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(a) The clock shows the time 1.00 am.
(i) Write down the mathematical name for the type of angle shown.
.................................................. [1]
(a)(ii) Explain why hand $H$ rotates through $360^\circ$ in 12 hours.
................................................................................................................... [1]
(a)(iii) Write down the calculation to show that the clockwise angle from hand $M$ to hand $H$ is $30^\circ$.
[1]
(a)(iv) Write down a calculation to show that the anticlockwise angle from hand $M$ to hand $H$ is $330^\circ$.
[1]
(b) This clock shows the time 4.00 am.
(i) Work out the clockwise angle from hand $M$ to hand $H$.
.................................................. [2]
(b)(ii) Work out the anticlockwise angle from hand $M$ to hand $H$.
.................................................. [2]
(c) Write down the clockwise angle from hand $M$ to hand $H$ at 6.00 am.
.................................................. [1]
(d) Complete the table using part (b) and part (c).
You may use the clock diagrams and patterns to help you.
$$\begin{array}{|c|c|c|}\hline \text{Hour shown by hand } H (x) & \text{Angle between hand } H \text{ and hand } M \text{ in degrees} \\ \hline \text{Clockwise angle} & \text{Anticlockwise angle} \\ \hline 1 & 30 & 330 \\ \hline 2 & & \\ \hline 3 & & \\ \hline \text{part (b)}: 4 & & \\ \hline 5 & & \\ \hline \text{part (c)}: 6 & & \\ \hline \end{array}$$
[4]
(e) Find an expression for the clockwise angle at hour x.
.................................................. [1]
(f) Write down the rule for continuing the sequence in the anticlockwise angle column.
................................................................................................................... [1]
(g) Find an expression for the anticlockwise angle at hour x.
.................................................. [1]
586
572
556
(a) In one hour, hand $H$ rotates clockwise from one number to the next number. For example, from 1.00 am to 2.00 am hand $H$ rotates from 1 to 2.
Show that hand $H$ rotates $0.5^\circ$ in one minute. [1]
(b) In one hour, hand $M$ rotates through a full circle.
Show that hand $M$ rotates $6^\circ$ in one minute. [1]
522
586
502
(a) This clock shows the time 1.10 am.
(i) Use Question 2(a) to find the angle that hand $H$ has rotated in the 10 minutes since 1.00 am. ..................................................... [2]
(ii) Use Question 2(b) to find the angle that hand $M$ has rotated in the 10 minutes since 1.00 am. ..................................................... [2]
(iii) Show that the clockwise angle from hand $H$ to hand $M$ at 1.10 am is 25$^{\circ}$. ..................................................... [1]
(b) Complete the table using your results from part (a)(i) and part (a)(ii).
You may use the clock diagrams and patterns to help you.
[Table_1]
$$\begin{array}{|c|c|c|c|}\hline\text{Number of minutes since 1.00 am} & \text{Angle rotated since 1.00 am in degrees} & \text{Clockwise angle between the hands} \\ (m) & \text{Hand H angle} & \text{Hand M angle} & \text{in degrees} \\ \hline 6 & & & \\ \hline 7 & 3.5 & 42 & 8.5 \\ \hline 8 & & & \\ \hline 9 & & & \\ \hline 10 & \text{part (a)(i)} & \text{part (a)(ii)} & 25 \\ \hline\end{array}$$ [6]
(c) Find an expression, in terms of $m$, for the clockwise angle between the hands. ..................................................... [2]
(d) Find how many minutes and seconds after 1.00 am the clockwise angle is 270$^{\circ}$.
Give your answer correct to the nearest second. ........................ minutes .................... seconds [5]
501
548
521
