SPM.2.SL.TZ0.9

pestleMathematicsAASLPaper 2SPM· sl-3-7-circular-functions-graphs-composites-transformationssource ↗

Consider a function $f$ , such that $f(x) = 5, 8\,{\text{sin}}\left( {\frac{\pi }{6}\left( {x + 1} \right)} \right) + b$ , 0 ≤ $x$ ≤ 10, $b \in \mathbb{R}$ .

The function $f$ has a local maximum at the point (2, 21.8) , and a local minimum at (8, 10.2).

A second function $g$ is given by $g(x) = p\,{\text{sin}}\left( {\frac{{2\pi }}{9}\left( {x - 3.75} \right)} \right) + q$ , 0 ≤ $x$ ≤ 10; $p$ , $q \in \mathbb{R}$ .

The function $g$ passes through the points (3, 2.5) and (6, 15.1).

Find the period of $f$ .

[2]

Find the value of $b$ .

[2]

b.i.

Hence, find the value of $f$ (6).

[2]

b.ii.

Find the value of $p$ and the value of $q$ .

[5]

Find the value of $x$ for which the functions have the greatest difference.

[2]

Markscheme / solution

correct approach A1

eg $\frac{\pi }{6} = \frac{{2\pi }}{{period}}$ (or equivalent)

period = 12 A1

[2 marks]

valid approach (M1)

eg $\frac{{{\text{max}} + {\text{min}}}}{2}\,\,b = {\text{max}} - {\text{amplitude}}$

$\frac{{21.8 + 10.2}}{2}$ , or equivalent

$b$ = 16 A1

[2 marks]

b.i.

attempt to substitute into their function (M1)

$5.8\,{\text{sin}}\left( {\frac{\pi }{6}\left( {6 + 1} \right)} \right) + 16$

$f$ (6) = 13.1 A1

[2 marks]

b.ii.

valid attempt to set up a system of equations (M1)

two correct equations A1

$p\,{\text{sin}}\left( {\frac{{2\pi }}{9}\left( {3 - 3.75} \right)} \right) + q = 2.5$ , $p\,{\text{sin}}\left( {\frac{{2\pi }}{9}\left( {6 - 3.75} \right)} \right) + q = 15.1$

valid attempt to solve system (M1)

$p$ = 8.4; $q$ = 6.7 A1A1

[5 marks]

attempt to use $\left| {f(x) - g(x)} \right|$ to find maximum difference (M1)

$x$ = 1.64 A1

[2 marks]

Examiners’ report

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b.i.

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b.ii.

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