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Date	May 2021	Marks available	2	Reference code	21M.2.SL.TZ1.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	2	Adapted from	N/A

Question

The diagram below shows a circular clockface with centre $O$ $O$ . The clock’s minute hand has a length of $10 cm$ $10 cm$ . The clock’s hour hand has a length of $6 cm$ $6 cm$ .

At $4 : 00$ $4 : 00$ pm the endpoint of the minute hand is at point $A$ $A$ and the endpoint of the hour hand is at point $B$ $B$ .

Between $4 : 00$ $4 : 00$ pm and $4 : 13$ $4 : 13$ pm, the endpoint of the minute hand rotates through an angle, $θ$ $θ$ , from point $A$ $A$ to point $C$ $C$ . This is illustrated in the diagram.

A second clock is illustrated in the diagram below. The clock face has radius $10 cm$ $10 cm$ with minute and hour hands both of length $10 cm$ $10 cm$ . The time shown is $6 : 00$ $6 : 00$ am. The bottom of the clock face is located $3 cm$ $3 cm$ above a horizontal bookshelf.

The height, $h$ $h$ centimetres, of the endpoint of the minute hand above the bookshelf is modelled by the function

$h (θ) = 10 cos θ + 13, θ \geq 0,$ $h (θ) = 10 \cos θ + 13, θ \geq 0,$

where $θ$ $θ$ is the angle rotated by the minute hand from $6 : 00$ $6 : 00$ am.

The height, $g$ $g$ centimetres, of the endpoint of the hour hand above the bookshelf is modelled by the function

$g (θ) = - 10 cos (\frac{θ}{12}) + 13, θ \geq 0,$ $g (θ) = - 10 \cos (\frac{θ}{12}) + 13, θ \geq 0,$

where $θ$ $θ$ is the angle in degrees rotated by the minute hand from $6 : 00$ $6 : 00$ am.

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

[2]

Find the distance between points $A$ $A$ and $B$ $B$ .

[3]

Find the size of angle $θ$ $θ$ in degrees.

[2]

Calculate the length of arc $AC$ $AC$ .

[2]

Calculate the area of the shaded sector, $AOC$ $AOC$ .

[2]

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

[1]

Find the value of $h$ $h$ when $θ = 160 °$ $θ = 160 °$ .

[2]

Write down the amplitude of $g (θ)$ $g (θ)$ .

[1]

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

Find the smallest possible value of $k$ $k$ .

[2]

Markscheme

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

$120 °$ $120 °$ A1

[2 marks]

substitution in cosine rule (M1)

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

$AB = 14$ $AB = 14$ $cm$ $cm$ A1

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

[3 marks]

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

$= 78 °$ $= 78 °$ A1

[2 marks]

substitution into the formula for arc length (M1)

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

$= 13.6 cm (13.6135 \dots, 4.33 π, \frac{13 π}{3})$ $= 13.6 cm (13.6135 \dots, 4.33 π, \frac{13 π}{3})$ A1

[2 marks]

substitution into the area of a sector (M1)

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