Mathematical Methods — Question 17

QCAA 2022, Paper 1 · 4 marks

Q17 · 2022 · Technology-freeComplex familiar4 marks

QUESTION 17 (4 marks)

Determine the value of

b

given

\int_a^b3x^2\,dx=117

and

\int_a^{b-1}3x^2\,dx=56

for

b>1

WORKED SOLUTION

4 marks

ANSWER

b=5

Worked solution

Using the first integral,

b^3-a^3=117.\tag{I}

Using the second,

(b-1)^3-a^3=56.\tag{II}

Subtracting (II) from (I),

b^3-(b-1)^3=61,

3b^2-3b-60=0,\qquad b^2-b-20=0,

hence

(b-5)(b+4)=0

. Thus

b=-4

5

. Given

b>1

b=5.

correctly establishes a formula for one of the integrals

[1 mark]

determines equation in

b

[1 mark]

determines values of

b

[1 mark]

evaluates the reasonableness of solutions

[1 mark]

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

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