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Date November 2013 Marks available 2 Reference code 13N.1.sl.TZ0.5
Level SL only Paper 1 Time zone TZ0
Command term Find Question number 5 Adapted from N/A

Question

Let \(f(x) = \sin \left( {x + \frac{\pi }{4}} \right) + k\). The graph of f passes through the point \(\left( {\frac{\pi }{4},{\text{ }}6} \right)\).

Find the value of \(k\).

[3]
a.

Find the minimum value of \(f(x)\).

[2]
b.

Let \(g(x) = \sin x\). The graph of g is translated to the graph of \(f\) by the vector \(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\).

Write down the value of \(p\) and of \(q\).

[2]
c.

Markscheme

METHOD 1

attempt to substitute both coordinates (in any order) into \(f\)     (M1)

eg     \(f\left( {\frac{\pi }{4}} \right) = 6,{\text{ }}\frac{\pi }{4} = \sin \left( {6 + \frac{\pi }{4}} \right) + k\)

correct working     (A1)

eg     \(\sin \frac{\pi }{2} = 1,{\text{ }}1 + k = 6\)

\(k = 5\)     A1     N2

[3 marks]

METHOD 2

recognizing shift of \(\frac{\pi }{4}\) left means maximum at \(6\)     R1)

recognizing \(k\) is difference of maximum and amplitude     (A1)

eg     \(6 - 1\)

\(k = 5\)     A1     N2

[3 marks] 

a.

evidence of appropriate approach     (M1)

eg     minimum value of \(\sin x\) is \( - 1,{\text{ }} - 1 + k,{\text{ }}f'(x) = 0,{\text{ }}\left( {\frac{{5\pi }}{4},{\text{ }}4} \right)\)

minimum value is \(4\)     A1     N2

[2 marks]

b.

\(p =  - \frac{\pi }{4},{\text{ }}q = 5{\text{     }}\left( {{\text{accept \(\left( \begin{array}{c} - {\textstyle{\pi  \over 4}}\\5\end{array} \right)\)}}} \right)\)     A1A1     N2

[2 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

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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