• Setting $\frac{dy}{dx} = 0$ M1

• $y \cos (xy) = 0$

• $y \neq 0 \Rightarrow \cos (xy) = 0$ A1

• $\Rightarrow \sin (xy) = \pm \sqrt{1 - \cos^2 (xy)} = \pm \sqrt{1 - 0} = \pm 1$

• OR $xy= (2n+1) \frac{\pi}{2}$ OR $xy = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$ A1

If they offer values for $xy$, award A1 for at least two correct values in two different 'quadrants' and no incorrect values.

• $y^2 = \sin (xy) > 0$ R1

• $\Rightarrow y^2 = 1$ A1

• $\Rightarrow y = \pm 1$ (as given)

5 marks total

• Attempt at implicit differentiation M1

• $2y\frac{dy}{dx}=\cos(xy) \left(x\frac{dy}{dx}+y \right)$ A1M1A1

Award A1 for LHS, M1 for attempt at chain rule, A1 for RHS.

• $2y\frac{dy}{dx}=x\frac{dy}{dx}\cos(xy)+y\cos(xy)$

• $2y\frac{dy}{dx}-x\frac{dy}{dx}\cos(xy)=y\cos(xy)$

• $\frac{dy}{dx}\left(2y-x\cos(xy)\right)=y\cos(xy)$ M1

Award M1 for collecting derivatives and factorising.

• $\frac{dy}{dx}=\frac{y\cos(xy)}{2y-x\cos(xy)}$ AG

5 marks total

• $y = \pm 1 \Rightarrow 1 = \sin (\pm x) \Rightarrow \sin x = \pm 1$

Method #1

• OR $y = \pm 1 \Rightarrow 0 = \cos (\pm x) \Rightarrow \cos x = 0$ M1

• $\left(\sin x = 1 \Rightarrow \left(\frac{\pi}{2}, 1\right)\right)$, $\left(\sin x = -1 \Rightarrow \left(\frac{3\pi}{2}, -1\right)\right)$, $\left(\sin x = -1 \Rightarrow \left(\frac{7\pi}{2}, -1\right)\right)$ A1A1

Allow 'coordinates' expressed as $x = \frac{\pi}{2}, y = 1$ for example.

Each of the A marks may be awarded independently and are not dependent on (M1) being awarded.

Mark only the candidate's first two attempts for each case of $\sin x$.

5 marks total