C3 June 2016 Q7

EdexcelOld spec5 marksFunctions (including |mod|)Trigonometry

(a) For \(-\dfrac{\pi}{2} \leqslant y \leqslant \dfrac{\pi}{2}\), sketch the graph of \(y = \mathrm{g}(x)\) where\[\mathrm{g}(x) = \arcsin x \qquad -1 \leqslant x \leqslant 1\] (2)

(b) Find the exact value of \(x\) for which\[3\mathrm{g}(x + 1) + \pi = 0\] (3)

Sketch of y = arcsin x: increasing curve through the origin, in the first and third quadrants, steep at each end — Mark scheme (a)

Scheme	Marks
Correct position or curvature	M1
Correct position and curvature	A1
	(2)

Mark scheme (b)

Scheme	Marks
\(3\arcsin(x + 1) + \pi = 0 \Rightarrow \arcsin(x + 1) = -\dfrac{\pi}{3}\)	M1
\(\Rightarrow (x + 1) = \sin\left(-\dfrac{\pi}{3}\right)\)
\(\Rightarrow x = -1 - \dfrac{\sqrt{3}}{2}\)	dM1A1
	(3)
	(5 marks)

Notes

M1: Substitutes \(\mathrm{g}(x + 1) = \arcsin(x + 1)\) in \(3\mathrm{g}(x + 1) + \pi = 0\) and attempts to make \(\arcsin(x + 1)\) the subject
Accept \(\arcsin(x + 1) = \pm\dfrac{\pi}{3}\) or even \(\mathrm{g}(x + 1) = \pm\dfrac{\pi}{3}\). Condone \(\dfrac{\pi}{3}\) in decimal form awrt1.047

dM1: Proceeds by evaluating \(\sin\left(\pm\dfrac{\pi}{3}\right)\) and making \(x\) the subject.
Accept for this mark \(\Rightarrow x = \pm\dfrac{\sqrt{3}}{2} \pm 1\). Accept decimal such as \(-1.866\)
Do not allow this mark if the candidate works in mixed modes (radians and degrees)
You may condone invisible brackets for both M's as long as the candidate is working correctly with the function

A1: \(-1 - \dfrac{\sqrt{3}}{2}\) oe with no other solutions. Remember to isw after a correct answer
Be careful with single fractions. \(-\dfrac{2 - \sqrt{3}}{2}\) and \(\dfrac{-2 + \sqrt{3}}{2}\) are incorrect but \(-\dfrac{2 + \sqrt{3}}{2}\) is correct

Note: It is possible for a candidate to change \(\dfrac{\pi}{3}\) to \(60^\circ\) and work in degrees for all marks