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Date May 2014 Marks available 3 Reference code 14M.2.sl.TZ1.9
Level SL only Paper 2 Time zone TZ1
Command term Find Question number 9 Adapted from N/A

Question

Let \(f(x) = \cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right),{\text{ for }} - 4 \leqslant x \leqslant 4.\)

Sketch the graph of \(f\).

[3]
a.

Find the values of \(x\) where the function is decreasing.

[5]
b.

The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(a\);

[3]
c(i).

The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(c\).

[4]
c(ii).

Markscheme


          A1A1A1     N3

 

Note:     Award A1 for approximately correct sinusoidal shape.

     Only if this A1 is awarded, award the following:

     A1 for correct domain,

     A1 for approximately correct range.

 

[3 marks]

 

a.

recognizes decreasing to the left of minimum or right of maximum,

eg     \(f'(x) < 0\)     (R1)

x-values of minimum and maximum (may be seen on sketch in part (a))     (A1)(A1)

eg     \(x =  - 3,{\text{ (1, 1.4)}}\)

two correct intervals     A1A1     N5

eg     \( - 4 < x <  - 3,{\text{ }}1 \leqslant x \leqslant 4;{\text{ }}x <  - 3,{\text{ }}x \geqslant 1\)

[5 marks]

b.

recognizes that \(a\) is found from amplitude of wave     (R1)

y-value of minimum or maximum     (A1)

eg     (−3, −1.41) , (1, 1.41)

\(a = 1.41421\)

\(a = \sqrt 2 {\text{,   (exact), 1.41,}}\)     A1     N3

[3 marks]

c(i).

METHOD 1

recognize that shift for sine is found at x-intercept     (R1)

attempt to find x-intercept    (M1)

eg     \(\cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right) = 0,{\text{ }}x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)

\(x =  - 1\)     (A1)

\(c = 1\)     A1     N4

 

METHOD 2

attempt to use a coordinate to make an equation     (R1)

eg     \(\sqrt 2 \sin \left( {\frac{\pi }{4}c} \right) = 1,{\text{ }}\sqrt 2 \sin \left( {\frac{\pi }{4}(3 - c)} \right) = 0\)

attempt to solve resulting equation     (M1)

eg     sketch, \(x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)

\(x =  - 1\)     (A1)

\(c = 1\)     A1     N4

[4 marks]

c(ii).

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c(i).
[N/A]
c(ii).

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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