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Date	May 2014	Marks available	2	Reference code	14M.2.sl.TZ2.6
Level	SL only	Paper	2	Time zone	TZ2
Command term	Find	Question number	6	Adapted from	N/A

Question

Let $f(x) = p\cos \left( {q(x + r)} \right) + 10$ , for $0 \leqslant x \leqslant 20$ . The following diagram shows the graph of $f$ .

The graph has a maximum at $(4, 18)$ and a minimum at $(16, 2)$ .

Write down the value of $r$ .

[2]

Find $p$ .

[2]

b(i).

Find $q$ .

[2]

b(ii).

Solve $f(x) = 7$ .

[2]

Markscheme

$r = - 4$ A2 N2

Note: Award A1 for $r = 4$ .

[2 marks]

evidence of valid approach (M1)

eg $\frac{{\max y{\text{ value -- }}y{\text{ value}}}}{2}$ , distance from $y = 10$

$p = 8$ A1 N2

[2 marks]

b(i).

valid approach (M1)

eg period is $24$ , $\frac{{360}}{{24}}$ , substitute a point into their $f(x)$

$q = \frac{{2\pi }}{{24}}\left( {\frac{\pi }{{12}},{\text{ exact}}} \right)$ , $0.262$ (do not accept degrees) A1 N2

[2 marks]

b(ii).

valid approach (M1)

eg line on graph at $y = 7,{\text{ }}8\cos \left( {\frac{{2\pi }}{{24}}(x - 4)} \right) + 10 = 7$

$x = 11.46828$

$x = 11.5$ (accept $(11.5, 7)$ ) A1 N2

[2 marks]

Note: Do not award the final A1 if additional values are given. If an incorrect value of $q$ leads to multiple solutions, award the final A1 only if all solutions within the domain are given.

Examiners report

[N/A]

b(i).

[N/A]

b(ii).

[N/A]

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.4 » The circular functions

$\sin x$ ,

$\cos x$ and

$\tan x$ : their domains and ranges; amplitude, their periodic nature; and their graphs.

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