CAIE A-Level Mathematics AS 2.4.3 Implicit & Parametric

[Maximum number: 7]

A curve is defined by the parametric equations

x=4 \cos ^{2} t, \quad y=\sqrt{3} \sin 2 t,

for values of t such that $0<t<\frac{1}{2} \pi$ .
Find the equation of the normal to the curve at the point for which $t=\frac{1}{6} \pi$ . Give your answer in the form a x+b y+c=0 where a, b and c are integers.

Obtain forms $\frac{\mathrm{d} x}{\mathrm{~d} t}=k_{1} \cos t \sin t$ or $\frac{\mathrm{d} y}{\mathrm{~d} t}=k_{2} \cos 2 t$

*M1

Obtain correct $-8 \cos t \sin t$ or $-4 \sin 2 t$ and $2 \sqrt{3} \cos 2 t$
Attempt value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $t=\frac{1}{6} \pi$

*DM1

Need to see attempt at substitution.

Obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{1}{2}$
State or imply gradient of normal is 2

**M1 FT

Following their value of the first derivative.

Attempt equation of normal

**DM1

Not tangent and with attempt to find coordinates $\left(3, \frac{3}{2}\right)$ .

Obtain 4 x-2 y-9=0
Or equivalent of requested form.

CAIE A-Level Mathematics AS 2.4.3 Differentiation Rules Questions

Exam points

CAIE A-Level Mathematics AS 2.4.3 Differentiation Rules Questions question 1