Further integration — Question 96

Original QCE Vault practice · 5 marks

Q96 · Practice questionTechnology-freeComplex unfamiliar5 marks

QUESTION 96 (5 marks)

The curves

y=k-x^2

and

y=x

, where

k>0

, enclose a region of area

\dfrac92

square units, as shown. Determine

k

WORKED SOLUTION

5 marks

ANSWER

k=2

Worked solution

The intersections are the roots of

x^2+x-k=0

, whose separation is

d=\sqrt{1+4k}

. Between the roots the vertical gap is a downward quadratic with zeros at the endpoints, and its area is

d^3/6

. Hence

(1+4k)^{3/2}/6=9/2

, so

(1+4k)^{3/2}=27

, giving

1+4k=9

and

k=2

Finds the intersection equation

x^2+x-k=0

[1 mark]

Identifies the root separation

\sqrt{1+4k}

[1 mark]

Forms an equivalent correct expression for the enclosed area.

[1 mark]

Sets the area equal to

9/2

and simplifies.

[1 mark]

Obtains

k=2

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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