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Date November 2017 Marks available 6 Reference code 17N.2.sl.TZ0.10
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 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]
a.

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]
c.

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

[6]
d.

Markscheme

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]

a.

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]

c.

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)

eg

33.7618     (A1)

33 (teeth)     A1     N2

[6 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.
[N/A]
d.

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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