Secondary 4 Elementary Mathematics Practice Paper 4

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TuitionGoWhere Practice Paper - Elementary Mathematics Secondary 4

TuitionGoWhere Practice Paper (AI)

Subject: Elementary Mathematics Level: Secondary 4 Paper: Practice Paper (Geometry & Trigonometry) Version: 4 of 5 Duration: 1 hour 30 minutes Total Marks: 60

Name: _________________________ Class: _________________________ Date: _________________________

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Instructions to Candidates

1. This paper consists of 20 questions divided into three sections.
2. Answer all questions.
3. Write your answers in the spaces provided.
4. Show all working clearly. Marks are awarded for method as well as final answers.
5. Unless otherwise stated, give non-exact numerical answers correct to 3 significant figures.
6. Diagrams are not necessarily drawn to scale.
7. You may use an approved scientific calculator.
8. The total mark for this paper is 60.

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Section A: Short Answer Questions (20 marks)

Answer all questions in this section. Each question carries 2 marks.

1. In the diagram, $O$ is the centre of the circle. $\angle AOB = 130^\circ$.

Find $\angle ACB$ and state the circle theorem used.

![Circle with centre O, points A, B, C on circumference, angle AOB marked 130°]

Answer: _________________________

Theorem: _________________________

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2. A sector of a circle has radius 8 cm and angle $\frac{5\pi}{6}$ radians.

Calculate the arc length of the sector.

Answer: _________________________ cm

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3. In $\triangle PQR$, $PQ = 10$ cm, $PR = 14$ cm, and $\angle QPR = 72^\circ$.

Find the area of $\triangle PQR$.

Answer: _________________________ cm²

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4. Convert $210^\circ$ to radians, leaving your answer in terms of $\pi$.

Answer: _________________________ radians

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5. In the diagram, $TA$ and $TB$ are tangents to the circle with centre $O$. $\angle ATB = 52^\circ$.

Find $\angle AOB$.

![Circle with centre O, tangents TA and TB from external point T]

Answer: _________________________

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6. A ladder of length 5 m leans against a vertical wall. The foot of the ladder is 2 m from the base of the wall.

Find the angle the ladder makes with the horizontal ground.

Answer: _________________________

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7. In $\triangle ABC$, $AB = 7$ cm, $BC = 9$ cm, and $\angle ABC = 110^\circ$.

Find the length of $AC$, correct to 3 significant figures.

Answer: _________________________ cm

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8. A chord $PQ$ of a circle is 16 cm long. The perpendicular distance from the centre $O$ to the chord is 6 cm.

Calculate the radius of the circle.

Answer: _________________________ cm

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9. The area of a sector is $45\pi$ cm² and its radius is 10 cm.

Find the angle of the sector in radians.

Answer: _________________________ radians

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10. From the top of a vertical cliff 80 m high, the angle of depression of a boat at sea is $28^\circ$.

How far is the boat from the base of the cliff?

Answer: _________________________ m

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Section B: Structured Questions (24 marks)

Answer all questions in this section. Marks are indicated in brackets.

11. In $\triangle XYZ$, $XY = 12$ cm, $YZ = 15$ cm, and $XZ = 9$ cm.

(a) Show that $\triangle XYZ$ is a right-angled triangle. [2]

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(b) Find $\angle YXZ$. [2]

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12. The diagram shows two triangles, $\triangle ABC$ and $\triangle ADE$, where $BC$ is parallel to $DE$.

$AB = 6$ cm, $BD = 4$ cm, $AC = 8$ cm, and $BC = 5$ cm.

![Triangle ADE with line BC parallel to DE, B on AD, C on AE]

(a) Explain why $\triangle ABC$ and $\triangle ADE$ are similar. [2]

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(b) Find the length of $DE$. [2]

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(c) Find the ratio of the area of $\triangle ABC$ to the area of $\triangle ADE$. [2]

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13. A, B, C, and D are points on a circle. $\angle ABC = 75^\circ$ and $\angle BCD = 110^\circ$.

(a) Find $\angle ADC$. [2]

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(b) Find $\angle BAD$. [2]

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(c) Explain why $ABCD$ is a cyclic quadrilateral. [1]

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14. In $\triangle PQR$, $PQ = 8$ cm, $QR = 10$ cm, and $\angle PQR = 60^\circ$.

(a) Find the length of $PR$ using the cosine rule. [2]

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(b) Hence find $\angle QPR$ using the sine rule. [3]

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15. A sector of a circle has radius $r$ cm and angle $\theta$ radians. The perimeter of the sector is 30 cm and its area is 50 cm².

(a) Write down an equation for the perimeter of the sector in terms of $r$ and $\theta$. [1]

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(b) Write down an equation for the area of the sector in terms of $r$ and $\theta$. [1]

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(c) Solve the equations to find the value of $r$ and $\theta$. [3]

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Section C: Problem-Solving Questions (16 marks)

Answer all questions in this section. Marks are indicated in brackets.

16. A ship sails from port $P$ on a bearing of $055^\circ$ for 12 km to point $Q$. It then changes course and sails on a bearing of $140^\circ$ for 9 km to point $R$.

(a) Draw a clearly labelled diagram showing the path of the ship. [2]

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(b) Calculate the distance $PR$. [3]

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(c) Find the bearing of $R$ from $P$. [3]

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17. A regular pentagon $ABCDE$ is inscribed in a circle with centre $O$.

(a) Calculate the size of $\angle AOB$. [1]

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(b) Find $\angle ACB$, where $C$ is a vertex of the pentagon. [2]

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(c) Prove that $\angle ABC = 108^\circ$. [2]

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18. The diagram shows a right pyramid with a square base $ABCD$ of side 6 cm. The vertex $V$ is vertically above the centre $O$ of the base. $VA = 10$ cm.

![Square-based pyramid with vertex V, base ABCD, centre O]

(a) Find the length of the diagonal $AC$. [1]

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(b) Find the height $VO$ of the pyramid. [2]

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(c) Calculate the angle between $VA$ and the base $ABCD$. [2]

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19. Two circles with centres $O$ and $P$ intersect at points $A$ and $B$. $OA = 5$ cm, $PA = 4$ cm, and $OP = 6$ cm.

(a) Find $\angle OAP$ using the cosine rule. [2]

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(b) Hence find the area of quadrilateral $OAPB$. [3]

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20. A triangular field $ABC$ has $AB = 120$ m, $BC = 150$ m, and $\angle ABC = 75^\circ$.

(a) Calculate the area of the field. [2]

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(b) A path runs from $B$ perpendicular to $AC$, meeting $AC$ at $D$. Find the length of $BD$. [3]

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END OF PAPER

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Check your work carefully. Ensure all answers are in the correct units and rounded as specified.

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TuitionGoWhere Practice Paper - Elementary Mathematics Secondary 4

Answer Key and Marking Scheme (Version 4)

Total Marks: 60

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Section A: Short Answer Questions (20 marks)

1. $\angle ACB = 65^\circ$ ✓ (1 mark) Theorem: Angle at centre is twice angle at circumference ✓ (1 mark) Accept: Angle subtended by an arc at the centre is twice the angle subtended at the circumference.

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2. Arc length $= r\theta = 8 \times \frac{5\pi}{6}$ ✓ (1 mark) $= \frac{40\pi}{6} = \frac{20\pi}{3} \approx 20.9$ cm ✓ (1 mark) Accept $\frac{20\pi}{3}$ cm or 20.9 cm (3 s.f.).

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3. Area $= \frac{1}{2} \times PQ \times PR \times \sin \angle QPR$ ✓ (1 mark) $= \frac{1}{2} \times 10 \times 14 \times \sin 72^\circ$ $= 70 \times 0.9511 = 66.6$ cm² (3 s.f.) ✓ (1 mark)

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4. $210^\circ \times \frac{\pi}{180^\circ}$ ✓ (1 mark) $= \frac{210\pi}{180} = \frac{7\pi}{6}$ radians ✓ (1 mark)

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5. $\angle AOB = 180^\circ - 52^\circ = 128^\circ$ ✓ (2 marks) Method: Tangents from external point are equal; $OA \perp TA$ and $OB \perp TB$; quadrilateral $AOBT$ has angles summing to $360^\circ$; $\angle OAT = \angle OBT = 90^\circ$; so $\angle AOB = 360^\circ - 90^\circ - 90^\circ - 52^\circ = 128^\circ$. Award 1 mark for correct method, 1 mark for correct answer.

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6. $\cos \theta = \frac{2}{5}$ ✓ (1 mark) $\theta = \cos^{-1}(0.4) = 66.4^\circ$ (3 s.f.) ✓ (1 mark) Accept 66.4° or 1.16 radians.

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7. $AC^2 = 7^2 + 9^2 - 2(7)(9)\cos 110^\circ$ ✓ (1 mark) $= 49 + 81 - 126(-0.3420)$ $= 130 + 43.09 = 173.09$ $AC = \sqrt{173.09} = 13.2$ cm (3 s.f.) ✓ (1 mark)

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8. Half-chord $= 8$ cm. By Pythagoras: $r^2 = 8^2 + 6^2$ ✓ (1 mark) $r^2 = 64 + 36 = 100$ $r = 10$ cm ✓ (1 mark)

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9. Area $= \frac{1}{2}r^2\theta$; $45\pi = \frac{1}{2}(10^2)\theta$ ✓ (1 mark) $45\pi = 50\theta$ $\theta = \frac{45\pi}{50} = \frac{9\pi}{10} = 0.9\pi$ radians ✓ (1 mark) Accept $\frac{9\pi}{10}$ or 2.83 radians (3 s.f.).

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10. $\tan 28^\circ = \frac{80}{d}$ ✓ (1 mark) $d = \frac{80}{\tan 28^\circ} = \frac{80}{0.5317} = 150$ m (3 s.f.) ✓ (1 mark)

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Section B: Structured Questions (24 marks)

11. (a) Check if $XY^2 + XZ^2 = YZ^2$: $12^2 + 9^2 = 144 + 81 = 225$; $15^2 = 225$ ✓ (1 mark) Since $XY^2 + XZ^2 = YZ^2$, by converse of Pythagoras' theorem, $\triangle XYZ$ is right-angled at $X$ ✓ (1 mark)

(b) $\sin(\angle YXZ) = \frac{YZ}{XY}$? No — in right triangle with right angle at $X$, $YZ$ is hypotenuse. $\cos(\angle YXZ) = \frac{XZ}{YZ} = \frac{9}{15} = 0.6$ ✓ (1 mark) $\angle YXZ = \cos^{-1}(0.6) = 53.1^\circ$ (3 s.f.) ✓ (1 mark) Alternative: $\tan(\angle YXZ) = \frac{XY}{XZ} = \frac{12}{9}$, $\angle YXZ = \tan^{-1}(1.333) = 53.1^\circ$.

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12. (a) $\angle BAC = \angle DAE$ (common angle) ✓ (1 mark) $\angle ABC = \angle ADE$ (corresponding angles, $BC \parallel DE$) ✓ (1 mark) Therefore $\triangle ABC \sim \triangle ADE$ (AA criterion).

(b) Scale factor $= \frac{AD}{AB} = \frac{6+4}{6} = \frac{10}{6} = \frac{5}{3}$ ✓ (1 mark) $DE = BC \times \frac{5}{3} = 5 \times \frac{5}{3} = \frac{25}{3} = 8.33$ cm (3 s.f.) ✓ (1 mark)

(c) Area ratio $= (\text{scale factor})^2 = \left(\frac{AB}{AD}\right)^2 = \left(\frac{6}{10}\right)^2 = \left(\frac{3}{5}\right)^2 = \frac{9}{25}$ ✓ (1 mark) Ratio of area of $\triangle ABC$ : area of $\triangle ADE = 9 : 25$ ✓ (1 mark)

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13. (a) In a cyclic quadrilateral, opposite angles sum to $180^\circ$. $\angle ADC + \angle ABC = 180^\circ$ ✓ (1 mark) $\angle ADC = 180^\circ - 75^\circ = 105^\circ$ ✓ (1 mark)

(b) $\angle BAD + \angle BCD = 180^\circ$ ✓ (1 mark) $\angle BAD = 180^\circ - 110^\circ = 70^\circ$ ✓ (1 mark)

(c) $ABCD$ is a cyclic quadrilateral because all four vertices lie on the circle (given) ✓ (1 mark) Accept: Because opposite angles sum to $180^\circ$ ($75^\circ + 105^\circ = 180^\circ$ and $110^\circ + 70^\circ = 180^\circ$).

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14. (a) $PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos \angle PQR$ ✓ (1 mark) $= 8^2 + 10^2 - 2(8)(10)\cos 60^\circ$ $= 64 + 100 - 160(0.5) = 164 - 80 = 84$ $PR = \sqrt{84} = 2\sqrt{21} \approx 9.17$ cm (3 s.f.) ✓ (1 mark)

(b) Using sine rule: $\frac{\sin(\angle QPR)}{QR} = \frac{\sin(\angle PQR)}{PR}$ ✓ (1 mark) $\frac{\sin(\angle QPR)}{10} = \frac{\sin 60^\circ}{\sqrt{84}}$ ✓ (1 mark) $\sin(\angle QPR) = \frac{10 \times \sin 60^\circ}{\sqrt{84}} = \frac{10 \times 0.8660}{9.165} = 0.9449$ $\angle QPR = \sin^{-1}(0.9449) = 70.9^\circ$ (3 s.f.) ✓ (1 mark)

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15. (a) Perimeter $= r\theta + 2r = 30$ ✓ (1 mark) $r(\theta + 2) = 30$

(b) Area $= \frac{1}{2}r^2\theta = 50$ ✓ (1 mark)

(c) From (a): $\theta = \frac{30}{r} - 2$ ✓ (1 mark) Substitute into (b): $\frac{1}{2}r^2\left(\frac{30}{r} - 2\right) = 50$ $\frac{1}{2}(30r - 2r^2) = 50$ $15r - r^2 = 50$ $r^2 - 15r + 50 = 0$ ✓ (1 mark) $(r - 5)(r - 10) = 0$ $r = 5$ or $r = 10$ If $r = 5$, $\theta = \frac{30}{5} - 2 = 4$ radians. If $r = 10$, $\theta = \frac{30}{10} - 2 = 1$ radian. ✓ (1 mark) Accept either valid pair: $(r, \theta) = (5, 4)$ or $(10, 1)$.

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Section C: Problem-Solving Questions (16 marks)

16. (a) Diagram showing:

• North line at $P$
• $PQ$ at bearing $055^\circ$, length 12 km ✓ (1 mark)
• North line at $Q$
• $QR$ at bearing $140^\circ$, length 9 km
• Triangle $PQR$ clearly labelled ✓ (1 mark)

(b) $\angle PQR = 180^\circ - 140^\circ + 55^\circ = 95^\circ$ ✓ (1 mark) Using cosine rule: $PR^2 = 12^2 + 9^2 - 2(12)(9)\cos 95^\circ$ ✓ (1 mark) $= 144 + 81 - 216(-0.08716) = 225 + 18.83 = 243.83$ $PR = \sqrt{243.83} = 15.6$ km (3 s.f.) ✓ (1 mark)

(c) Using sine rule: $\frac{\sin(\angle QPR)}{9} = \frac{\sin 95^\circ}{15.62}$ ✓ (1 mark) $\sin(\angle QPR) = \frac{9 \times \sin 95^\circ}{15.62} = \frac{9 \times 0.9962}{15.62} = 0.5740$ $\angle QPR = \sin^{-1}(0.5740) = 35.0^\circ$ ✓ (1 mark) Bearing of $R$ from $P = 055^\circ + 35.0^\circ = 090.0^\circ$ (or $090^\circ$) ✓ (1 mark)

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17. (a) A regular pentagon divides the circle into 5 equal arcs. $\angle AOB = \frac{360^\circ}{5} = 72^\circ$ ✓ (1 mark)

(b) $\angle ACB$ is an angle at the circumference subtended by arc $AB$. $\angle ACB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 72^\circ = 36^\circ$ ✓ (2 marks) Award 1 mark for identifying angle at circumference, 1 mark for correct calculation.

(c) Interior angle of regular pentagon $= \frac{(5-2) \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ$ ✓ (1 mark) Therefore $\angle ABC = 108^\circ$ ✓ (1 mark) Alternative: $\angle ABC$ is sum of angles subtended by arcs; or using isosceles triangles and circle theorems.

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18. (a) $AC = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \approx 8.49$ cm ✓ (1 mark)

(b) $O$ is centre of base, so $AO = \frac{AC}{2} = 3\sqrt{2}$ cm ✓ (1 mark) In right triangle $VOA$: $VO^2 + AO^2 = VA^2$ $VO^2 + (3\sqrt{2})^2 = 10^2$ $VO^2 + 18 = 100$ $VO = \sqrt{82} \approx 9.06$ cm (3 s.f.) ✓ (1 mark)

(c) Angle between $VA$ and base is $\angle VAO$ ✓ (1 mark) $\cos(\angle VAO) = \frac{AO}{VA} = \frac{3\sqrt{2}}{10} = \frac{4.243}{10} = 0.4243$ $\angle VAO = \cos^{-1}(0.4243) = 64.9^\circ$ (3 s.f.) ✓ (1 mark)

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19. (a) In $\triangle OAP$: $OA = 5$, $PA = 4$, $OP = 6$. $\cos(\angle OAP) = \frac{OA^2 + PA^2 - OP^2}{2(OA)(PA)}$ ✓ (1 mark) $= \frac{5^2 + 4^2 - 6^2}{2(5)(4)} = \frac{25 + 16 - 36}{40} = \frac{5}{40} = 0.125$ $\angle OAP = \cos^{-1}(0.125) = 82.8^\circ$ (3 s.f.) ✓ (1 mark)

(b) Area of $\triangle OAP = \frac{1}{2} \times OA \times PA \times \sin(\angle OAP)$ $= \frac{1}{2} \times 5 \times 4 \times \sin 82.8^\circ$ ✓ (1 mark) $= 10 \times 0.9922 = 9.922$ cm² By symmetry, $\triangle OBP \cong \triangle OAP$ ✓ (1 mark) Area of quadrilateral $OAPB = 2 \times 9.922 = 19.8$ cm² (3 s.f.) ✓ (1 mark) Alternative: Find $\angle AOP$ and $\angle APO$, then use area formulas.

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20. (a) Area $= \frac{1}{2} \times AB \times BC \times \sin \angle ABC$ ✓ (1 mark) $= \frac{1}{2} \times 120 \times 150 \times \sin 75^\circ$ $= 9000 \times 0.9659 = 8690$ m² (3 s.f.) ✓ (1 mark)

(b) First find $AC$ using cosine rule: $AC^2 = 120^2 + 150^2 - 2(120)(150)\cos 75^\circ$ ✓ (1 mark) $= 14400 + 22500 - 36000(0.2588)$ $= 36900 - 9317 = 27583$ $AC = \sqrt{27583} = 166.1$ m

Area also $= \frac{1}{2} \times AC \times BD$ ✓ (1 mark) $8693 = \frac{1}{2} \times 166.1 \times BD$ $BD = \frac{2 \times 8693}{166.1} = \frac{17386}{166.1} = 105$ m (3 s.f.) ✓ (1 mark)

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End of Answer Key

Marking notes: Award method marks where working is shown and logically correct, even if final answer contains arithmetic error. Deduct 1 mark for incorrect or missing units where units are specified. Accept equivalent forms of answers (e.g., exact surd form or decimal).