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    SLPaper 2

    Consider a function fff, such that f(x)=5.8sin⁡(π6(x+1))+bf(x) = 5.8\sin\left(\frac{\pi}{6}(x + 1)\right) + bf(x)=5.8sin(6π​(x+1))+b, 0≤x≤100 \leq x \leq 100≤x≤10, b∈Rb \in \mathbb{R}b∈R.

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

    A second function ggg is given by g(x)=psin⁡(2π9(x−3.75))+qg(x) = p\sin\left(\frac{2\pi}{9}(x - 3.75)\right) + qg(x)=psin(92π​(x−3.75))+q, 0≤x≤100 \leq x \leq 100≤x≤10; ppp, q∈Rq \in \mathbb{R}q∈R.

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

    1.

    Find the period of f{f}f.

    [2]
    Verified
    Solution

    correct approach A1 eg π6=2πperiod{\frac{\pi }{6} = \frac{{2\pi }}{{period}}}6π​=period2π​ (or equivalent) period = 12 A1 [2 marks]

    Note: It seems like the LaTeX equation in the original HTML was invalid.

    2.

    Find the value of bbb.

    [2]
    Verified
    Solution

    validapproach (M1) egmax+min2  b=max−amplitude\frac{{\text{max}} + \text{min}}{2}\,\,b = \text{max} - \text{amplitude}2max+min​b=max−amplitude 21.8+10.22\frac{{21.8 + 10.2}}{2}221.8+10.2​, or equivalent bbb = 16 A1 [2 marks]

    3.

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

    [2]
    Verified
    Solution

    attempt to substitute into their function (M1) 5.8 sin⁡(π6(6+1))+165.8\,\sin\left( {\frac{\pi }{6}\left( {6 + 1} \right)} \right) + 165.8sin(6π​(6+1))+16 f(6)=13.1f(6) = 13.1f(6)=13.1 (A1) [2 marks]

    4.

    Find the value of p{p}p and the value of q{q}q.

    [5]
    Verified
    Solution

    valid attempt to set up a system of equations (M1) two correct equations A1 p⋅sin⁡(2π9(3−3.75))+q=2.5p\cdot \sin\left( \frac{2\pi }{9}\left( 3 - 3.75 \right) \right) + q = 2.5p⋅sin(92π​(3−3.75))+q=2.5, p⋅sin⁡(2π9(6−3.75))+q=15.1p\cdot \sin\left( \frac{2\pi }{9}\left( 6 - 3.75 \right) \right) + q = 15.1p⋅sin(92π​(6−3.75))+q=15.1 valid attempt to solve system (M1) p=8.4; q=6.7p = 8.4;\ q = 6.7p=8.4; q=6.7 A1A1 [5 marks]

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    5.

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

    [2]
    Verified
    Solution

    attempt to use∣(f(x)−g(x))∣|\left( f(x) - g(x) \right)|∣(f(x)−g(x))∣ to find maximum difference (M1) xxx = 1.64 A1

    [2 marks]

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