Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 6 2024 Summer Zone 2

In Questions 1 to 4, the number of branches, $n$, is always even.

This tree has 6 branches with 2 leaves on each branch.

[Image]

Find the value of $A$, the value of $B$ and the value of $T$.

$A = \text{..............................}$
$B = \text{..............................}$
$T = \text{..............................}$ [3]

(a) Complete the table for trees with two leaves on each branch. Use your answers to Question 1 to help you.

[Table_1]

Number of branches $(n)$ | $A$ | $B$ | $T$ | $A−B$ | $A+B−T$
--- | --- | --- | --- | --- | ---
2 | 6 | 3 | 5 | 3 | 4
4 | 18 | 12 | 22 | 6 | 8
6 |   |   |   | 9 |
8 | 60 |   |   |   | 16
[3]

(b) (i) Use values from the table to give an example to show that $A−B=1.5n$.
............................................................................................................................. ............................................................................................................................... [1]
(ii) Write an expression for $A+B−T$ in terms of $n$.
................................................................. [1]

In this question trees have 3 leaves on each branch. This tree has 6 branches.



(a) Complete the table.

[Table_1]

(b) (i) Write an expression for $A - B$ in terms of $n$.

.................................................. [1]

(ii) Write an expression for $A + B - T$ in terms of $n$.

.................................................. [1]

(a) Complete the table below.
Use the table above and your answers from Questions 2(b)(ii) and 3(b) to help you.



(b) A tree has an even number of branches, $n$, with $x$ leaves on each branch.

Complete the expressions for $A - B$ and $A + B - T$ in terms of $n$ and $x$.

$ A - B = \frac{n}{2} \text{(....................................................)}$

$ A + B - T = \text{...............................................} $ [2]

In this question trees have an odd number of branches. Trees have more than one branch. This tree has 5 branches with 2 leaves on each branch. (a) Find the value of $A-B$ and the value of $A+B-T$ for this tree. $A-B =$ ........................................... $A+B-T =$ ........................................... [2] (b) In Question 4(b) you found expressions for trees with an even number of branches. Do your expressions in Question 4(b) give correct values for this tree? Show how you decide. $A-B$ .......................................................................................................................... A+B−T ...... (c) The table shows some values of $A-B$ and $A+B-T$ for trees with odd numbers of branches. [Table_1] A tree has an odd number of branches, $n$, with $x$ leaves on each branch. Find expressions for $A-B$ and $A+B-T$ in terms of $n$ and $x$. $A-B =$ ........................................... $A+B-T =$ ........................................... [3]

(a) An expression for the outer leaf on the second branch is $k + 2p + 1$. Show that this statement is correct.

(b) Find expressions for the third integer on the trunk and the outer leaf on the third branch. Write your expressions, in their simplest forms, on the tree.

(c) (i) Find expressions for $A + B - T$ when a tree has 2 branches, 3 branches, and 4 branches. Give each expression in its simplest form.
(ii) Find an expression for $A + B - T$ when a tree has $n$ branches.

B MODELLING (QUESTIONS 7 to 11)

CARBON FOOTPRINT OF AIR TRAVEL (30 marks)

You are advised to spend no more than 50 minutes on this part.

This task looks at modelling the personal carbon footprint of someone travelling by air.

Carbon footprint is the amount of carbon dioxide that an activity puts into the atmosphere.

Nikita wants to estimate her carbon footprint when she travels by air.
Her carbon footprint depends on the amount of fuel used by the aircraft, the length of the journey and the number of people on the aircraft.

Average fuel use is the amount of fuel, in kilograms, used for each kilometre travelled.

An aircraft uses $T ext{kg}$ of fuel for a journey of length $L$ km.
A formula for the average fuel use, $F ext{kg/km}$, is

$F = \frac{T}{L}$.

An aircraft makes a journey of 850 km and uses 4760 kg of fuel.

Find the average fuel use for this journey.

Nikita finds this data for an aircraft making journeys of different lengths.

[Table_1]



(a) What happens to the average fuel use as the journey length increases?
......................................................................................................................................................................... [1]

(b) Nikita thinks a model for the data is $F = a + \frac{b}{L}$ where $a$ and $b$ are constants.

(i) Use the values of $F$ from the table for $L = 400$ and $L = 1200$ to write down two equations in terms of $a$ and $b$.
................................................
................................................ [2]

(ii) Solve your equations to find the value of $a$ and the value of $b$ and write down the model.
$a =$ .....................................
$b =$ .....................................
Model ..................................... [3]

The model gives these values of $F$, correct to one decimal place.
[Table_1]
(a) Plot these values on the grid in Question 8 and draw the graph of the model. [2]
(b) How well does the model fit Nikita’s data?
............................................................................................................ [1]
(c) The aircraft makes a journey of $10\ 000$ km.
Find the average fuel use that the model gives.
.................................................. [2]
(d) Is your answer to part (c) valid? Give a reason.
............................................................................................................ [1]

The amount of carbon dioxide given out by an aircraft depends on the amount of fuel used.

A kilogram of aircraft fuel makes 3200 grams of carbon dioxide.

(a) The aircraft makes a journey of 800 km and uses 5.3 kg of fuel per km.

Work out the amount of carbon dioxide, in grams, that this aircraft gives out for each kilometre of the journey.

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

(b) The personal carbon footprint, $C$, for an aircraft passenger is the amount of carbon dioxide, in grams, that the aircraft gives out per kilometre divided by the number of passengers.

Nikita assumes that this aircraft has 200 passengers.

(i) Use your model for $F$ in Question 8(b)(ii) to show that a model for $C$ for a passenger on this aircraft with a journey length $L$ is

$$C = 48 + \frac{28800}{L}.$$
[2]

(ii) Sketch the graph of this model for $200 \leq L \leq 1600$.

[3]




(c) Nikita travels 900 km on this aircraft.

Use the model in part (b)(i) to calculate the amount of carbon dioxide, in kg, that the aircraft gives out for each passenger for the whole journey.

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

Nikita wants to compare her carbon footprint for air travel and train travel.

(a) She finds that the personal carbon footprint, $C$, for train travel is 40 g/km for each passenger for any length of journey.

Sketch the graph of the model for $C$ for train travel on the axes in Question 10(b)(ii). [2]

(b) One train company claims:

When you travel by train, your carbon footprint will never be more than 30\% of your carbon footprint for the same journey by air.

For what lengths of journey is this claim correct?................................................................................................................................................................................................. [4]