Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 6 2022 Summer Zone 3

The diagrams show the number of regions inside a circle when 1 radius and 2 radii are drawn.
The regions inside the circle are numbered.


(a) Complete the table.

\[ \begin{array}{|c|c|} \hline \text{Number of radii} & \text{Number of regions} \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & \\ 4 & \\ 5 & \\ 6 & \\ \hline \end{array} \]
[1]

(b) Write a formula, in terms of $n$, for the number of regions, $R$, when there are $n$ radii.
.................................................. [1]

Diameters
The diagrams show the number of regions inside a circle when 1 diameter and 2 diameters are drawn.

(a) Complete the table for 3, 4 and 5 diameters. You may use the empty circle to help you.
[Table_1]
Number of diameters | Number of regions
1 | 2
2 | 4
3 |
4 |
5 |

(b) Write a formula, in terms of $n$, for the number of regions, $R$, when there are $n$ diameters.

Chords

In this investigation:
• each chord must cut every other chord
• only two chords may intersect at any point.

The diagrams show the number of regions inside a circle when 1 chord, 2 chords and 3 chords are drawn.



(a) Count the number of regions in the circle when 4 chords are drawn.

[1]

(b) Complete this table. You may use the empty circle to help you.

[Table_1]
[1]

(c) Find a formula, in terms of $n$, for the number of regions, $R$, when there are $n$ intersecting chords.

[4]

Tangents
A region can be inside or outside the circle when the lines are tangents.
These two diagrams both show a circle with 2 tangents and the regions numbered.
The maximum number of regions for a circle with 2 tangents is 6.

(a) Give a reason why the first diagram does not have the maximum number of regions with 2 tangents.
..........................................................................................................................
.......................................................................................................................... [1]
(b) Use this diagram to find the maximum number of regions when there are 3 tangents.
[1]
(c) Draw a fourth tangent on the diagram below to find the maximum number of regions.
Complete the table.

| Number of tangents | Maximum number of regions |
| ------------------ | ------------------------- |
| 1 | 3 |
| 2 | 6 |
| 3 | |
| 4 | |
| 5 | 21 |
[2]
(d) This is a formula for the maximum number of regions, $R$, when there are $n$ tangents.
$$R = \frac{1}{2} n^2 + bn + 1$$
Find the value of $b$. [2]

A secant is a straight line that intersects a circle at two points and extends outside the circle.

In this investigation:
• each secant must cut every other secant
• only 2 secants may intersect at any point
• secants must not intersect on the circumference of the circle.

The diagram shows the number of regions with 2 secants drawn on a circle.

(a) Find the number of regions when there are 3 secants.
Complete the table.
[Table_1: Number of secants and Number of regions]

(b) This is a formula for the number of regions, $R$, when there are $n$ secants.
$$ R = \frac{1}{2} n^2 + bn + c $$
Find the value of $b$ and the value of $c$.
$$ b = \text{.................................} $$
$$ c = \text{.................................} $$

Tangents are drawn on a circle to give the maximum number of regions. There are 1225 regions.
Find the number of tangents.

There are two circles.
The first circle has chords drawn on it.
The second circle has secants drawn on it.
The number of chords on the first circle is the same as the number of secants on the second circle.
Each circle has the maximum number of regions.
One circle has 60 more regions than the other.

(a) Find the number of straight lines on each diagram. .................................................. [2]

(b) Find the larger number of regions.
.................................................. [2]

AIRPORT RUNWAY (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at the factors that affect the decision to build a second runway at an airport.
The factors are:
• the number of seconds a plane waits over the airport before it can start to land
• the number of seconds it then takes to land.
A plane cannot begin to land until the runway is free but must wait over the airport.
As soon as the runway is free the plane begins to land.
The number of seconds between one plane and the next plane arriving over the airport is called the $\text{inter-arrival time}$.
The number of seconds from when a plane begins to land and when it stops is called the $\text{landing time}$.

This table shows the data for the first 5 planes arriving at the airport for the 1080 seconds after 8 am on day 1.
For example: Plane B arrives 120 seconds after plane A. Plane A has not ended its landing.
Plane B starts its landing 180 seconds after 8 am, as soon as plane A has ended its landing.
[Table_1]

(a) Complete the table.
[5]

(b) A plane uses the runway for the whole of its landing time.
Calculate the total time that the runway was $not$ used during these 1080 seconds.
[2]

(a) Complete the table.

[Table]

Inter-arrival time $(t \text{ seconds})$ | Number of planes | Percentage of planes
\hline
$0 < t \leq 60$ | 42 | 23
$60 < t \leq 120$ | 34 | 19
$120 < t \leq 180$ | 29 | $\text{...............}$
$180 < t \leq 240$ | 23 | $\text{...............}$
$240 < t \leq 300$ | 16 | $\text{...............}$
$300 < t \leq 360$ | 11 | 6
$360 < t \leq 420$ | 11 | 6
$420 < t \leq 480$ | 7 | 4
$480 < t \leq 540$ | 4 | 2
$540 < t \leq 600$ | 2 | 1
$600 < t \leq 660$ | 0 | 0
$660 < t \leq 720$ | 0 | 0
$720 < t \leq 780$ | 1 | 1
$780 < t \leq 840$ | 0 | 0
$840 < t \leq 900$ | 0 | 0

Inter-arrival time $(t \text{ seconds})$ | Cumulative percentage of planes $(p)$
\hline
$t \leq 60$ | 23
$t \leq 120$ | 42
$t \leq 180$ | $\text{...............}$
$t \leq 240$ | 71
$t \leq 300$ | 80
$t \leq 360$ | 86
$t \leq 420$ | $\text{...............}$
$t \leq 480$ | $\text{...............}$
$t \leq 540$ | $\text{...............}$
$t \leq 600$ | 99
$t \leq 660$ | 99
$t \leq 720$ | 99
$t \leq 780$ | 100
$t \leq 840$ | 100
$t \leq 900$ | 100

(b) On the grid below, complete the cumulative percentage curve.



(c) Use the graph to estimate the inter-arrival time for a cumulative percentage of 50.

........................................................

(a) Use the points (60, 23) and (360, 86) to write down two equations in terms of $a$ and $k$.

(b) Show that $52.3 = \frac{360-a}{60-a}$, where 52.3 is correct to 1 decimal place.

(c) Solve the equation in to show that $a = 54$, correct to the nearest integer.

(d) Find the value of $k$, correct to the nearest integer, and complete the model. $$p = \text{.............} (t - \text{.............})^{\frac{1}{3}}$$

(e) Use the model to find the inter-arrival time for a cumulative percentage of 50.

(f) Sketch the model on the axes in .

(g) Comment on the validity of this model.

(a) Use the model to find the percentage of planes that arrived over the airport within 120 seconds of the previous plane.
...................................................... [1]

(b) The table shows information about landing times for the 180 planes. All values are given correct to the nearest integer.

[Table_1]

Find the percentage of planes where the landing time is more than 120 seconds.
...................................................... [1]

(c) Based on your answers to part (a) and part (b), should a second runway be built at the airport? Give a reason for your answer.
...................................................................................................
................................................................................................... [1]

[Table_1]
| Landing time ($t$ seconds) | Number of planes | Percentage of planes | | Landing time ($t$ seconds) | Cumulative percentage of planes ($p$) |
|----------------------------|------------------|-----------------------|-----------------------|-----------------------------|---------------------------------------|
| $0 < t \leq 60$ | 2 | 1 | | $t \leq 60$ | 1 |
| $60 < t \leq 120$ | 7 | 4 | | $t \leq 120$ | 5 |
| $120 < t \leq 180$ | 11 | 6 | | $t \leq 180$ | 11 |
| $180 < t \leq 240$ | 15 | 8 | | $t \leq 240$ | 19 |
| $240 < t \leq 300$ | 20 | 11 | | $t \leq 300$ | 30 |
| $300 < t \leq 360$ | 34 | 19 | | $t \leq 360$ | 49 |
| $360 < t \leq 420$ | 42 | 23 | | $t \leq 420$ | 72 |
| $420 < t \leq 480$ | 30 | 17 | | $t \leq 480$ | 89 |
| $480 < t \leq 540$ | 18 | 10 | | $t \leq 540$ | 99 |
| $540 < t \leq 600$ | 1 | 1 | | $t \leq 600$ | 100 |