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

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

Find the equation of $L$.

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

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

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]

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]

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)

eg

33.7618     (A1)

33 (teeth)     A1     N2

[6 marks]