Secondary 3 Elementary Mathematics Semestral Assessment 2 (End of Year) Paper 2

SA2 Practice Paper — Version 2 of 5

Elementary Mathematics Secondary 3 — Answer Key

Section A

1. Using Pythagoras' theorem: $QR = \sqrt{PR^2 - PQ^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \text{ cm}$

Answer: $QR = 24$ cm ✓ (2 marks)

2. Using Pythagoras: $AC = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ cm

$\tan(\angle ACB) = \frac{AB}{BC} = \frac{5}{12}$ $\angle ACB = \tan^{-1}\left(\frac{5}{12}\right) = 22.6198...^\circ$

Answer: $\angle ACB = 22.6^\circ$ ✓ (2 marks)

Common mistake: Students may confuse which ratio to use. $\angle ACB$ is at $C$, so opposite = $AB$, adjacent = $BC$.

3. Let $\theta$ be the angle the ladder makes with the ground.

$\cos\theta = \frac{2.4}{6} = 0.4$ $\theta = \cos^{-1}(0.4) = 66.4218...^\circ$

Answer: $\theta = 66.4^\circ$ ✓ (2 marks)

4. Using Pythagoras: $XZ = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$ cm

$\tan(\angle XZY) = \frac{XY}{YZ} = \frac{8}{15}$ $\angle XZY = \tan^{-1}\left(\frac{8}{15}\right) = 28.0724...^\circ$

Answer: $\angle XZY = 28.1^\circ$ ✓ (2 marks)

5. Let $\theta$ be the angle of elevation of the sun.

$\tan\theta = \frac{12}{9} = \frac{4}{3}$ $\theta = \tan^{-1}\left(\frac{4}{3}\right) = 53.1301...^\circ$

Answer: $\theta = 53.1^\circ$ ✓ (2 marks)

6. In $\triangle DEF$, $\angle E = 90^\circ$, so:

$\tan(38.2^\circ) = \frac{EF}{DE} = \frac{EF}{11}$ $EF = 11 \times \tan(38.2^\circ) = 11 \times 0.7869... = 8.656...$

Answer: $EF = 8.7$ cm ✓ (2 marks)

Common mistake: Students may use $\sin$ or $\cos$ instead of $\tan$. Since $DE$ is adjacent to $\angle DFE$ and $EF$ is opposite, $\tan$ is the correct ratio.

7. Let the height of the building be $h$ m and let the distance from point $B$ to the base of the building be $x$ m.

From point $A$ (which is $x - 40$ m from the building): $\tan 35^\circ = \frac{h}{x - 40} \quad \Rightarrow \quad h = (x - 40)\tan 35^\circ \quad \text{...(1)}$

From point $B$: $\tan 20^\circ = \frac{h}{x} \quad \Rightarrow \quad h = x\tan 20^\circ \quad \text{...(2)}$

Equating (1) and (2): $(x - 40)\tan 35^\circ = x\tan 20^\circ$ $x\tan 35^\circ - 40\tan 35^\circ = x\tan 20^\circ$ $x(\tan 35^\circ - \tan 20^\circ) = 40\tan 35^\circ$ $x = \frac{40\tan 35^\circ}{\tan 35^\circ - \tan 20^\circ} = \frac{40 \times 0.7002}{0.7002 - 0.3640} = \frac{28.008}{0.3362} = 83.31...$

$h = 83.31 \times \tan 20^\circ = 83.31 \times 0.3640 = 30.32...$

Answer: Height of building $= 30.3$ m ✓ (2 marks)

Marking: 1 mark for correct setup of two equations; 1 mark for correct answer.

8. Using the cosine rule: $LN^2 = LM^2 + MN^2 - 2(LM)(MN)\cos(\angle LMN)$ $LN^2 = 9^2 + 14^2 - 2(9)(14)\cos 52^\circ$ $LN^2 = 81 + 196 - 252 \times 0.6157$ $LN^2 = 277 - 155.15 = 121.85$ $LN = \sqrt{121.85} = 11.038...$

Answer: $LN = 11.0$ cm ✓ (2 marks)

Section B

9. (a) Using the cosine rule: $AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle ABC)$ $AC^2 = 13^2 + 10^2 - 2(13)(10)\cos 62^\circ$ $AC^2 = 169 + 100 - 260 \times 0.4695$ $AC^2 = 269 - 122.07 = 146.93$ $AC = \sqrt{146.93} = 12.121...$

Answer: $AC = 12.1$ cm ✓ (3 marks)

Marking: 1 mark for correct cosine rule setup; 1 mark for correct substitution; 1 mark for correct answer.

(b) Using area formula: $\text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle ABC)$ $\text{Area} = \frac{1}{2} \times 13 \times 10 \times \sin 62^\circ$ $\text{Area} = 65 \times 0.8829 = 57.39...$

Answer: Area $= 57.4$ cm² ✓ (2 marks)

10. (a) At point $Q$, the ship turns to a bearing of $150^\circ$. The interior angle $\angle PQR = 180^\circ - (150^\circ - 90^\circ) = 180^\circ - 60^\circ = 120^\circ$.

Explanation: Bearing of $150^\circ$ from $Q$ means the direction is $60^\circ$ south of east. Since $PQ$ is due east, the angle between $PQ$ (extended) and $QR$ is $180^\circ - 60^\circ = 120^\circ$.

Using the cosine rule: $PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(\angle PQR)$ $PR^2 = 45^2 + 60^2 - 2(45)(60)\cos 120^\circ$ $PR^2 = 2025 + 3600 - 5400 \times (-0.5)$ $PR^2 = 5625 + 2700 = 8325$ $PR = \sqrt{8325} = 91.241...$

Answer: $PR = 91.2$ km ✓ (3 marks)

Marking: 1 mark for finding $\angle PQR = 120^\circ$; 1 mark for correct cosine rule setup and substitution; 1 mark for correct answer.

(b) Using the sine rule in $\triangle PQR$: $\frac{\sin(\angle QPR)}{QR} = \frac{\sin(\angle PQR)}{PR}$ $\frac{\sin(\angle QPR)}{60} = \frac{\sin 120^\circ}{91.241}$ $\sin(\angle QPR) = \frac{60 \times 0.8660}{91.241} = \frac{51.96}{91.241} = 0.5695$ $\angle QPR = \sin^{-1}(0.5695) = 34.72^\circ$

The bearing of $R$ from $P$: Since $PQ$ is due east (bearing $090^\circ$), and $\angle QPR = 34.72^\circ$ measured south of east: $\text{Bearing} = 90^\circ + 34.72^\circ = 124.72^\circ$

Answer: Bearing of $R$ from $P = 125^\circ$ ✓ (3 marks)

Marking: 1 mark for correct sine rule setup; 1 mark for finding $\angle QPR$; 1 mark for correct bearing.

11. (a) Using the cosine rule: $\cos(\angle PQR) = \frac{PQ^2 + QR^2 - PR^2}{2(PQ)(QR)}$ $\cos(\angle PQR) = \frac{7^2 + 11^2 - 9^2}{2(7)(11)} = \frac{49 + 121 - 81}{154} = \frac{89}{154} = 0.5779...$ $\angle PQR = \cos^{-1}(0.5779) = 54.685...^\circ$

Answer: $\angle PQR = 54.7^\circ$ ✓ (3 marks)

Marking: 1 mark for correct cosine rule setup; 1 mark for correct substitution; 1 mark for correct answer.

(b) Area of $\triangle PQR$ using the sine formula: $\text{Area} = \frac{1}{2} \times PQ \times QR \times \sin(\angle PQR)$ $\text{Area} = \frac{1}{2} \times 7 \times 11 \times \sin 54.7^\circ = 38.5 \times 0.8162 = 31.42... \text{ cm}^2$

Also, area using base $QR$ and height $PS$: $\text{Area} = \frac{1}{2} \times QR \times PS = \frac{1}{2} \times 11 \times PS$

Equating: $\frac{1}{2} \times 11 \times PS = 31.42$ $PS = \frac{31.42 \times 2}{11} = \frac{62.84}{11} = 5.713...$

Answer: $PS = 5.71$ cm ✓ (3 marks)

Marking: 1 mark for finding area using sine formula; 1 mark for equating with base-height formula; 1 mark for correct answer.

Section C

12. (a) In right-angled $\triangle ABC$ ($\angle ABC = 90^\circ$): $AC = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ cm}$

Answer: $AC = 17$ cm ✓ (2 marks)

(b) In $\triangle ACD$, using the cosine rule: $\cos(\angle DAC) = \frac{AD^2 + AC^2 - CD^2}{2(AD)(AC)}$ $\cos(\angle DAC) = \frac{12^2 + 17^2 - 10^2}{2(12)(17)} = \frac{144 + 289 - 100}{408} = \frac{333}{408} = 0.8162...$ $\angle DAC = \cos^{-1}(0.8162) = 35.287...^\circ$

Answer: $\angle DAC = 35.3^\circ$ ✓ (3 marks)

Marking: 1 mark for correct cosine rule setup; 1 mark for correct substitution; 1 mark for correct answer.

(c) Area of quadrilateral $ABCD$ = Area of $\triangle ABC$ + Area of $\triangle ACD$

Area of $\triangle ABC$: $\text{Area}_{ABC} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2$

Area of $\triangle ACD$: $\text{Area}_{ACD} = \frac{1}{2} \times AD \times AC \times \sin(\angle DAC)$ $\text{Area}_{ACD} = \frac{1}{2} \times 12 \times 17 \times \sin 35.3^\circ = 102 \times 0.5780 = 58.96... \text{ cm}^2$

Alternatively, using $\angle ADC = 55^\circ$: $\text{Area}_{ACD} = \frac{1}{2} \times AD \times CD \times \sin(\angle ADC) = \frac{1}{2} \times 12 \times 10 \times \sin 55^\circ = 60 \times 0.8192 = 49.15... \text{ cm}^2$

Wait — there is an inconsistency. Let me recalculate using the given data directly.

Using the given $\angle ADC = 55^\circ$: $\text{Area}_{ACD} = \frac{1}{2} \times AD \times CD \times \sin(\angle ADC) = \frac{1}{2} \times 12 \times 10 \times \sin 55^\circ = 60 \times 0.8192 = 49.15 \text{ cm}^2$

Total area: $\text{Area}_{ABCD} = 60 + 49.15 = 109.15...$

Answer: Area of $ABCD = 109$ cm² ✓ (3 marks)

Marking: 1 mark for area of $\triangle ABC$; 1 mark for area of $\triangle ACD$ using given angle; 1 mark for correct total.

(d) The shortest distance from $B$ to diagonal $AC$ is the perpendicular distance.

Area of $\triangle ABC = 60$ cm² (from part c)

Also: $\text{Area}_{ABC} = \frac{1}{2} \times AC \times h$ where $h$ is the perpendicular distance from $B$ to $AC$.

$60 = \frac{1}{2} \times 17 \times h$ $h = \frac{60 \times 2}{17} = \frac{120}{17} = 7.0588...$

Answer: Shortest distance $= 7.06$ cm ✓ (2 marks)

Marking: 1 mark for using area = ½ × base × height; 1 mark for correct answer.

Mark Summary