Give your answer in the form \(a\sqrt{2}\) where \(a\) is an integer. [3 marks]
Mark scheme
Answer
Mark
Comments
Alternative method 1: sine rule
\(\dfrac{10}{\sin 45} = \dfrac{AC}{\sin 30}\)
M1
oe could be seen by implication in further working
\(\sin 30 = \dfrac{1}{2}\) and \(\sin 45 = \dfrac{1}{\sqrt{2}}\) or \(\dfrac{\sqrt{2}}{2}\)
B1
oe could be seen by implication in working must be clearly selected if part of a list of values
\(5\sqrt{2}\)
A1
Alternative method 2: splitting the triangle into 2 right angled triangles with a perpendicular from C to the side AB at a point D
\(CD = 10\sin 30\) and \(AC = \dfrac{CD}{\sin 45}\) or \(\sqrt{CD^2 + CD^2}\)
M1
oe
\(\sin 30 = \dfrac{1}{2}\) and \(\sin 45 = \dfrac{1}{\sqrt{2}}\) or \(\dfrac{\sqrt{2}}{2}\)
B1
oe could be seen by implication in further working must be clearly selected if part of a list of values if Pythagoras used in M mark sin45 not needed so \(\sin 30 = \dfrac{1}{2}\) is enough for B1
(a) Write down the value of \(x\) for \(\quad 0^\circ \leqslant x \leqslant 360^\circ \quad\) when \(\quad \sin x = -1\) [1 mark]
(b) Work out the values of \(y\) for \(\quad 0^\circ \leqslant y \leqslant 360^\circ \quad\) when \(\quad \sqrt{3}\tan y = 1\) [3 marks]
Mark scheme (a)
Answer
Mark
Comments
(\(x =\)) \(270^\circ\)
B1
Additional guidance
Do not accept answers outside of the given range or a choice of answers
Condone \(^\circ\) missing
Mark scheme (b)
Answer
Mark
Comments
\(\tan y = \dfrac{1}{\sqrt{3}}\) or \(\tan y = \dfrac{\sqrt{3}}{3}\) or \(y = \tan^{-1}\dfrac{\sqrt{3}}{3}\) or \(y = \tan^{-1}\dfrac{1}{\sqrt{3}}\)
M1
(\(y =\)) \(30^\circ\) or \(210^\circ\)
M1dep
could include up to two further incorrect values or \(30^\circ\) may be seen as part of a further calculation or on a diagram or embedded in \(\tan 30^\circ\) \(\tan 30^\circ\) embedded in a table with other exact trigonometric values (and not identified) would not gain M1dep
Work out the exact value of \(\sin\theta\). [3 marks]
Mark scheme
Answer
Mark
Comments
\(\sin 135^\circ = \dfrac{\sqrt{2}}{2}\) or \(\dfrac{1}{\sqrt{2}}\)
B1
oe could be embedded in the calculation
\(\dfrac{\sin\theta}{x} = \dfrac{\sin 135}{4x}\)
M1
oe \(x\) may be a different letter (would need to be the same letter for both sides) or numerical values using multiples of 1 and 4 used
(\(\sin\theta =\)) \(\dfrac{\sqrt{2}}{8}\)
A1
oe eg \(\dfrac{1}{4\sqrt{2}}\)
Additional guidance
Penalise incorrect further working Condone \(\theta = \sin^{-1}\dfrac{\sqrt{2}}{8}\) seen in answer as long as \(\sin\theta = \dfrac{\sqrt{2}}{8}\) seen in working
15 Here is a sketch graph of \(\quad y = \cos x \quad\) for \(\quad 0^\circ \leqslant x \leqslant 360^\circ\)
You are given that \(\quad \cos 36^\circ = 0.8090\)
Solve \(\quad \cos x = -0.8090 \quad\) for \(\quad 0^\circ \leqslant x \leqslant 360^\circ\) [2 marks]
Mark scheme
Answer
Mark
Comments
\(144^\circ\)
B1
answers should be on answer line but can be accepted if they are the only angles written on the diagram (other than \(36^\circ\) which is the question so fine) condone missing degree sign
\(216^\circ\)
B1
Additional guidance
Don’t accept \(\cos 144^\circ\), \(\cos 216^\circ\), \(\cos x = 144^\circ\), \(\cos x = 216^\circ\)
B0
Accept \(\cos 144^\circ = -0.8090\) and \(\cos 216^\circ = -0.8090\)
B1, B1
If more than 2 angles offered this is choice 4 or more angles 2 wrong 1 right 1 wrong 2 right 1 wrong 1 right
\((x =)\ \dfrac{6\sqrt{2} \times \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}\) or \(\dfrac{6\sqrt{2} \times \frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}}\)
M1dep
oe a correct expression or equation coming from both surds correct eg \(\ \dfrac{\sqrt{3}}{2}x = 6\sqrt{2} \times \dfrac{1}{\sqrt{2}}\) or \(\dfrac{6}{x} = \dfrac{\sqrt{3}}{2}\)
\((x =)\ \dfrac{12}{\sqrt{3}}\) or \(\sqrt{3}x = 12\)
A1
\((x =)\ 4\sqrt{3}\)
A1
Alternative method 2
\(\sin 45 = \dfrac{1}{\sqrt{2}}\) or \(\dfrac{\sqrt{2}}{2}\) or \(\sin 60 = \dfrac{\sqrt{3}}{2}\)
B1
look for these in their working … one of them correct will gain the mark at any stage of their working
\(\dfrac{AD}{6\sqrt{2}} = \sin 45\) or \((AD =)\ 6\sqrt{2} \times \sin 45\) or \((AD =)\ 6\)
M1
where D is the foot of the perpendicular from A to BC
\((x =)\ \dfrac{6\sqrt{2} \times \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}\) or \(\dfrac{6\sqrt{2} \times \frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}}\)
M1dep
oe a correct expression or equation coming from both surds correct eg \(\ \dfrac{\sqrt{3}}{2}x = 6\sqrt{2} \times \dfrac{1}{\sqrt{2}}\) or \(\dfrac{6}{x} = \dfrac{\sqrt{3}}{2}\)
\((x =)\ \dfrac{12}{\sqrt{3}}\) or \(\sqrt{3}x = 12\)
A1
\((x =)\ 4\sqrt{3}\)
A1
Additional guidance
If one surd value is incorrect they can only score a maximum of 2 marks
SC1 for a final answer of \(\dfrac{2\sqrt{3}\sin 15^\circ}{\sin 135^\circ}\), possibly with \(\dfrac{1}{\sqrt{2}}\) or \(\dfrac{\sqrt{2}}{2}\) for \(\sin 135^\circ\)
