November 2018 Paper 2 Foundation

In 542.3 the digits are 5 hundreds, 4 tens, 2 ones and 3 tenths.

So the 4 is worth 40.

Square numbers: 1, 4, 9, 16, 25, ...

The odd ones are 1, 9, 25, 49, ... so, for example, 9 works ($3^2 = 9$).

(a) 1 kg = 1000 g, so $4560 \div 1000 = 4.56$ kg.

(b) 1 m = 1000 mm, so $7.3 \times 1000 = 7300$ mm.

$4 \times 4 \times 4 = 64$, so $\sqrt[3]{64} = 4$.

0.31 is 31 hundredths, so $0.31 = \frac{31}{100}$.

Change each to a decimal:

$\frac{3}{4} = 0.75$, $\frac{5}{7} = 0.714...$, $\frac{19}{25} = 0.76$, $\frac{11}{15} = 0.733...$

Smallest first: $\frac{5}{7}, \frac{11}{15}, \frac{3}{4}, \frac{19}{25}$

(a) $3m - m - m + 3m = (3 - 1 - 1 + 3)m = 4m$

(b) Multiply the numbers: $2 \times 4 = 8$, then write the letters: $8np$.

Each 1 cm stands for 14 km.

$18.8 \times 14 = 263.2$ km

(a) Terms: 7, 10, 13, 16, 19, 22, ... It jumps from 19 to 22, so 21 is missed.

Or: $3n + 4 = 21 \Rightarrow 3n = 17 \Rightarrow n = 5.67$, not a whole number.

(b) Doubling: 1, 2, 4, 8, 16. Or the gaps grow by 1: 1, 2, 4, 7, 11.

(a) $8 \times 5 = 40$, $40 - 2 = 38$

(b) Work backwards: $28 + 2 = 30$, $30 \div 5 = 6$

Adam: $0.3 \times 80 = £24$

Difference: $28 - 24 = £4$

Blue counters: $49 - 20 = 29$

P(blue) $= \frac{29}{49}$

(a) Side $= \sqrt{81} = 9$ cm, so perimeter $= 4 \times 9 = 36$ cm.

(b) Triangle: $\frac{1}{2} \times 16 \times 9 = 72$ cm²

Parallelogram: $5 \times 72 = 360$ cm²

$30h = 360$, so $h = 12$

(a) A fair dice gives each number a $\frac{1}{6}$ chance, so 3 and 6 are equally likely.

(b) Two sixes: $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$, not $\frac{2}{6}$.

(c) Each of 6 numbers pairs with H or T: $6 \times 2 = 12$ outcomes, 1H to 6H and 1T to 6T.

Interest each year: $75 \div 5 = £15$

Rate: $\frac{15}{600} \times 100 = 2.5\%$

Each point of A has the same $x$-coordinate in B, with the $y$-coordinate's sign changed, e.g. $(1, 1) \to (1, -1)$.

So it is a **reflection in the $x$-axis**.

Bags needed (round up):

Cement: $180 \div 25 = 7.2 \to 8$ (has 10)

Sand: $375 \div 22.5 = 16.7 \to 17$ (has 20)

Stone: $1080 \div 50 = 21.6 \to 22$ (has 20)

He only needs to buy 2 bags of stone.

(a) $1.3 = 130\%$, which adds 30%. A 3% increase uses $1.03$: $150 \times 1.03 = 154.5$.

(b) Decrease by 3%: $100\% - 3\% = 97\% = 0.97$

$150 \times 0.97 = 145.5$

(a) $3(x - 4) = 12 \Rightarrow x - 4 = 4 \Rightarrow x = 8$

(b) The highest common factor of $9b$ and $3b^2$ is $3b$.

$9b - 3b^2 = 3b(3 - b)$

(a) $\mathcal{E} = \{2, 4, 6, ..., 24\}$, 12 numbers.

8 is in all three sets. 20 is in B and C only. 6 is B only; 2, 10, 14 are A only; 18, 22 are C only.

4, 12, 16, 24 go outside the circles.

(b) $A \cap B = \{8\}$, so P $= \frac{1}{12}$

Check the line: it should pass through the middle of the crosses with roughly equal numbers either side, which Sean's does not.

Check the axes: the height axis jumps from 140 to 160 in the same space as 160 to 170, so the scale is uneven.

Angle $ABE = 180 - 110 = 70°$ (co-interior angles add up to 180°)

Angle $EBG = 180 - 70 - 35 = 75°$ (angles on a straight line add up to 180°)

Angle $BEG = 180 - 110 - 25 = 45°$ (angles on a straight line)

$x = 180 - 75 - 45 = 60°$ (angles in a triangle add up to 180°)

(a) Ali: $2000 \times 1.025^3 = £2153.78$, interest $£153.78$

Ben: $1600 \times 1.035^3 = £1773.95$, interest $£173.95$

Ben gets more.

(b) Ben: $1600 \times 1.035^2 \times 1.04 = £1782.52$, interest $£182.52$. Still Ben, so no change.

Area $= \frac{1}{2}(10 + 16) \times 7 = 91$ m²

Paint $= 91 \div 2 = 45.5$ litres

Tins $= 45.5 \div 5 = 9.1$, so 10 tins

Cost $= 10 \times 16.99 = £169.90 > £160$, so No.

Gradient $= \frac{\text{change in } y}{\text{change in } x}$

Change in $y = 15 - 9 = 6$, so change in $x = 6 \div 3 = 2$

$d = 5 + 2 = 7$

(a) $(5x + 2)(2x - 3) = 10x^2 - 15x + 4x - 6 = 10x^2 - 11x - 6$

(b) Two numbers that multiply to 3 and add to 4: 1 and 3.

$x^2 + 4x + 3 = (x + 1)(x + 3)$

(a) Move the decimal point 5 places right: $7.547 \times 10^{-5}$

(b) $3.42 \times 10^4 = 34\,200$

(c) Top: $2.3 \times 10^4 \times 6.7 \times 10^3 = 1.541 \times 10^8$

$\frac{1.541 \times 10^8}{5 \times 10^{-8}} = 0.3082 \times 10^{16} = 3.082 \times 10^{15}$