(a)     Sketch the curve \(f(x) = \sin 2x\) , \(0 \leqslant x \leqslant \pi \) .

(b)     Hence sketch on a separate diagram the graph of \(g(x) = \csc 2x\) , \(0 \leqslant x \leqslant \pi \) , clearly stating the coordinates of any local maximum or minimum points and the equations of any asymptotes.

(c)     Show that tan \(x + \cot x \equiv 2\csc 2x\) .

(d)     Hence or otherwise, find the coordinates of the local maximum and local minimum points on the graph of \(y = \tan 2x + \cot 2x\) , \(0 \leqslant x \leqslant \frac{\pi }{2}\) .

(e)     Find the solution of the equation \(\csc 2x = 1.5\tan x - 0.5\) , \(0 \leqslant x \leqslant \frac{\pi }{2}\) .

(a)         A2

 

Note: Award A1 for shape.

A1 for scales given on each axis.

 

[2 marks]

 

(b)         A5

Asymptotes \(x = 0,{\text{ }}x = \frac{\pi }{2},{\text{ }}x = \pi \)

\({\text{Max }}\left( {\frac{{3\pi }}{4}, - 1} \right)\) , \({\text{Min }}\left( {\frac{\pi }{4},1} \right)\)

Note: Award A1 for shape

A2 for asymptotes, A1 for one error, A0 otherwise.

A1 for max.

A1 for min.

 

[5 marks]

 

(c)     \(\tan x + \cot x \equiv \frac{{\sin x}}{{\cos x}} + \frac{{\cos x}}{{\sin x}}\)     M1

\( \equiv \frac{{{{\sin }^2}x + {{\cos }^2}x}}{{\sin x\cos x}}\)     A1

\( \equiv \frac{1}{{\frac{1}{2}\sin 2x}}\)     A1

\( \equiv 2\csc 2x\)     AG

[3 marks]

 

(d)     \(\tan 2x + \cot 2x \equiv 2\csc 4x\)     (M1)

Max is at \(\left( {\frac{{3\pi }}{8}, - 2} \right)\)     A1A1

Min is at \(\left( {\frac{\pi }{8},2} \right)\)     A1A1

[5 marks]

 

(e)     \(\csc 2x = 1.5\tan x - 0.5\)

\(\frac{1}{2}\tan x + \frac{1}{2}\cot x = \frac{3}{2}\tan x - \frac{1}{2}\)     M1

\(\tan x + \cot x = 3\tan x - 1\)

\(2\tan x - \frac{1}{{\tan x}} - 1 = 0\)     M1

\(2{\tan ^2}x - \tan x - 1 = 0\)     A1

\((2\tan x + 1)(\tan x - 1) = 0\)     M1

\(\tan x = - \frac{1}{2}{\text{ or 1}}\)     A1

\({\text{x = }}\frac{\pi }{4}\)     A1

Note: Award A0 for answer in degrees or if more than one value given for x.

[6 marks]

Total [21 marks]

Although the better candidates scored well on this question, it was disappointing to see that a number of candidates did not appear to be well prepared and made little progress. It was disappointing that a small minority of candidates were unable to sketch \(y = \sin 2x\) . Most candidates who completed part (a) attempted part (b), although not always successfully. In many cases the coordinates of the local maximum and minimum points and the equations of the asymptotes were not clearly stated. Part (c) was attempted by the vast majority of candidates. The responses to part (d) were disappointing with a significant number of candidates ignoring the hence and attempting differentiation which more often than not resulted in either arithmetic or algebraic errors. A reasonable number of candidates gained the correct answer to part (e), but a number tried to solve the equation is terms of sin x and cos x and made little progress.