November 2022 Paper 1 Higher

$500 = 2 \times 250 = 2 \times 2 \times 125$

$125 = 5 \times 5 \times 5$

So $500 = 2^2 \times 5^3$

(a) $1 + 2 = 3$; $\frac{3}{5} + \frac{1}{4} = \frac{12}{20} + \frac{5}{20} = \frac{17}{20}$

Answer $3\frac{17}{20}$

(b) $2\frac{2}{3} = \frac{8}{3}$

$\frac{8}{3} \div 6 = \frac{8}{3} \times \frac{1}{6} = \frac{8}{18} = \frac{4}{9}$

Inside the bracket add the powers: $2^{-5} \times 2^8 = 2^{3}$

Square by multiplying the power: $(2^3)^2 = 2^6$

$4 \times 32 = 128$

There are $3 + 2 = 5$ decimal places in the question, so the answer is $0.00128$.

15 out of 80 people chose B.

$\frac{15}{80} \times 40\,000 = 7500$

(a)(i) Double $1 : 3$ to $2 : 6$ so $b$ is 6 in both. $a : b : c = 2 : 6 : 5$

(a)(ii) Total parts $2 + 6 + 5 = 13$, so $a$ is $\frac{2}{13}$.

(b) $p = 5n = 5 \times 2m = 10m$, so $m : p = 1 : 10$

Area $= 4 \times 2 = 8\text{ m}^2$

Pressure $= \frac{10\,000}{8} = 1250$ N/m$^2$

m must be a multiple of both 5 and 7, so try $m = 35$.

n must be a multiple of both 2 and 7, so try $n = 14$.

$35 = 5 \times 7$ and $14 = 2 \times 7$, so the HCF is 7.

(a) $x = -2$: $6(-2) - (-2)^3 = -12 + 8 = -4$

$x = -1$: $-6 + 1 = -5$; $x = 0$: $0$; $x = 1$: $6 - 1 = 5$

(b) Plot $(-3, 9), (-2, -4), (-1, -5), (0, 0), (1, 5), (2, 4), (3, -9)$ and join with a smooth curve.

(a) A 5 came up 10 times out of 40, so $P(5) \approx \frac{10}{40} = \frac{1}{4}$

Two fives: $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$

(b) A 1 came up 6 times out of 40: $\frac{6}{40} = 0.15 = 15\%$

Q's top edge is 3 units, P's is 9 units, so the scale factor is $\frac{3}{9} = \frac{1}{3}$.

Draw lines through matching corners, e.g. $(6, 14)$ to $(2, 6)$ and $(15, 14)$ to $(5, 6)$: they meet at $(0, 2)$.

Enlargement, scale factor $\frac{1}{3}$, centre $(0, 2)$.

Multiply the first by 3 and the second by 2:

$15x + 6y = 33$ and $8x + 6y = 12$

Subtract: $7x = 21$, so $x = 3$

$5(3) + 2y = 11 \Rightarrow 2y = -4 \Rightarrow y = -2$

$p \propto \frac{1}{t}$ so $p \times t$ is constant: $1 \times 100 = 100$

$t = 25$: $p = \frac{100}{25} = 4$

$p = 5$: $t = \frac{100}{5} = 20$

$t = 2$: $p = \frac{100}{2} = 50$

Frequency density = frequency ÷ class width

$50$–$70$: $20 \div 20 = 1$

$70$–$80$: $50 \div 10 = 5$

$80$–$90$: $60 \div 10 = 6$

$90$–$110$: $30 \div 20 = 1.5$

Draw bars of these heights with the correct widths.

Full circumference $= 2\pi \times 18 = 36\pi$

Fraction of the circle: $\frac{4\pi}{36\pi} = \frac{1}{9}$

$x = \frac{1}{9} \times 360 = 40$

(a) $(2m + 1)^2 - (2n - 1)^2 = 4m^2 + 4m + 1 - (4n^2 - 4n + 1)$

$= 4m^2 - 4n^2 + 4m + 4n = 4[(m + n)(m - n) + (m + n)]$

$= 4(m + n)(m - n + 1)$

(b) Yes. $2m + 1$ and $2n - 1$ stand for any odd numbers, and the result has a factor of 4, so it is always a multiple of 4.

The denominator 3 means cube root: $\sqrt[3]{\frac{8}{27}} = \frac{2}{3}$

The numerator 4 means raise to the power 4: $\left(\frac{2}{3}\right)^4 = \frac{16}{81}$

$OA = OB$ (radii), so triangle $AOB$ is isosceles: $OBA = \frac{180 - x}{2} = 90 - \frac{x}{2}$

A tangent meets a radius at $90°$, so $OBC = 90°$

$ABC = 90 - \left(90 - \frac{x}{2}\right) = \frac{1}{2}x$

Combine the fractions: $\frac{(x + 1) - x}{x(x + 1)} = 4$, so $\frac{1}{x(x + 1)} = 4$

$1 = 4x^2 + 4x$, so $4x^2 + 4x - 1 = 0$

$x = \frac{-4 \pm \sqrt{16 + 16}}{8} = \frac{-4 \pm 4\sqrt{2}}{8}$

$x = -\frac{1}{2} \pm \frac{1}{2}\sqrt{2}$

Same colour: $\frac{3}{11} \times \frac{2}{10} + \frac{7}{11} \times \frac{6}{10} + \frac{1}{11} \times 0 = \frac{6 + 42}{110} = \frac{48}{110}$

Different colours: $1 - \frac{48}{110} = \frac{62}{110} = \frac{31}{55}$

The cosine graph starts at $(0, 1)$ and falls to its lowest value of $-1$ at $x = 180°$.

So $P = (180, -1)$

Sine rule: $\frac{\sin ABC}{6.5} = \frac{\sin 30°}{10.7}$

$\sin ABC = \frac{6.5 \times 0.5}{10.7} = \frac{3.25}{10.7}$

Multiply top and bottom by 20: $\frac{65}{214}$

(a) Ratio: $10 \div \sqrt{5} = 2\sqrt{5}$

Next term: $400\sqrt{5} \times 2\sqrt{5} = 800 \times 5 = 4000$

(b) $r^2 = \frac{5\sqrt{2}}{8} \div \frac{5\sqrt{2}}{4} = \frac{1}{2}$, so $r = \frac{1}{\sqrt{2}}$ (terms positive)

$a = \frac{5\sqrt{2}}{4} \div r^3 = \frac{5\sqrt{2}}{4} \times 2\sqrt{2} = \frac{20}{4} = 5$

(a) $\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h \Rightarrow 4r = h$, so $r : h = 1 : 4$

(b) $4\pi r^2 = \pi r^2 + \pi r l \Rightarrow 3r = l$

$l^2 = r^2 + h^2 \Rightarrow 9r^2 = r^2 + h^2 \Rightarrow h^2 = 8r^2$

$h = \sqrt{8}\,r$, so $r : h = 1 : \sqrt{8}$