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WINNING LINES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the number of winning lines on a grid.
In a game you win by making a straight line of three ⭕s on a 3 by 3 grid.
Examples of a winning line with three ⭕s.
There are 8 different winning lines.
(a) Another grid is 4 by 4. You now need four ⭕s in a line to win.
Find the number of winning lines on a 4 by 4 grid.
You may use the grids to help you.
[2 marks]
(b) Another grid is 5 by 5. You now need five ⭕s in a line to win.
Find the number of winning lines on a 5 by 5 grid.
You may use the grids to help you.
[1 mark]
(c) A grid is $n$ by $n$. You need $n$ ⭕s in a line to win.
Find an expression, in terms of $n$, for the number of winning lines.
Give your answer in its simplest form.
[2 marks]
512
528
595
(a) In a 3 by 3 grid you need two O\'s in a line to win.
Complete the table to find the number of winning lines with two O\'s.
You may use the grids below the table to help you.
[Table_1]
Size of grid | Number of winning lines
Horizontal | Vertical | Diagonal | Total
3 by 3 | | 8 | |
(b) In a 4 by 4 grid you need three O\'s in a line to win.
Complete the table to find the number of winning lines with three O\'s.
You may use the grids below the table to help you.
[Table_2]
Size of grid | Number of winning lines
Horizontal | Vertical | Diagonal | Total
4 by 4 | | | |
(c) Copy your results from part (a) and part (b) into this table.
Complete the table.
You may use the grids below the table to help you.
[Table_3]
Size of grid | Number of winning lines
Horizontal | Vertical | Diagonal | Total
3 by 3 | | 8 | |
4 by 4 | | | |
5 by 5 | | 8 | |
n by n | | | |
(d) In an n by n grid you need (n-1) O\'s in a line to win.
n must be at least 3.
In one grid the total number of winning lines is a square number less than 50.
Find the grid size.
564
542
563
A rectangular grid has height 2 and width at least 2. You need two ○s in a line to win.
These diagrams show all the winning lines with two ○s on a 2 by 3 grid.
(a) Complete the table for the number of winning lines with two ○s.
You may use the grid below the table to help you.
[Table_1]
Size of grid | Number of winning lines
| Horizontal | Vertical | Diagonal | Total
| 2 by 2 | 2 | 2 | 2 | 6
| 2 by 3 | 4 | 3 | 4 | 11
| 2 by 4 | 6 | 4 |
| 2 by 5 | 5 | 8 |
| 2 by \( w \) | \( w \)
[Grid_Image]
(b) A rectangular grid has height \( n \) and width \( w \) where \( n \geq 2 \) and \( w \geq n \).
Any winning line of ○s has \( n \) ○s.
(i) Give a reason why there are always \( w \) vertical winning lines.
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............................................................................................................................... [1]
(ii) Find an expression, in terms of \( n \) and \( w \), for the total number of horizontal winning lines.
You may use the grid to help you.
[Grid_Image]
........................................................ [2]
(iii) Use part (i) and part (ii) and the geometry of the grid to find an expression for the total number of winning lines.
Do not simplify your expression.
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............................................................................................................................... [1]
(iv) Using your answer to part (iii) show that, when \( w = 2n \), the expression for the total number of winning lines is \( n^2 + 5n + 2 \).
...............................................................................................................................
595
535
571
A rectangular grid has height $n$ and width $w$, where $n$ must be at least 3 and $w \geq n$. Any winning line of Os has $(n - 1)$ Os.
(a) Complete the table with an expression for the number of diagonal winning lines in terms of $n$ and $w$.
You do not need to simplify your expression.
You may use the grid below the table to help you.
[Table_1]
(b) A grid of height 4 contains 54 winning lines with three Os.
Find the width of the grid.
523
534
523
(a) Plot these points on the axes below and draw a line of best fit. The first three points have been plotted for you.
[Table]
Temperature ($T^\circ C$): -20, -10, 0, 10, 20, 30, 40
Speed of sound (S/m/s): 319, 325, 331, 337, 343, 349, 355
(b) Find a model for $S$ in terms of $T$.
................................................ [2]
(c) (i) Find the speed of sound when the air temperature is 27$^\circ C$.
................................................ [2]
(ii) Ellie watches a storm approaching. The air temperature is 27$^\circ C$. She sees a flash of lightning as it happens and hears the sound of its thunder 5 seconds later.
Use your answer to part (i) to find how far the flash is from Ellie in kilometres.
................................................ [2]
566
540
567
Humid air contains water vapour.
Sound travels at a different speed in humid air compared to its speed in dry air.
Air with 0\% humidity is dry air.
Air with 100\% humidity holds all the water vapour that it can.
The table shows the speed of sound, correct to 2 decimal places, for different values of humidity, when the temperature is 20\,^{\circ}\text{C}.
[Table_1]
| Humidity (H\%) | 0 | 20 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|---|
| Speed of sound (S\text{m/s}) | 343.37 | 343.62 | 343.87 | 344.12 | 344.37 | 344.62 |
(a) Show that a linear relationship connects $H$ and $S$.
[2]
(b) (i) $I$ is the increase in the speed of sound with humidity when the temperature is 20\,^{\circ}\text{C}.
Show that $I = 0.0125H$.
[1]
(ii) Assume that the increase in part (i) is the same for all temperatures.
When $T = 0$ and $H = 0$, the speed of sound, correct to 2 decimal places, is 331.37 m/s.
Use this information and part (i) to change your model for $S$ in Question 5(b).
Give your answer in the form $S = x + yT + zH$, where $x, y$ and $z$ are constants.
..................................................................................... [1]
(c) A firework makes a loud bang.
The sound travels 1 km.
The temperature is 20\,^{\circ}\text{C}.
Find the difference between the time for the sound to travel 1 km in dry air and the time to travel 1 km when the humidity is 63\%.
Give your answer in seconds, correct to 5 decimal places.
................................................... [4]
596
540
522
In Question (b)(ii) you wrote a new model for the speed of sound using $I = 0.0125H$ that assumed $I$ was the same at all temperatures. In fact the increase in speed, $I$, does change as temperature changes.
(a) This table shows values of $I$ for some temperatures.
[Table_1]
$$\begin{array}{|c|c|c|c|c|} \hline \text{Temperature (}T^\circ \text{C)} & -20 & 0 & 10 & 40 \\ \hline \text{Increase (}I\text{)} & 0.0006H & 0.0031H & 0.0056H & 0.0412H \\ \hline \end{array}$$
This is a new model for $I$.
$$I = \frac{kT}{350}H$$
When $T = 40$, find the value of $k$. Give your answer correct to 3 decimal places. ............................................................. [3]
(b) Use part (a) and your answer to Question (b)(ii) to write down another model for $S$ in terms of $T$ and $H$. ............................................................. [1]
(c) For places where humans live, air temperatures are usually between $-40^\circ \text{C}$ and $50^\circ \text{C}$ but can reach above $100^\circ \text{C}$ near volcanic eruptions and wildfires.
On the axes, sketch the graph of your model when $H = 25$. [3]
(d) There is a loud explosion in a desert at exactly 1218. The temperature is $45^\circ \text{C}$ and the humidity is $25\%$. The explosion is heard 100 km away.
Find the time when the explosion is heard. Give your time correct to the nearest second. ............................................................. [6]
552
527
503
