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The clock shows the time 1.00 am.
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.
[2]
In one hour, hand $M$ rotates through a full circle.
Show that hand $M$ rotates $6^\circ$ in one minute.
[1]
532
546
502
m is the number of minutes after the last hour. In this question the last hour is 1.00 am.
Examples
At 1.10 am, $m = 10$.
At 1.45 am, $m = 45$.
(a) This clock shows the time 1.10 am.
Show that the clockwise angle from hand $H$ to hand $M$ at 1.10 am is $25^\circ$.
[2]
(b) Complete the table. You may use the clock diagrams to help you.
[Table_1]
Number of minutes after the last hour ($m$) | Angle rotated since 1.00 am in degrees | Clockwise angle between the hands in degrees
| Hand $H$ angle | Hand $M$ angle
6 | | 8.5
7 | |
8 | |
9 | |
10 | | 25
[4]
(c) Find an expression, in terms of $m$, for the clockwise angle between the hands when the last hour is 1.00 am.
...............................................................
[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
[4]
502
508
562
In this question the last hour is 2.00 am.
(a) This clock shows the time 2.15 am.
Show that the clockwise angle between the hands is 22.5°. [1]
(b) Complete the table.
You may use the clock diagrams to help you.
[Table_1]
| Number of minutes after the last hour (m) | Clockwise angle between the hands in degrees |
|---|---|
| 15 | 22.5 |
| 16 | |
| 17 | |
| 18 | |
| 19 | |
[2]
(c) Find an expression, in terms of $m$, for the clockwise angle between the hands when the last hour is 2.00 am. ............................................................ [2]
568
555
562
(a) $h$ is the number of hours in the time. $m$ is the number of minutes after the last hour.
Examples
At 1.30 am, $h = 1$ and $m = 30$.
At 8.45 am, $h = 8$ and $m = 45$.
Complete the table using expressions of the form $am + b$.
Use your expressions from Question 2(c) and Question 3(c).
[Table_1: Number of hours in the time (h) / Clockwise angle in degrees between the hands m minutes after h]
1
2
3
4
(b) Find an expression, in terms of $m$ and $h$, for the clockwise angle between the hands. [3]
(c) You may use the clock diagrams to help you in this part.
(i) Use your expression from part (b) to find the two angles between the hands at 10.12 am.
........................ and ........................ [3]
(ii) There are two times between 7.00 am and 8.00 am when an angle between the hands is 100°. Find these times correct to the nearest minute. [4]
[Image of two clock diagrams]
558
554
598
This task looks at models for the lengths of arches. An arch is the curved part of a tunnel or bridge. Engineers use the dimensions of an arch to calculate its strength.
In this task, each arch has:
• width $w$ metres
• height $h$ metres
• curved length $l$ metres.
Semicircular arch model
This arch is a semicircle. The width is 8 m.
(a) Write down the height of the arch.
[2]
(b) Find the curved length of the arch. Give your answer correct to the nearest centimetre.
[2]
500
586
520
Segmental arch model
This arch is an arc of a circle with radius $r$ metres. The width of the arch is 8 metres.
(a) The formula for the radius of the circle is $r = \frac{4h^2 + w^2}{8h}$.
(i) Show that $r = \frac{h}{2} + \frac{8}{h}$.
[2]
(ii) On the diagram, sketch the graph of $r$ for $0 < h \le 8$.
[2]
(iii) Find the height that gives the minimum radius.
[1]
(b) (i) The height of the arch is 1 metre. Find the radius of the arch.
[1]
(ii)
The angle at the centre of the circle that forms the arch is $\theta$. Find the value of $\theta$ correct to the nearest degree.
[3]
(iii) Find the curved length of the arch.
[2]
579
523
500
Lancet arch model
This arch is made using two equal arcs.
The equal arcs are parts of the circumferences of two identical circles both of radius $r$ metres.
The base of the arch is on the line joining the centres of the two circles.
The height is on the line of symmetry of the arch.
(a) Use Pythagoras’ Theorem to show that the model for $r$ in terms of $h$ and $w$ is $r = \frac{h^2}{w} + \frac{w}{4}$.
[4]
(b) The width of the arch is 8 metres.
(i) Show that $r = \frac{h^2}{8} + 2$.
[1]
(ii) On the diagram in Question 6(a)(ii), sketch the graph of $r$ for $0 < h \le 8$.
[2]
(iii) Find the value of $h$ and the value of $r$ when the graphs from Question 6(a)(ii) and Question 7(b)(ii) intersect.
Explain what your answers show about the shapes of the two arches at this point.
\[ h = \text{..................................................}\]
\[ r = \text{..................................................}\]
[3]
(c) The width of the lancet arch is 8 metres.
The radius is 10 metres.
(i) Find its height.
\[ \text{..................................................} \] [1]
(ii) Use trigonometry to help you find the curved length of the arch.
\[ \text{..................................................} \] [4]
577
554
587
