Find the size of angle $\text{A}\hat{\text{O}}\text{B}$ in degrees.

Find the distance between points $\text{A}$ and $\text{B}$.

Find the size of angle $\theta$ in degrees.

Calculate the length of arc $\text{AC}$.

Calculate the area of the shaded sector, $\text{AOC}$.

Write down the height of the endpoint of the minute hand above the bookshelf at $6:00$ am.

Find the value of $h$ when $\theta =160°$.

Write down the amplitude of $g\left(\theta \right)$.

The endpoints of the minute hand and hour hand meet when $\theta =k$.

Find the smallest possible value of $k$.

$4\times \frac{360°}{12}$  OR  $4\times 30°$                   (M1)

$120°$                  A1

[2 marks]

substitution in cosine rule                   (M1)

${\text{AB}}^2={10}^2+6^2-2\times 10\times 6\times \cos \left(120°\right)$                  (A1)

$\text{AB}=14$ $\text{cm}$                  A1

Note: Follow through marks in part (b) are contingent on working seen.

[3 marks]

$\theta =13\times 6$                   (M1)

$=78°$                A1

[2 marks]

substitution into the formula for arc length                 (M1)

$l=\frac{78}{360}\times 2\times \mathrm{\pi }\times 10$  OR  $l=\frac{13\mathrm{\pi }}{30}\times 10$

$=13.6 \text{cm} \left(13.6135\dots , 4.33\mathrm{\pi }, \frac{13\mathrm{\pi }}{3}\right)$                A1

[2 marks]

substitution into the area of a sector                (M1)

$A=\frac{78}{360}\times \mathrm{\pi }\times {10}^2$  OR  $l=\frac{1}{2}\times \frac{13\mathrm{\pi }}{30}\times {10}^2$

$=68.1 {\text{cm}}^2 \left(68.0678\dots , 21.7\mathrm{\pi }, \frac{65\mathrm{\pi }}{3}\right)$                A1

[2 marks]

$23$              A1

[1 mark]

correct substitution              (M1)

$h=10 \cos \left(160°\right)+13$

$=3.60 \text{cm} \left(3.60307\dots \right)$              A1

[2 marks]

$10$           A1

[1 mark]

EITHER

$10\times \cos \left(\theta \right)+13=-10\times \cos \left(\frac{\theta }{12}\right)+13$              (M1)

OR

                             (M1)

Note: Award M1 for equating the functions. Accept a sketch of $h\left(\theta \right)$ and $g\left(\theta \right)$ with point(s) of intersection marked.

THEN

$k=196° \left(196.363\dots \right)$          A1

Note: The answer $166.153\dots$ is incorrect but the correct method is implicit. Award (M1)A0.

[2 marks]