Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 6 2024 Winter Zone 3

This question is about grids with $h = 1$.
A 1 by 2 grid has one diagonal square, so $D(1, 2) = 1$.
A 1 by 3 grid has two diagonal squares, so $D(1, 3) = 2$.
A 1 by 4 grid has three diagonal squares, so $D(1, 4) = 3$.

(a) Write down $D(1, 7)$. ............................................................... [1]
(b) Find an expression for $D(1, w)$. ............................................................... [1]
(c) $D(1, w) = 20$. Write down the value of $w$. ............................................................... [1]
(d) Explain why $D(1, n) = D(n, 1)$. ......................................................................................... [1]

DIAGONAL SQUARES (30 marks)

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

This investigation looks at the number of diagonal squares in a grid where the diagonals of each square have been drawn.
This grid is $h$ by $w$, where $h$ is the number of rows in the grid and $w$ is the number of columns in the grid.


Example

This grid is 3 by 5.
It has 3 rows and 5 columns, so $h = 3$ and $w = 5$.
Some of the diagonal squares are shaded.


Throughout the investigation the number of diagonal squares in an $h$ by $w$ grid is called $D(h, w)$.

This question is about grids with $h = 2$.
A 2 by 2 grid has 2 rows each with 1 diagonal square and 1 row of 2 diagonal squares.


The table shows this.

| 2 by 2 grid | | | |
|--------------|----------------------|-------------|-------|
| | Number of rows of | Number of | Total |
| | diagonal squares | diagonal | |
| | | squares in | |
| | | each row | |
|--------------|----------------------|-------------|-------|
| | 2 | 1 | 2 × 1 = 2 |
| | 1 | 2 | 1 × 2 = 2 |

$D(2, 2) = 2 + 2 = 4$

A 2 by 4 grid has 2 rows of 3 diagonal squares and 1 row of 4 diagonal squares.


The table shows this.

| 2 by 4 grid | | | |
|--------------|----------------------|-------------|-------|
| | Number of rows of | Number of | Total |
| | diagonal squares | diagonal | |
| | | squares in | |
| | | each row | |
|--------------|----------------------|-------------|-------|
| | 2 | 3 | 2 × 3 = 6 |
| | 1 | 4 | 1 × 4 = 4 |

$D(2, 4) = 6 + 4 = 10$

(a) Complete the table.
You may use the grids to help you.

| 2 by 8 grid | | | |
|--------------|----------------------|-------------|-------|
| | Number of rows of | Number of | Total |
| | diagonal squares | diagonal | |
| | | squares in | |
| | | each row | |
|--------------|----------------------|-------------|-------|
| | 2 | | |
| | 1 | | |

$D(2, 8) = + = 22$


[3]

(b) A 2 by $w$ grid has $h = 2$ and width $w$.

Complete the table with expressions in terms of $w$.

| 2 by $w$ grid | | | |
|---------------|----------------------|-------------|-------|
| | Number of rows of | Number of | Total |
| | diagonal squares | diagonal | |
| | | squares in | |
| | | each row | |
|---------------|----------------------|-------------|-------|
| | 2 | | |
| | 1 | | |

$D(2, w) =$
[3]

DIAGONAL SQUARES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the number of diagonal squares in a grid where the diagonals of each square have been drawn.
This grid is $h$ by $w$, where $h$ is the number of rows in the grid and $w$ is the number of columns in the grid.

Example
This grid is 3 by 5.
It has 3 rows and 5 columns, so $h = 3$ and $w = 5$.
Some of the diagonal squares are shaded.

Throughout the investigation the number of diagonal squares in an $h$ by $w$ grid is called $D(h, w)$.
This question is about grids with $h = 3$.
A 3 by 5 grid has 3 rows each with 4 diagonal squares and 2 rows each with 5 diagonal squares.

The table shows this.
[Table_1]
(a) Work out $D(3, 8)$.
.................................................. [2]
(b) A 3 by $w$ grid has $h = 3$ and width $w$.
Complete the table with expressions in terms of $w$.
[Table_2] .................................................. [3]
(c) In a 3 by $w$ grid, $D(3, w) = 97$.
Find the value of $w$.
.................................................. [2]

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

This investigation looks at the number of *diagonal squares* in a grid where the diagonals of each square have been drawn.
This grid is $h$ by $w$, where $h$ is the number of rows in the grid and $w$ is the number of columns in the grid.



Example

This grid is 3 by 5.
It has 3 rows and 5 columns, so $h = 3$ and $w = 5$.
Some of the diagonal squares are shaded.



Throughout the investigation the number of diagonal squares in an $h$ by $w$ grid is called $D(h, w)$.

In an $h$ by $w$ grid there are:

• $h$ rows with $(w-1)$ diagonal squares
• $(h-1)$ rows with $w$ diagonal squares.

DIAGONAL SQUARES (30 marks)

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

This investigation looks at the number of diagonal squares in a grid where the diagonals of each square have been drawn.
This grid is $h$ by $w$, where $h$ is the number of rows in the grid and $w$ is the number of columns in the grid.



Example

This grid is 3 by 5.
It has 3 rows and 5 columns, so $h = 3$ and $w = 5$.
Some of the diagonal squares are shaded.



Throughout the investigation the number of diagonal squares in an $h$ by $w$ grid is called $D(h, w)$.

This question is about a square grid with $n$ rows and $n$ columns.
This is an $n$ by $n$ grid.

(a) $D(h, w) = 2hw - (h + w)$
Show that the number of diagonal squares in an $n$ by $n$ grid, $D(n, n)$, is always an even number. [2]

(b) The width of square grid $A$ is one more than the width of square grid $B$.
The width of square grid $B$ is $n$.
The difference between the number of diagonal squares in the two square grids is 36.
Find the value of $n$. [4]

DIAGONAL SQUARES (30 marks)

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

This investigation looks at the number of extit{diagonal squares} in a grid where the diagonals of each square have been drawn.
This grid is $h$ by $w$, where $h$ is the number of rows in the grid and $w$ is the number of columns in the grid.



Example

This grid is 3 by 5.
It has 3 rows and 5 columns, so $h = 3$ and $w = 5$.
Some of the diagonal squares are shaded.



Throughout the investigation the number of diagonal squares in an $h$ by $w$ grid is called $D(h, w)$.

A grid is a rectangle.
It has 31 diagonal squares.

$D(h, w) = 2hw - (h + w)$

Find the sizes of the possible rectangular grids when $h < w$.
........................................................................................................... [5]

The table shows the distance of a car from a fixed point on the road at time t seconds.

[Table_1]

The graph shows this information.



(a) Find the speed of the car. .................................................. [2]

(b) Find a model for $d$ in terms of $t$. .................................................. [2]

(c) Find the distance of the car from the fixed point when $t = 7$ seconds. .................................................. [1]

A car and a truck travel in the same direction along a road.
The sketch graph shows their distances from a fixed point on the road.



(a) Write down the number of metres the truck is in front of the car when $t = 0$.
.................................................. [1]

(b) A model for the distance the car travels, in terms of $t$, is $d = 25t$.
The speed of the truck is $20\text{ m/s}$.
Find a model for the distance the truck travels in terms of $t$.
.................................................. [1]

(c) The car overtakes the truck at point $P$.
Use algebra to find the coordinates of point $P$.
( .................... , .................... ) [3]

(d) When $t = 5$ the car changes its speed to $30\text{ m/s}$.
The two lines now meet at a different point.

(i) On the axes, sketch the new graph for the car. [1]
(ii) Find the number of seconds the car has travelled from the fixed point when it reaches the truck.
.................................................. [4]

This question is about what happens when a car overtakes a truck.
The diagrams are not to scale.
In these diagrams, the direction of travel is from left to right.

The car has length 5 metres.
The truck has length 15 metres.

The car moves into the outside lane to overtake the truck when there is a gap of 45 metres between the front of the car and the back of the truck.



The car moves back into the inside lane when there is a gap of 50 metres between the back of the car and the front of the truck.



(a) When the car starts overtaking the distance between the front of the car and the front of the truck is 60m.
Show that when the car finishes overtaking the distance between the front of the car and the front of the truck is 55m.

(b) The car starts overtaking the truck at time $t = 8$ seconds.
The car finishes overtaking the truck at time $t = k$ seconds.
The sketch graph shows their distances from a fixed point on the road.



The distance travelled, $d$ metres, and the time taken, $t$ seconds, are measured from a fixed point on the road.
A model for the distance travelled by the car, in terms of $t$, is $d = 25t$.

Find the speed of the truck and write down a model for the distance travelled by the truck.

Speed ..........................................
Model ..........................................

(c) When $t = 8$, the truck is 60 metres in front of the car.
Find the time, $k$ seconds, when the car is 55 metres in front of the truck.
..............................................

(d) Work out the total distance that the car travels when overtaking the truck.
..............................................

(e) When the car overtakes the truck it travels further than the truck.
(i) Show that the difference between the distances travelled by the car and the truck is 115 metres.
[1]
(ii) The speed of the car is now $x$ m/s and the speed of the truck is 20 m/s.
Write down an expression for the difference in speed between the car and the truck.
..............................................
[1]
(iii) Show that a model for the total distance, $D$ metres, the car travels when overtaking the truck is $D = \frac{115x}{x-20}$.
[2]
(iv) The car travels 690 metres when overtaking the truck.
Find the speed of the car when overtaking the truck.
..............................................
[3]