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

REVERSE DIFFERENCES (30 marks)

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

This investigation looks at what happens when you reverse the digits of a number and then find the difference between the new number and the original number. This is called the reverse difference.

STEP 1 Write down a 2-digit number.
STEP 2 Reverse the digits of the number.
STEP 3 Find the positive difference between the two numbers.

Example 1 Example 2
STEP 1 Write a number 52 STEP 1 13
STEP 2 Reverse the digits 25 STEP 2 31
STEP 3 Find the difference $52 - 25 = 27$ STEP 3 $31 - 13 = 18$

(a) Complete the three steps for each 2-digit number in the table.

[Table_1]
STEP 1 12 13 14 15 16 17 18
STEP 2 21 31 41 51
STEP 3 18 45 63
[2]

(b) Complete this table of 2-digit numbers and their reverse differences. Use part (a) and any patterns you notice to help you.

[Table_2]
| Number | Reverse difference | Number | Reverse difference | Number | Reverse difference |
| 10 | 9 | 20 | 18 | 30 | 27 | 40 36 |
| 11 | 21 | 9 | 31 | 18 | 41 | 27 |
| 12 | 22 | 32 | 9 | 42 ||
| 13 | 18 | 23 | 9 | 33 ||
| 14 | 24 | 34 || ||
| 15 | 25 | 27 | 35 ||
| 16 | 45 | 26 | 36 | 36 ||
| 17 | 27 | 45 | 37 | 36 ||
| 18 | 63 | 28 | 54 | 38 | 45 |
| 19 | 29 | 63 | 39 | 54 | 49 | 45 |[3]

(c) Complete the statement with the largest number possible.

The reverse difference is always a multiple of ........................................... [1]

(d) The table in part (b) is extended to the right. These two columns are part of the extended table.

[Table_3]
| Number | Reverse difference |
| $A$ ||
| ||
| ||
| ||
| ||
| 9 ||

What is the value of the 2-digit number $A$?
........................................... [2]

You are advised to spend no more than 50 minutes on this part.
This investigation looks at what happens when you reverse the digits of a number and then find the difference between the new number and the original number. This is called the reverse difference.

STEP 1 Write down a 2-digit number.
STEP 2 Reverse the digits of the number.
STEP 3 Find the positive difference between the two numbers.

Example 1 Example 2
STEP 1 Write a number 52 STEP 1 13
STEP 2 Reverse the digits 25 STEP 2 31
STEP 3 Find the difference $52 - 25 = 27$ STEP 3 $31 - 13 = 18$

You can find reverse differences for 3-digit numbers using the same steps.

Example
STEP 1 Write a number 138
STEP 2 Reverse the digits 831
STEP 3 Find the difference $831 - 138 = 693$

(a) The table continues to the right until 199.
Complete the first column of reverse differences.

[Table]
| Number | Reverse difference | Number |
| ----- | ----- | ----- |
| 100 | 99 | 110 |
| 101 | 0 | 111 |
| 102 | | |
| 103 | 198 | |
| 104 | 297 | |
| 105 | 396 | |
| 106 | | |
| 107 | | |
| 108 | 693 | |
| 109 | 792 | |

[2]
(b) Explain why each column of reverse differences has the same sequence of reverse differences.
...................................................................................................................................................................................... [1]

(c) Complete the statement with the largest number possible.
The reverse difference is always a multiple of ............................................................. [1]

(d) A 3-digit number $abc$ has first digit $a$, second digit $b$ and third digit $c$. In this part $a > c$.

So the number 601 has $a = 6$, $b = 0$ and $c = 1$.

(i) Anna says that $a$ is the hundreds digit, $b$ is the tens digit and $c$ is the units digit. She says the value of $abc$ is $100a + 10b + c$.
Anna writes the value of the reverse number $cba$. $100c + 10b + a$
She writes the difference between the two numbers. $(100a + 10b + c) - (100c + 10b + a)$
Complete Anna’s working and factorise the result.
.......................................................... [2]

(ii) A 3-digit number has $a = 8$ and a reverse difference of 594.
Find three possible 3-digit numbers.
.......................................................... [2]

REVERSE DIFFERENCES (30 marks)

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

This investigation looks at what happens when you reverse the digits of a number and then find the difference between the new number and the original number. This is called the reverse difference.

STEP 1 Write down a 2-digit number.
STEP 2 Reverse the digits of the number.
STEP 3 Find the positive difference between the two numbers.

Example 1

STEP 1 Write a number 52
STEP 2 Reverse the digits 25
STEP 3 Find the difference $52 - 25 = 27$

Example 2

STEP 1 Write a number 13
STEP 2 Reverse the digits 31
STEP 3 Find the difference $31 - 13 = 18$

Use your result from Question 2(d)(i) to answer both parts of this question.

Anna uses 3-digit numbers to find reverse differences.

(a) The reverse difference for a 3-digit number is 99. Comment on the values of each of the three digits.
[3]

(b) The table in Question 2(a) is extended to 999. Find all the possible reverse differences for 3-digit numbers.
[2]

REVERSE DIFFERENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.

This investigation looks at what happens when you reverse the digits of a number and then find the difference between the new number and the original number. This is called the reverse difference.

STEP 1 Write down a 2-digit number.
STEP 2 Reverse the digits of the number.
STEP 3 Find the positive difference between the two numbers.

A sheet of newsprint measures 560 mm by 430 mm. Each square metre of newsprint has a mass of 42 g.
(a) Show that the mass of one sheet of newsprint is 0.0101 kilograms, correct to 3 significant figures. [3]

(b) A newspaper uses 20 sheets of newsprint.
Work out the mass of this newspaper in kilograms. .................................................. [1]

(c) Give a practical reason why your answer to part (b) is slightly less than the actual mass of the newspaper. You may assume all measurements are accurate. ...................................................................................................................................................... [1]

(d) \textit{Circulation} is the number of copies of a newspaper made in a day. The newspaper has an average circulation of 950 000.
(i) Use your answer to \textbf{Question 5(b)} to work out the total mass of these newspapers. Give your answer in tonnes. .................................................. [2]

(ii) The newspaper is made every day from Monday to Friday each week. Work out the mass of newsprint, in tonnes, used in a year of 52 weeks. Give your answer correct to the nearest thousand. .................................................. [2]

There are different sizes of newsprint.
A sheet of newsprint measures $L$ mm by $W$ mm.
The mass of one square metre of newsprint is $d$ grams.
Each newspaper has $S$ sheets of newsprint.
The circulation is $C$ newspapers per day, Monday to Friday.

(a) Write a model for the mass of newsprint, $M$ tonnes, that is used to make the newspaper in a year. ................................................................. [2]

(b) The company that makes the newspaper now uses newsprint with a mass of 43 g per square metre. All other figures remain the same as in Question 5.
Calculate the mass, in tonnes, of newsprint that the company now uses in a year. ................................................................. [2]

(a) Complete this statement.
The best mathematical shape to model the trunk of a fir tree is a cone because ...................
......................................................................................................................................... [1]

(b) Trees are cut down when the diameter at the base is 21 cm.
The average height of the trunk is 14 metres.
The mass of one cubic metre of wood is 530 kg.



Volume, V, of cone of radius r, height h. \( V = \frac{1}{3}\pi r^2 h \)

Show that it takes approximately 12 trees to make one tonne of wood pulp. [5]

Trees are usually planted in grids to make them easier to look after. This is part of a plan showing some planted trees. Each dot is a tree.
The distance between a tree and its nearest neighbour is $D$ metres.
Find a model, in terms of $D$, for the number of trees, $N$, in a 100m by 100m square.

(a) Sketch your model from Question 8, for values of $D$ between 0 and 10. [2]
(b) An internet site recommends a distance of 4.2 m between trees. A company plants a grid of 620 trees in a 100 m by 100 m square.
Use your graph to find if the company has used the recommended distance between trees. ........................................................................................................................ [2]

When wood pulp is turned into newsprint the mass remains the same.
12 trees make 1 tonne of wood pulp.
There are 620 trees in each 100 m by 100 m square.

The company needs the mass of newsprint in Question 6(b).

Work out the area of trees that the company needs.