QCE Vault / Specialist Maths Specialist Maths — Question 16 QCAA 2023, Paper 1 · 5 marks
Browse all questions Original practice exam Report an issue Q16 · 2023 · Technology-free Complex familiar 5 marks
QUESTION 16 (5 marks) A curve is defined by the parametric equations x = 2 tan ( θ ) x=2\tan(\theta) x = 2 tan ( θ ) and y = 3 sin ( 2 θ ) y=3\sin(2\theta) y = 3 sin ( 2 θ ) , where 0 ≤ θ < π 2 0\le\theta<\frac{\pi}{2} 0 ≤ θ < 2 π .
Given that d y d x \frac{dy}{dx} d x d y can be expressed in the form a cos 4 ( θ ) + b cos 2 ( θ ) a\cos^4(\theta)+b\cos^2(\theta) a cos 4 ( θ ) + b cos 2 ( θ ) , where a , b ∈ R a,b\in\mathbb R a , b ∈ R , determine the values of a a a and b b b . WORKED SOLUTION
QCAA guide · typeset solution 5 marks Worked solution
d x / d θ = 2 sec 2 θ dx/d\theta=2\sec^2\theta d x / d θ = 2 sec 2 θ and d y / d θ = 6 cos 2 θ dy/d\theta=6\cos2\theta d y / d θ = 6 cos 2 θ . Therefore
d y d x = 3 cos ( 2 θ ) cos 2 θ = 3 ( 2 cos 2 θ − 1 ) cos 2 θ = 6 cos 4 θ − 3 cos 2 θ . \frac{dy}{dx}=3\cos(2\theta)\cos^2\theta
=3(2\cos^2\theta-1)\cos^2\theta
=6\cos^4\theta-3\cos^2\theta. d x d y = 3 cos ( 2 θ ) cos 2 θ = 3 ( 2 cos 2 θ − 1 ) cos 2 θ = 6 cos 4 θ − 3 cos 2 θ .
Thus a = 6 a=6 a = 6 and b = − 3 b=-3 b = − 3 . determines d x / d θ dx/d\theta d x / d θ [1 mark] determines d y / d θ dy/d\theta d y / d θ [1 mark] determines an expression for d y / d x dy/dx d y / d x [1 mark] expresses d y / d x dy/dx d y / d x in the required form [1 mark] states the values of a a a and b b b [1 mark] One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
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