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Date	May 2022	Marks available	2	Reference code	22M.2.SL.TZ1.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that	Question number	2	Adapted from	N/A

Question

Eddie decides to construct a path across his rectangular grass lawn using pairs of tiles.

Each tile is $10 cm$ wide and $20 cm$ long. The following diagrams show the path after Eddie has laid one pair and three pairs of tiles. This pattern continues until Eddie reaches the other side of his lawn. When $n$ pairs of tiles are laid, the path has a width of $w_{n}$ centimetres and a length $l_{n}$ centimetres.

The following diagrams show this pattern for one pair of tiles and for three pairs of tiles, where the white space around each diagram represents Eddie’s lawn.

The following table shows the values of $w_{n}$ and $l_{n}$ for the first three values of $n$ .

Find the value of

Write down an expression in terms of $n$ for

Eddie’s lawn has a length $740 cm$ .

The tiles cost $$ 24.50$ per square metre and are sold in packs of five tiles.

To allow for breakages Eddie wants to have at least $8 %$ more tiles than he needs.

There is a fixed delivery cost of $$ 35$ .

$a$ .

[1]

a.i.

$b$ .

[1]

a.ii.

$w_{n}$ .

[2]

b.i.

$l_{n}$ .

[1]

b.ii.

Show that Eddie needs $144$ tiles.

[2]

c.i.

Find the value of $w_{n}$ for this path.

[1]

c.ii.

Find the total area of the tiles in Eddie’s path. Give your answer in the form $a \times 10^{k}$ where $1 \leq a < 10$ and $k$ is an integer.

[3]

Find the cost of a single pack of five tiles.

[3]

Find the minimum number of packs of tiles Eddie will need to order.

[3]

Find the total cost for Eddie’s order.

[2]

Markscheme

$30$ A1

[1 mark]

a.i.

$40$ A1

[1 mark]

a.ii.

arithmetic formula chosen (M1)

$w_{n} = 20 + (n - 1) 10 (= 10 + 10 n)$ A1

[2 marks]

b.i.

arithmetic formula chosen

$l_{n} = 30 + (n - 1) 10 (= 20 + 10 n)$ A1

[1 mark]

b.ii.

$740 = 30 + (n - 1) 10$ OR $740 = 20 - 10 n$ M1

$n = 72$ A1

$144$ tiles AG

Note: The AG line must be stated for the final A1 to be awarded.

[2 marks]

c.i.

$w_{72} = 730$ A1

[1 mark]

c.ii.

$(10 \times 20) \times 144$ (M1)

$= 28800$ (A1)

$2.88 \times 10^{4} {cm}^{2}$ A1

Note: Follow through within the question for correctly converting their intermediate value into standard form (but only if the pre-conversion value is seen).

[3 marks]

EITHER

$1$ square metre $= 100 cm \times 100 cm$ (M1)

(so, $50$ tiles) and hence $10$ packs of tiles in a square metre (A1)

(so each pack is $\frac{$ 24.50}{10 packs}$ )

area covered by one pack of tiles is $(0.2 m \times 0.1 m \times 5 =) 0.1 m^{2}$ (A1)

$24.5 \times 0.1$ (M1)

THEN

$$ 2.45$ per pack (of $5$ tiles) A1

[3 marks]

$\frac{1.08 \times 144}{5} (= 31.104)$ (M1)(M1)

Note: Award M1 for correct numerator, M1 for correct denominator.

$32$ (packs of tiles) A1

[3 marks]

$35 + (32 \times 2.45)$ (M1)

$$ 113 (113.4)$ A1

[2 marks]

Examiners report

In part (a), most candidates were able to find the correct values for $a$ and $b$ .

In part (b), most candidates were able to write down the correct expressions for $w_{n}$ and $l_{n}$ .

In part (c)(i), candidates continued to struggle with “show that” questions. Some substituted 144 for $n$ and worked backwards, however this is never the intention of the question; candidates should progress towards (not “away from”) the given result. Part (ii) was well answered by many candidates.

Part (d) was poorly answered with many candidates multiplying 720 with 730 instead of 10 with 20. However, the majority managed to convert their answer correctly to standard form which gained them a mark for that particular skill.

Part (e) saw very few candidates find the cost of one packet of tiles. The main reason was the failure to convert cm² to m².

In part (f), about half of the candidates managed to find the correct number of packets. Some gained a mark for finding 8% or dividing by 5.

In part (g), most candidates could use their answers to parts (e) and (f) to score “follow through” marks and find the total cost of their order.

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.