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Date	November 2017	Marks available	3	Reference code	17N.1.AHL.TZ0.H_11
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find and Hence or otherwise	Question number	H_11	Adapted from	N/A

Question

Consider the function ${f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }$ .

Determine whether ${f_n}$ is an odd or even function, justifying your answer.

[2]

a.

By using mathematical induction, prove that

${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .

[8]

b.

Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .

[3]

c.

Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .

[8]

d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

even function A1

since $\cos kx = \cos ( - kx)$ and ${f_n}(x)$ is a product of even functions R1

OR

even function A1

since $(\cos 2x)(\cos 4x) \ldots = \left( {\cos ( - 2x)} \right)\left( {\cos ( - 4x)} \right) \ldots$ R1

Note: Do not award A0R1.

[2 marks]

a.

consider the case $n = 1$

$\frac{{\sin 4x}}{{2\sin 2x}} = \frac{{2\sin 2x\cos 2x}}{{2\sin 2x}} = \cos 2x$ M1

hence true for $n = 1$ R1

assume true for $n = k$ , ie, $(\cos 2x)(\cos 4x) \ldots (\cos {2^k}x) = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}$ M1

Note: Do not award M1 for “let $n = k$ ” or “assume $n = k$ ” or equivalent.

consider $n = k + 1$ :

${f_{k + 1}}(x) = {f_k}(x)(\cos {2^{k + 1}}x)$ (M1)

$= \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}\cos {2^{k + 1}}x$ A1

$= \frac{{2\sin {2^{k + 1}}x\cos {2^{k + 1}}x}}{{{2^{k + 1}}\sin 2x}}$ A1

$= \frac{{\sin {2^{k + 2}}x}}{{{2^{k + 1}}\sin 2x}}$ A1

so $n = 1$ true and $n = k$ true $\Rightarrow n = k + 1$ true. Hence true for all $n \in {\mathbb{Z}^ + }$ R1

Note: To obtain the final R1, all the previous M marks must have been awarded.

[8 marks]

b.

attempt to use $f’ = \frac{{vu' - uv'}}{{{v^2}}}$ (or correct product rule) M1

${f’_n}(x) = \frac{{({2^n}\sin 2x)({2^{n + 1}}\cos {2^{n + 1}}x) - (\sin {2^{n + 1}}x)({2^{n + 1}}\cos 2x)}}{{{{({2^n}\sin 2x)}^2}}}$ A1A1

Note: Award A1 for correct numerator and A1 for correct denominator.

[3 marks]

c.

${f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{\left( {{2^n}\sin \frac{\pi }{2}} \right)\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right) - \left( {\sin {2^{n + 1}}\frac{\pi }{4}} \right)\left( {{2^{n + 1}}\cos \frac{\pi }{2}} \right)}}{{{{\left( {{2^n}\sin \frac{\pi }{2}} \right)}^2}}}$ (M1)(A1)

${f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{({2^n})\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right)}}{{{{({2^n})}^2}}}$ (A1)

$= 2\cos {2^{n + 1}}\frac{\pi }{4}{\text{ }}( = 2\cos {2^{n - 1}}\pi )$ A1

${f’_n}\left( {\frac{\pi }{4}} \right) = 2$ A1

${f_n}\left( {\frac{\pi }{4}} \right) = 0$ A1

Note: This A mark is independent from the previous marks.

$y = 2\left( {x - \frac{\pi }{4}} \right)$ M1A1

$4x - 2y - \pi = 0$ AG

[8 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

Topic 5 —Calculus » SL 5.4—Tangents and normal

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The line $y = kx - 5$ is a tangent to the curve of $f$ . Find the values of $k$ .
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21N.1.SL.TZ0.5c:
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Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .
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Show that the equation of $L$ is $y = - x + 2 \ln 45 + 2$ .
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Show that $\frac{d y}{d x} = \frac{3 x^{2}}{2 e^{2 y} - 1}$ .
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Write down the coordinates of C, the midpoint of line segment AB.
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Find the equation of this tangent. Give your answer in the form y = mx + c.
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Find the value of $k$ .
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Find the gradient of the line DC.
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Write down the $y$ -intercept of the graph.
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Show that $a = 8$ .
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Find $f'(x)$ .
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Write down $f' (x)$ .
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18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
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Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
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Write down the range of $f(x)$ .

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