June 2019 Paper 3 Higher

(a) In both A and B: 6, 9

A only: 1, 5, 8. B only: 2. Neither: 3, 4, 7.

(b) $A \cap B = \{6, 9\}$, 2 numbers out of 9.

$P = \frac{2}{9}$

Multiplier for 1.5% interest: 1.015

$200\,000 \times 1.015^4 = 212\,272.71$

Interest: $212\,272.71 - 200\,000 = £12\,272.71$

(a) 80 plants, so the median is about the 40th–41st value.

Running totals: 7, 20, 34, 46, ... The 40th value is in $40 < h \le 50$.

(b) Plot each frequency at the middle of its class: (15, 7), (25, 13), (35, 14), (45, 12), (55, 16), (65, 18). Join with straight lines.

Things wrong with the graph:

1. The points are joined with curves; a time series should use straight lines.
2. The vertical axis starts at 6, which exaggerates the changes.

Other valid points: the first quarter of each year is missing, and it is not clear what 2, 3, 4 on the horizontal axis mean.

Interior angles of a hexagon: $(6 - 2) \times 180 = 720^\circ$

By symmetry $A = B = 117^\circ$; let $D = E = x$ and $C = F = 2x$.

$117 + 117 + 2x + 2x + x + x = 720$, so $6x = 486$ and $x = 81$

Angle $AFE = 2 \times 81 = 162^\circ$

Radius $= 0.8$ m

Top and bottom: $2 \times \pi \times 0.8^2 = 4.021...$

Curved surface: $2 \times \pi \times 0.8 \times 1.8 = 9.047...$

One tank: $13.069...$ m²; three tanks: $39.21$ m²

Paint covers $7 \times 5 = 35$ m² $< 39.21$, so No, he does not have enough.

Numerator: $2.5 \times \sin 43° = 1.70499$

Denominator: $8.2^2 - 50.5 = 67.24 - 50.5 = 16.74$

$\sqrt{1.70499 \div 16.74} = \sqrt{0.10185} = 0.3191...$

Answer: $0.319$

(a) The right angle is at $B$, so the longest side is $AC$. Sarah treated $BC$ as the hypotenuse. Correct: $BC^2 = 8^2 - 6^2 = 28$, $BC = 5.29$ cm.

(b) Count squares from $C$: $CR = 4$ but $CZ = 10$, so $10 \div 4 = 2.5$. Roy enlarged by 2.5, not $1\frac{1}{2}$.

Total work: $6 \times 9 = 54$ machine-days

First 3 days: $3 + 4 + 5 = 12$ machine-days

Left: $54 - 12 = 42$, done by 6 machines in $42 \div 6 = 7$ days

Total: $3 + 7 = 10$ days

Tax is 20% of the interest, so interest $= 28.80 \div 0.2 = £144$

Rate $= \frac{144}{8000} \times 100 = 1.8\%$

(a) Time $= \frac{3.9 \times 10^7}{3 \times 10^5} = 1.3 \times 10^2 = 130$ s

(b) A lower speed over the same distance means more time, so the real time is longer than 130 s.

A power of $\frac{1}{4}$ means the fourth root, not dividing by 4.

$64^{\frac{1}{4}} = \sqrt[4]{64} = \sqrt{\sqrt{64}} = \sqrt{8} = 2\sqrt{2} \approx 2.83$

Work per cm³ scaled by litres (the conversion factor cancels):

Ethanol: $1.09 \times 60 = 65.4$

Propylene: $0.97 \times 128 = 124.16$

Density $= \frac{65.4 + 124.16}{188} = \frac{189.56}{188} = 1.008... \approx 1.01$ g/cm³

Area of triangle $AFE = \frac{1}{2} \times 24 \times AE \times \sin 30° = 6\,AE$

Two triangles: $12\,AE$

Rectangle: $AB \times AE = 12\,AE$, so $AB = 12$

$AE = 3 \times AB = 36$ cm

$f(-x)$ reflects in the $y$-axis: $(7, 2) \to (-7, 2)$

$-3$ moves it 3 down: $(-7, 2) \to (-7, -1)$

First differences: 6, 10, 14, 18, 22. Second differences: 4, so the sequence starts $2n^2$.

$2n^2$: 2, 8, 18, 32, ...

Sequence minus $2n^2$: −3, −3, −3, ... so the nth term is $2n^2 - 3$.

- $f(x) \propto \frac{1}{x}$: two branches that never touch the axes, B.
- Trigonometric: a wave through $O$, A.
- Exponential: flat on the left, cuts the $y$-axis above 0 and shoots up, D.
- $f(x) \propto \sqrt{x}$: starts at $O$ and rises more and more slowly, C.

(a) $(2x+1)(x+3) = 2x^2 + 7x + 3$

$(2x^2 + 7x + 3)(3x + 7) = 6x^3 + 14x^2 + 21x^2 + 49x + 9x + 21 = 6x^3 + 35x^2 + 58x + 21$

So $a = 6, b = 35, c = 58, d = 21$, all integers.

(b) $-\frac{3}{5} < 1 - x < \frac{3}{5}$

$\frac{2}{5} < x < \frac{8}{5}$

(a) $u$: 26.15 to 26.25. $a$: 4.25 to 4.35.

Largest $D$: biggest top, smallest bottom.

$D_{max} = \frac{26.25^2}{2 \times 4.25} = \frac{689.0625}{8.5} = 81.0662$

(b) Lower bound 78.6003, upper bound 81.0662. They differ at 1 s.f. in the units but both round to 80 to the nearest 10, so $D = 80$.

From $3x + 4y = 7$: $4y = 7 - 3x$, so $4y^2 = \frac{(7-3x)^2}{4}$

$x^2 - \frac{49 - 42x + 9x^2}{4} = 9 \Rightarrow 4x^2 - 49 + 42x - 9x^2 = 36$

$5x^2 - 42x + 85 = 0 \Rightarrow (5x - 17)(x - 5) = 0$

$x = 5 \Rightarrow y = \frac{7 - 15}{4} = -2$

$x = \frac{17}{5} \Rightarrow y = \frac{7 - 10.2}{4} = -\frac{4}{5}$

Frequency = frequency density × width:

0–60: $2 \times 60 = 120$; 60–90: $5 \times 30 = 150$; 90–120: $9 \times 30 = 270$; 120–135: $6 \times 15 = 90$; 135–180: $2 \times 45 = 90$

Total $= 720$. Factory (60–120 g): $150 + 270 = 420$

$x = \frac{420}{720} \times 360 = 210$

Tangent meets radius at 90°, so triangle $OAB$ is right-angled at $A$.

$OA = 16 \sin 30° = 8$, so the circle is $x^2 + y^2 = 64$

$(3p)^2 + p^2 = 64 \Rightarrow 10p^2 = 64 \Rightarrow p^2 = 6.4$

$p = 2.529... = 2.5$ (1 d.p.)

The back-bearing of $A$ from $B$ is $217°$, so $ABC = 217 - 150 = 67°$.

$AC^2 = 8^2 + 9^2 - 2 \times 8 \times 9 \cos 67° = 88.73$, $AC = 9.420$ km

$\sin BAC = \frac{9 \sin 67°}{9.420} \Rightarrow BAC = 61.58°$

Bearing $= 37 + 61.58 = 098.6°$