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Date	November 2017	Marks available	3	Reference code	17N.2.SL.TZ0.S_10
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_10	Adapted from	N/A

Question

Note: In this question, distance is in millimetres.

Let $f(x) = x + a\sin \left( {x - \frac{\pi }{2}} \right) + a$ , for $x \geqslant 0$ .

The graph of $f$ passes through the origin. Let ${{\text{P}}_k}$ be any point on the graph of $f$ with $x$ -coordinate $2k\pi$ , where $k \in \mathbb{N}$ . A straight line $L$ passes through all the points ${{\text{P}}_k}$ .

Diagram 1 shows a saw. The length of the toothed edge is the distance AB.

N17/5/MATME/SP2/ENG/TZ0/10.d_01

The toothed edge of the saw can be modelled using the graph of $f$ and the line $L$ . Diagram 2 represents this model.

N17/5/MATME/SP2/ENG/TZ0/10.d_02

The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of $f$ and the line $L$ , between ${{\text{P}}_k}$ and ${{\text{P}}_{k + 1}}$ .

Show that $f(2\pi ) = 2\pi$ .

[3]

Find the coordinates of ${{\text{P}}_0}$ and of ${{\text{P}}_1}$ .

[3]

b.i.

Find the equation of $L$ .

[3]

b.ii.

Show that the distance between the $x$ -coordinates of ${{\text{P}}_k}$ and ${{\text{P}}_{k + 1}}$ is $2\pi$ .

[2]

A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.

[6]

Markscheme

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

substituting $x = 2\pi$ M1

eg $\,\,\,\,\,$ $2\pi + a\sin \left( {2\pi - \frac{\pi }{2}} \right) + a$

$2\pi + a\sin \left( {\frac{{3\pi }}{2}} \right) + a$ (A1)

$2\pi - a + a$ A1

$f(2\pi ) = 2\pi$ AG N0

[3 marks]

substituting the value of $k$ (M1)

${{\text{P}}_0}(0,{\text{ }}0),{\text{ }}{{\text{P}}_1}(2\pi ,{\text{ }}2\pi )$ A1A1 N3

[3 marks]

b.i.

attempt to find the gradient (M1)

eg $\,\,\,\,\,$ $\frac{{2\pi - 0}}{{2\pi - 0}},{\text{ }}m = 1$

correct working (A1)

eg $\,\,\,\,\,$ $\frac{{y - 2\pi }}{{x - 2\pi }} = 1,{\text{ }}b = 0,{\text{ }}y - 0 = 1(x - 0)$

y = x A1 N3

[3 marks]

b.ii.

subtracting $x$ -coordinates of ${{\text{P}}_{k + 1}}$ and ${{\text{P}}_k}$ (in any order) (M1)

eg $\,\,\,\,\,$ $2(k + 1)\pi - 2k\pi ,{\text{ }}2k\pi - 2k\pi - 2\pi$

correct working (must be in correct order) A1

eg $\,\,\,\,\,$ $2k\pi + 2\pi - 2k\pi ,{\text{ }}\left| {2k\pi - 2(k + 1)\pi } \right|$

distance is $2\pi$ AG N0

[2 marks]

METHOD 1

recognizing the toothed-edge as the hypotenuse (M1)

eg $\,\,\,\,\,$ ${300^2} = {x^2} + {y^2}$ , sketch

correct working (using their equation of $L$ (A1)

eg $\,\,\,\,\,$ ${300^2} = {x^2} + {x^2}$

$x = \frac{{300}}{{\sqrt 2 }}$ (exact), 212.132 (A1)

dividing their value of $x$ by $2\pi {\text{ }}\left( {{\text{do not accept }}\frac{{300}}{{2\pi }}} \right)$ (M1)

eg $\,\,\,\,\,$ $\frac{{212.132}}{{2\pi }}$

33.7618 (A1)

33 (teeth) A1 N2

METHOD 2

vertical distance of a tooth is $2\pi$ (may be seen anywhere) (A1)

attempt to find the hypotenuse for one tooth (M1)

eg $\,\,\,\,\,$ ${x^2} = {(2\pi )^2} + {(2\pi )^2}$

$x = \sqrt {8{\pi ^2}}$ (exact), 8.88576 (A1)

dividing 300 by their value of $x$ (M1)

33.7618 (A1)

33 (teeth) A1 N2

[6 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations

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EXM.3.AHL.TZ0.5e.i:
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EXM.3.AHL.TZ0.5b.ii:
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17N.2.SL.TZ0.S_10d:
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EXM.3.AHL.TZ0.5e.iii:
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EXM.3.AHL.TZ0.5a.i:
Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$
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19M.1.SL.TZ2.S_7b:
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22M.1.SL.TZ1.6b:
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19M.2.AHL.TZ1.H_10b:
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22M.2.AHL.TZ2.10b:
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Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .
17M.2.SL.TZ1.S_8b.ii:
Find the value of $q$ ;
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EXN.1.SL.TZ0.9a:
Find the height of the ball above the ground when it is released.
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Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.
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Find ${p_{av}}$ (0.007).
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Find the height of point $A$ above the ground.
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(i) Find $w$ .

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21M.2.SL.TZ2.7e:
If the observer stands directly under point $C$ for one minute, point $D$ will pass over his head $n$ times.

Find the value of $n$ .
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.1.SL.TZ1.6a:
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21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
EXM.3.AHL.TZ0.5a.ii:
Write down the amplitude of this graph
EXM.3.AHL.TZ0.5d:
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18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
17M.2.SL.TZ1.S_8b.i:
Find the value of $p$ ;
EXM.3.AHL.TZ0.5e.ii:
$B$ .
EXM.3.AHL.TZ0.5e.iv:
$D$ .
17M.2.SL.TZ1.S_8b.iii:
Find the value of $r$ .
17M.2.SL.TZ1.S_8c:
There are two high tides on 12 December 2017. At what time does the second high tide occur?
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
SPM.2.SL.TZ0.9d:
Find the value of $x$ for which the functions have the greatest difference.
SPM.2.SL.TZ0.9c:
Find the value of $p$ and the value of $q$ .
21M.2.SL.TZ1.4:
A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $P$ is a point on the wheel. The wheel starts moving with $P$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $P$ above the ground after $t$ minutes is given by $h (t) = a \cos (b t) + c$ , where $a, b, c \in ℝ$ .

Find the values of $a$ , $b$ and $c$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

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Topic 3— Geometry and trigonometry

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