Specialist Maths — Question 16

QCAA 2023, Paper 2 · 6 marks

Q16 · 2023 · Technology-activeComplex familiar6 marks

QUESTION 16 (6 marks)

A curve modelled by the relation

xy^2-y+\cos^{-1}(2x)=1

, where

-0.35\le x\le0.27

and

0\le y\le1

, intersects the

y

-axis at point A. Determine the equation of the tangent to the curve at point A.

WORKED SOLUTION

6 marks

ANSWER

y\approx-1.67x+0.57

Worked solution

x=0

, the relation gives

y=\pi/2-1\approx0.57

. Implicit differentiation gives

y^2+2xy\frac{dy}{dx}-\frac{dy}{dx}-\frac{1}{\sqrt{0.25-x^2}}=0,

\frac{dy}{dx}=\frac{1/\sqrt{0.25-x^2}-y^2}{2xy-1}.

At A the gradient is approximately

-1.67

. Thus the tangent is

y\approx-1.67x+0.57.

determines the

y

-intercept

[1 mark]

correctly differentiates

\cos^{-1}(2x)

[1 mark]

correctly differentiates

xy^2

[1 mark]

determines an expression for

dy/dx

[1 mark]

determines the gradient at A

[1 mark]

determines the equation of the tangent

[1 mark]

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

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