20N.1.AHL.TZ0.H_11

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt at implicit differentiation       M1

$2y\frac{\text{d}y}{\text{d}x}= \cos \left(xy\right)⌊x\frac{\text{d}y}{\text{d}x}+y⌋$       A1M1A1

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

$2y\frac{\text{d}y}{\text{d}x}= x\frac{\text{d}y}{\text{d}x}\cos \left(xy\right)+y \cos \left(xy\right)$

$2y\frac{\text{d}y}{\text{d}x}- x\frac{\text{d}y}{\text{d}x}\cos \left(xy\right)=y \cos \left(xy\right)$

$\frac{\text{d}y}{\text{d}x}\left(2y-x \cos \left(xy\right)\right)=y \cos \left(xy\right)$       M1

Note: Award M1 for collecting derivatives and factorising.

$\frac{\text{d}y}{\text{d}x}=\frac{y \cos \left(xy\right)}{2y-x \cos \left(xy\right)}$       AG

[5 marks]

setting $\frac{\text{d}y}{\text{d}x}=0$

$y \cos \left(xy\right)=0$       (M1)

$\left(y\ne 0\right)\Rightarrow \cos \left(xy\right)=0$       A1

$\Rightarrow \sin \left(xy\right)\left(=\pm \sqrt{1-{\cos }^2 \left(xy\right)}=\pm \sqrt{1-0}\right)=\pm 1$  OR  $xy=\left(2n+1\right)\frac{\mathrm{\pi }}{2} \left(n\in \mathrm{\mathbb{Z}}\right)$  OR  $xy=\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2},\dots$       A1

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

$y^2 \left(=\sin \left(xy\right)\right)>0$       R1

$\Rightarrow y^2=1$       A1

$\Rightarrow y=\pm 1$       AG

[5 marks]

$y=\pm 1\Rightarrow 1=\sin \left(\pm x\right)\Rightarrow \sin x=\pm 1$  OR  $y=\pm 1\Rightarrow 0=\cos \left(\pm x\right)\Rightarrow \cos x=0$       (M1)

$\left(\sin x=1\Rightarrow \right)\left(\frac{\mathrm{\pi }}{2}, 1\right), \left(\frac{5\mathrm{\pi }}{2}, 1\right)$       A1A1

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

Note: Allow ‘coordinates’ expressed as $x=\frac{\mathrm{\pi }}{2}, y=1$ for example.
Note: Each of the A marks may be awarded independently and are not dependent on (M1) being awarded.

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

[5 marks]

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