(3x^3-\frac{2}{x^3}\right)^2=9x^6-12+4x^{-6}.

Hence

y=\int(9x^6-12+4x^{-6})\,dx=\frac97x^7-12x-\frac45x^{-5}+c.

correctly integrates the exponential term

[1 mark]

correctly integrates the trigonometric term

[1 mark]

determines the constant of integration

[1 mark]

correctly expands the squared bracket

[1 mark]

simplifies the expanded squared bracket into separate terms

[1 mark]

determines y in terms of x by integrating all terms

[1 mark]

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

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