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HOUSE OF CARDS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the number of cards in a house of cards.
The diagram shows a house of cards with three rows.
Rows are counted down from the top of the house.
In this investigation
[Image_1: Horizontal and diagonal card notation]
Example 1
This house of cards has 3 rows of cards.
[Image_2: Diagram with 3 rows]
Row 1
Row 2
Row 3
Example 2
This house of cards has 5 rows of cards.
[Image_3: Diagram with 5 rows]
Row 1
Row 2
Row 3
Row 4
Row 5
This is Row 1 in a house of cards.
[Image_4: Diagram of Row 1]
There are 0 horizontal cards.
There are 2 diagonal cards.
There are 2 cards in total.
This is Row 2 in a house of cards.
[Image_5: Diagram of Row 2]
There is 1 horizontal card.
There are 4 diagonal cards.
There are 5 cards in total.
This is Row 3 in a house of cards.
[Image_6: Diagram of Row 3]
(a) Complete the table.
[Table_1: Row (n) - Number of horizontal cards, Number of diagonal cards, Total number of cards]
Row (n) | Number of horizontal cards | Number of diagonal cards | Total number of cards
1 | 0 | 2 | 2
2 | 1 | 4 | 5
3 | [ ] | [ ] | [ ]
4 | [ ] | [ ] | [ ]
5 | [ ] | [ ] | [ ]
[3]
(b) Find an expression, in terms of $n$, for the total number of cards in Row $n$.
.............................................. [3]
(c) The total number of cards in Row $p$ is 368.
Work out how many \textit{diagonal} cards are in Row $p$.
.............................................. [3]
510
555
525
(a) Complete the table.
You may use the grid to help you.
[Table_1: House, Number of horizontal cards, Number of diagonal cards, Total number of cards]
| House (h) | Number of horizontal cards | Number of diagonal cards | Total number of cards |
| --- | --- | --- | --- |
| 1 | 0 | 2 | 2 |
| 2 | 1 | 6 | 7 |
| 3 | | | |
| 4 | | | |
| 5 | | | |
(b) Find an expression, in terms of $h$, for the number of diagonal cards in House $h$.
(c) This is an expression for the number of horizontal cards in House $h$:
$$0.5h(h-1)$$
Use this expression and your answer from part (b) to find an expression for the total number of cards in House $h$.
Give your answer in its simplest form.
(d) The total number of cards in House $k$ is 737.
Find the number of rows in House $k$.
569
561
533
The investigation now looks at the total number of cards in a sequence of houses of cards.
This is the first diagram in the sequence of houses. There is 1 house.
There are 0 horizontal cards.
There are 2 diagonal cards.
There are 2 cards in total.
This is the second diagram in the sequence of houses.
There are 2 houses.
There is 1 horizontal card.
There are 8 diagonal cards.
There are 9 cards in total.
This is the third diagram in the sequence of houses.
There are 3 houses.
There are 4 horizontal cards.
There are 20 diagonal cards.
There are 24 cards in total.
(a) Complete the table. You may use the table in Question 2(a) to help you.
[Table]
| Total number of houses (t) | Number of horizontal cards (H) | Number of diagonal cards | Total number of cards |
|---------------------------|------------------------------|--------------------------|------------------------|
| 1 | 0 | 2 | 2 |
| 2 | 1 | 8 | 9 |
| 3 | 4 | 20 | 24 |
| 4 | | | |
| 5 | | | |
(b) This is a formula for the number of horizontal cards, $H$, in a sequence of $t$ houses of cards.
$$H = \frac{1}{6}t(t+a)(t-a),$$
where $a$ is a positive constant.
Find the value of $a$ and write down the formula.
$a = ..........................................................
H = ...........................................................$ [3 marks]
(c) The $n$th diagram, with $n$ houses, in the sequence of houses has 2925 horizontal cards.
Use part (b) and Question 2(c) to find the total number of cards in the last house in the diagram.
[5 marks]
583
552
514
The table shows the age of athletes running 100 m and their fastest times.
[Table_1]
(a) Complete the scatter diagram to show the results. The first seven points have been plotted for you.
(b) A straight line through the points (38, 10) and (74, 13) models the data.
(i) On the grid, draw the model. [1]
(ii) Find the equation of the model.
............................................................... [3]
(iii) The fastest time for a certain age is 12 seconds. Use the model to find this age.
............................................................... [2]
(iv) The fastest recorded time to run 100 m is 9.58 seconds. Comment on the validity of the model for an athlete aged 20 years.
....................................................................................... [2]
587
534
583
For athletes younger than 20 years there is a different model for the fastest time to run 100 m. This is a graph of the model.
(a) An athlete aged 13 years runs 100 m.
Use the graph to write down the fastest time for this age.
......................................................... [2]
(b) The model for the fastest times for athletes younger than 20 years is
$y = 268 + c \times x^{0.0139}$
where $c$ is a constant.
Use your answer to part (a) to find the value of $c$ correct to the nearest integer. Write down the model.
$c = .......................................................$
$y = .......................................................$ [3]
(c) 10.0 seconds is the fastest time for a certain age that is below 20 years.
Using your model in part (b), solve an equation to show that this age is 17 years. [4]
577
514
564
For athletes aged from 82 years to 105 years there is a different model for the fastest time to run 100m. This model is $y = 0.0381x^2 - 6.23x + 269$.
(a) On the axes, sketch the graph of the model for $82 \leq x \leq 105$.
[3]
(b) The fastest time for an athlete aged 100 years to run 100 m is 26.99 seconds. Find the difference between this time and the time that the model predicts. ......................................................... [2]
(c) 18.32 seconds is the fastest time for a certain age between 82 years and 105 years. Use the model to find this age. ......................................................... [1]
558
572
598
For any age, the fastest recorded time to run 100 m is 9.58 seconds.
Use each model to find the possible ages of the athlete who ran this fastest time.
574
536
510
