Mathematical Methods — Question 18

QCAA 2021, Paper 1 · 4 marks

Q18 · 2021 · Technology-freeComplex familiar4 marks

QUESTION 18 (4 marks)

The graph of

y=f(x)

, where

f(x)

is the quadratic function

f(x)=ax^2+bx+4

, is shown. Two regions of the area between the graph of

y=f(x)

and the

x

-axis are shaded. Region

P

has an area of

\dfrac{13}{6}

units

^2

and Region

Q

has an area of

\dfrac{43}{6}

units

^2

. Determine the values of

a

and

b

WORKED SOLUTION

4 marks

ANSWER

a=2

b=5

Worked solution

\int_{-1}^{0}f(x)\,dx=\frac{13}{6},\qquad \int_0^1f(x)\,dx=\frac{43}{6}.

With antiderivative

\frac{ax^3}{3}+\frac{bx^2}{2}+4x+C

, these give

2a-3b=-11

and

2a+3b=19

. Solving gives

b=5

and

a=2

, so

f(x)=2x^2+5x+4

correctly identifies the use of integrals

[1 mark]

correctly determines the two equations in the two unknowns a and b

[1 mark]

determines b

[1 mark]

determines a

[1 mark]

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

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