Secondary 4 Elementary Mathematics Practice Paper 5

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TuitionGoWhere Practice Paper - Elementary Mathematics Secondary 4

TuitionGoWhere Practice Paper (AI)

Subject: Elementary Mathematics Level: Secondary 4 Paper: Practice Paper (Version 5 of 5) Duration: 2 hours 15 minutes Total Marks: 90

Name: _________________________ Class: _________________________ Date: _________________________

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Instructions to Candidates

1. This paper consists of two sections. Answer all questions.
2. Write your answers in the spaces provided.
3. Show all working clearly. Marks are awarded for method, not just final answers.
4. Unless stated otherwise, give numerical answers correct to 3 significant figures.
5. Diagrams are not necessarily drawn to scale.
6. You are expected to use a scientific calculator where appropriate.
7. The total mark for this paper is 90.

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Section A: Short Answer Questions (45 marks)

Answer all questions in this section. Each question carries the marks indicated.

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1. In the diagram, $O$ is the centre of the circle. $A$, $B$, $C$, and $D$ are points on the circumference. $\angle AOB = 130^\circ$ and $\angle BDC = 25^\circ$.

Find (a) $\angle ACB$, [1] (b) $\angle ABD$. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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2. In $\triangle PQR$, $PQ = 8.4$ cm, $QR = 11.2$ cm, and $\angle PQR = 72^\circ$.

Find (a) the length of $PR$, [2] (b) the area of $\triangle PQR$. [2]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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3. A sector of a circle has radius 15 cm and angle $\frac{5\pi}{6}$ radians.

Find (a) the arc length of the sector, [1] (b) the area of the sector. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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4. In $\triangle ABC$, $AB = 7$ cm, $BC = 9$ cm, and $AC = 12$ cm.

Find $\angle ABC$. [3]

Answer space:

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5. From the top of a vertical cliff 85 m high, the angle of depression of a boat is $28^\circ$.

Find the horizontal distance from the base of the cliff to the boat. [3]

Answer space:

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6. In the diagram, $A$, $B$, $C$, and $D$ are points on a circle. $AD$ and $BC$ intersect at $X$. $\angle BAD = 55^\circ$, $\angle ABC = 70^\circ$, and $\angle BCD = 95^\circ$.

Find (a) $\angle ADC$, [1] (b) $\angle AXC$. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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7. A ladder of length 6.5 m leans against a vertical wall. The foot of the ladder is 2.5 m from the wall.

Find (a) the height the ladder reaches up the wall, [2] (b) the angle the ladder makes with the ground. [2]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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8. In $\triangle XYZ$, $XY = 10$ cm, $XZ = 14$ cm, and $\angle YXZ = 35^\circ$.

Find the area of $\triangle XYZ$. [2]

Answer space:

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9. A ship sails from port $P$ on a bearing of $055^\circ$ for 12 km to point $Q$. It then sails on a bearing of $145^\circ$ for 9 km to point $R$.

Find (a) the distance $PR$, [3] (b) the bearing of $R$ from $P$. [3]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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10. In the diagram, $O$ is the centre of the circle. $PT$ is a tangent to the circle at $T$. $\angle OPT = 34^\circ$.

Find (a) $\angle OTP$, [1] (b) $\angle POT$, [1] (c) $\angle PQT$, where $Q$ is a point on the circle on the major arc $PT$. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

(c) _________________________________________________________

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11. A triangle has sides of length 8 cm, 11 cm, and 14 cm.

Find the largest angle of the triangle. [3]

Answer space:

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12. In $\triangle ABC$, $AB = 6$ cm, $AC = 8$ cm, and $\angle BAC = 120^\circ$.

Find the length of $BC$. [3]

Answer space:

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Section B: Structured Questions (45 marks)

Answer all questions in this section. Each question carries the marks indicated.

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13. In the diagram, $A$, $B$, $C$, and $D$ are points on a circle with centre $O$. $AC$ is a diameter. $\angle BAC = 28^\circ$ and $\angle CAD = 42^\circ$.

Find (a) $\angle ABC$, [1] (b) $\angle BCA$, [2] (c) $\angle BDC$, [2] (d) $\angle BOC$. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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(c) _________________________________________________________

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(d) _________________________________________________________

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14. A vertical flagpole $AB$ of height 12 m stands on horizontal ground. From a point $C$ on the ground, the angle of elevation of the top of the flagpole $A$ is $31^\circ$. From another point $D$ on the ground, which is further from the flagpole than $C$ and in line with $C$ and $B$, the angle of elevation of $A$ is $19^\circ$.

Find (a) the distance $BC$, [2] (b) the distance $BD$, [2] (c) the distance $CD$. [2]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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(c) _________________________________________________________

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15. In $\triangle PQR$, $PQ = 15$ cm, $QR = 18$ cm, and $PR = 12$ cm.

Find (a) $\angle PQR$, [3] (b) the area of $\triangle PQR$, [3] (c) the shortest distance from $P$ to $QR$. [2]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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(c) _________________________________________________________

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16. The diagram shows a circle with centre $O$ and radius 10 cm. $A$ and $B$ are points on the circumference such that $\angle AOB = 1.2$ radians.

Find (a) the length of the minor arc $AB$, [1] (b) the area of the minor sector $AOB$, [2] (c) the area of the minor segment cut off by chord $AB$. [3]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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(c) _________________________________________________________

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17. Two ships, $X$ and $Y$, leave a port $O$ at the same time. Ship $X$ sails at 15 km/h on a bearing of $120^\circ$. Ship $Y$ sails at 20 km/h on a bearing of $210^\circ$.

Find (a) the distance between the two ships after 2 hours, [4] (b) the bearing of ship $Y$ from ship $X$ at this time. [3]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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18. In the diagram, $ABCD$ is a cyclic quadrilateral. $AB$ is parallel to $DC$. $\angle DAB = 75^\circ$ and $\angle ABD = 40^\circ$.

Find (a) $\angle BCD$, [1] (b) $\angle BDC$, [2] (c) $\angle ADC$, [2] (d) $\angle DBC$. [2]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

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(c) _________________________________________________________

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(d) _________________________________________________________

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19. A triangle has vertices $A(2, 1)$, $B(8, 5)$, and $C(4, 9)$.

Find (a) the length of $AB$, [1] (b) the length of $BC$, [1] (c) the length of $AC$, [1] (d) the area of $\triangle ABC$. [3]

Answer space:

(a) _________________________________________________________

(b) _________________________________________________________

(c) _________________________________________________________

(d) _________________________________________________________

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20. In $\triangle DEF$, $DE = 9$ cm, $DF = 13$ cm, and $\angle EDF = 50^\circ$. $G$ is a point on $DF$ such that $DG = 5$ cm.

Find (a) the length of $EF$, [2] (b) the area of $\triangle DEF$, [2] (c) the area of $\triangle DEG$. [2]

Answer space:

(a) _________________________________________________________

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(b) _________________________________________________________

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(c) _________________________________________________________

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END OF PAPER

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TuitionGoWhere Practice Paper - Elementary Mathematics Secondary 4

Answer Key and Marking Scheme (Version 5)

Total Marks: 90

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Section A: Short Answer Questions (45 marks)

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1. (a) $\angle ACB = 65^\circ$ [1] (Angle at centre = 2 × angle at circumference: $\angle ACB = \frac{1}{2} \times 130^\circ = 65^\circ$)

(b) $\angle ABD = 40^\circ$ [2]

• $\angle ADB = \angle ACB = 65^\circ$ (angles in same segment) [1]
• In $\triangle ABD$, $\angle ABD = 180^\circ - 65^\circ - 25^\circ - 65^\circ = 25^\circ$? Wait, recalculate.
• $\angle ADB = 65^\circ$ (angles in same segment as $\angle ACB$)
• $\angle BAD = \frac{1}{2} \times (360^\circ - 130^\circ) = 115^\circ$? No.
• Alternative: $\angle ACD = \frac{1}{2} \times 130^\circ = 65^\circ$. $\angle ABD = \angle ACD = 65^\circ$ (angles in same segment). Wait, check.
• $\angle ABD$ subtends arc $AD$. $\angle ACD$ also subtends arc $AD$. $\angle ACD = \angle ACB + \angle BCD$? Not given.
• Better: $\angle AOB = 130^\circ$, so reflex $\angle AOB = 230^\circ$. $\angle ADB = \frac{1}{2} \times 230^\circ = 115^\circ$ (angle at centre).
• In $\triangle ABD$, $\angle ABD = 180^\circ - 115^\circ - 25^\circ = 40^\circ$. [1] for method, [1] for answer.

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2. (a) $PR^2 = 8.4^2 + 11.2^2 - 2(8.4)(11.2)\cos 72^\circ$ [1] $PR^2 = 70.56 + 125.44 - 188.16(0.3090) = 196 - 58.14 = 137.86$ $PR = \sqrt{137.86} = 11.7$ cm (3 s.f.) [1]

(b) Area $= \frac{1}{2} \times 8.4 \times 11.2 \times \sin 72^\circ$ [1] $= 47.04 \times 0.9511 = 44.7$ cm$^2$ (3 s.f.) [1]

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3. (a) Arc length $= r\theta = 15 \times \frac{5\pi}{6} = \frac{75\pi}{6} = \frac{25\pi}{2} = 39.3$ cm (3 s.f.) [1]

(b) Sector area $= \frac{1}{2}r^2\theta = \frac{1}{2} \times 15^2 \times \frac{5\pi}{6}$ [1] $= \frac{1}{2} \times 225 \times \frac{5\pi}{6} = \frac{1125\pi}{12} = \frac{375\pi}{4} = 295$ cm$^2$ (3 s.f.) [1]

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4. Using cosine rule: $\cos B = \frac{7^2 + 9^2 - 12^2}{2 \times 7 \times 9}$ [1] $= \frac{49 + 81 - 144}{126} = \frac{-14}{126} = -0.1111$ [1] $\angle ABC = \cos^{-1}(-0.1111) = 96.4^\circ$ (1 d.p.) [1]

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5. Let $d$ be the horizontal distance. $\tan 28^\circ = \frac{85}{d}$ [1] $d = \frac{85}{\tan 28^\circ}$ [1] $d = \frac{85}{0.5317} = 160$ m (3 s.f.) [1]

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6. (a) $\angle ADC = 180^\circ - 55^\circ = 125^\circ$ (opposite angles of cyclic quadrilateral sum to $180^\circ$) [1]

(b) In $\triangle ABX$: $\angle AXB = 180^\circ - 55^\circ - 70^\circ = 55^\circ$ [1] $\angle AXC = 180^\circ - 55^\circ = 125^\circ$ (angles on a straight line) [1]

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7. (a) Let height be $h$. $h^2 + 2.5^2 = 6.5^2$ [1] $h^2 = 42.25 - 6.25 = 36$ $h = 6$ m [1]

(b) Let $\theta$ be the angle. $\cos \theta = \frac{2.5}{6.5}$ or $\sin \theta = \frac{6}{6.5}$ or $\tan \theta = \frac{6}{2.5}$ [1] $\theta = \cos^{-1}(\frac{2.5}{6.5}) = 67.4^\circ$ (1 d.p.) [1]

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8. Area $= \frac{1}{2} \times 10 \times 14 \times \sin 35^\circ$ [1] $= 70 \times 0.5736 = 40.2$ cm$^2$ (3 s.f.) [1]

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9. (a) $\angle PQR = 145^\circ - 55^\circ = 90^\circ$ (or using bearings: $180^\circ - 55^\circ - (180^\circ - 145^\circ) = 90^\circ$) [1] $PR^2 = 12^2 + 9^2 = 144 + 81 = 225$ [1] $PR = 15$ km [1]

(b) $\tan(\angle QPR) = \frac{9}{12} = 0.75$ [1] $\angle QPR = 36.87^\circ$ [1] Bearing of $R$ from $P = 055^\circ + 36.87^\circ = 092^\circ$ (nearest degree) [1]

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10. (a) $\angle OTP = 90^\circ$ (tangent perpendicular to radius) [1]

(b) In $\triangle OPT$: $\angle POT = 180^\circ - 90^\circ - 34^\circ = 56^\circ$ [1]

(c) $\angle PQT = \frac{1}{2} \times \angle POT = \frac{1}{2} \times 56^\circ = 28^\circ$ (angle at centre = 2 × angle at circumference) [1] Or: $\angle PQT = \angle OPT = 34^\circ$? No, alternate segment theorem: $\angle PQT = \angle OPT$? Check: tangent $PT$, chord $QT$. $\angle PQT$ is angle in alternate segment to $\angle OPT$? Actually $\angle PQT = \angle PTQ$? Let's use angle at centre. $\angle PQT = 28^\circ$. [1] for correct theorem, [1] for answer.

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11. Largest angle is opposite longest side (14 cm). $\cos \theta = \frac{8^2 + 11^2 - 14^2}{2 \times 8 \times 11}$ [1] $= \frac{64 + 121 - 196}{176} = \frac{-11}{176} = -0.0625$ [1] $\theta = \cos^{-1}(-0.0625) = 93.6^\circ$ (1 d.p.) [1]

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12. $BC^2 = 6^2 + 8^2 - 2(6)(8)\cos 120^\circ$ [1] $= 36 + 64 - 96(-0.5) = 100 + 48 = 148$ [1] $BC = \sqrt{148} = 12.2$ cm (3 s.f.) [1]

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Section B: Structured Questions (45 marks)

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13. (a) $\angle ABC = 90^\circ$ (angle in a semicircle) [1]

(b) In $\triangle ABC$: $\angle BCA = 180^\circ - 90^\circ - 28^\circ = 62^\circ$ [1] for method, [1] for answer.

(c) $\angle BDC = \angle BAC = 28^\circ$ (angles in same segment) [1] for theorem, [1] for answer.

(d) $\angle BOC = 2 \times \angle BAC = 2 \times 28^\circ = 56^\circ$ (angle at centre = 2 × angle at circumference) [1] for theorem, [1] for answer.

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14. (a) $\tan 31^\circ = \frac{12}{BC}$ [1] $BC = \frac{12}{\tan 31^\circ} = \frac{12}{0.6009} = 20.0$ m (3 s.f.) [1]

(b) $\tan 19^\circ = \frac{12}{BD}$ [1] $BD = \frac{12}{\tan 19^\circ} = \frac{12}{0.3443} = 34.9$ m (3 s.f.) [1]

(c) $CD = BD - BC = 34.9 - 20.0 = 14.9$ m [1] for method, [1] for answer.

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15. (a) $\cos(\angle PQR) = \frac{15^2 + 18^2 - 12^2}{2 \times 15 \times 18}$ [1] $= \frac{225 + 324 - 144}{540} = \frac{405}{540} = 0.75$ [1] $\angle PQR = \cos^{-1}(0.75) = 41.4^\circ$ (1 d.p.) [1]

(b) Area $= \frac{1}{2} \times 15 \times 18 \times \sin 41.4^\circ$ [1] $= 135 \times 0.6613 = 89.3$ cm$^2$ (3 s.f.) [1] for substitution, [1] for answer.

(c) Shortest distance $h$ from $P$ to $QR$: Area $= \frac{1}{2} \times QR \times h$ [1] $89.3 = \frac{1}{2} \times 18 \times h$ $h = \frac{89.3}{9} = 9.92$ cm (3 s.f.) [1]

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16. (a) Arc length $= r\theta = 10 \times 1.2 = 12.0$ cm (3 s.f.) [1]

(b) Sector area $= \frac{1}{2}r^2\theta = \frac{1}{2} \times 10^2 \times 1.2$ [1] $= 50 \times 1.2 = 60.0$ cm$^2$ [1]

(c) Triangle area $= \frac{1}{2}r^2\sin\theta = \frac{1}{2} \times 10^2 \times \sin 1.2$ [1] $= 50 \times 0.9320 = 46.6$ cm$^2$ [1] Segment area $= 60.0 - 46.6 = 13.4$ cm$^2$ (3 s.f.) [1]

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17. (a) After 2 hours: $OX = 30$ km, $OY = 40$ km. [1] Angle between paths: Bearing $120^\circ$ and $210^\circ$. Difference $= 210^\circ - 120^\circ = 90^\circ$. [1] $XY^2 = 30^2 + 40^2 = 900 + 1600 = 2500$ [1] $XY = 50$ km [1]

(b) $\tan(\angle OXY) = \frac{40}{30} = \frac{4}{3}$ [1] $\angle OXY = 53.13^\circ$ [1] Bearing of $Y$ from $X$: $X$ is on bearing $120^\circ$ from $O$. The line $XY$ makes angle $53.13^\circ$ with $OX$. Bearing $= 120^\circ + 180^\circ + 53.13^\circ = 353^\circ$ (nearest degree). Or: $120^\circ + 180^\circ - 53.13^\circ = 247^\circ$? Need careful vector approach. Vector $\vec{OX} = (30\cos 120^\circ, 30\sin 120^\circ) = (-15, 25.98)$ Vector $\vec{OY} = (40\cos 210^\circ, 40\sin 210^\circ) = (-34.64, -20)$ $\vec{XY} = \vec{OY} - \vec{OX} = (-19.64, -45.98)$ Bearing $= 180^\circ + \tan^{-1}(\frac{45.98}{19.64}) = 180^\circ + 66.9^\circ = 247^\circ$ (nearest degree). [1]

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18. (a) $\angle BCD = 180^\circ - 75^\circ = 105^\circ$ (opposite angles of cyclic quadrilateral) [1]

(b) $\angle ADB = \angle ABD = 40^\circ$? No. $\angle ADB$ and $\angle ACB$ subtend same arc. Not helpful. Since $AB \parallel DC$, $\angle BDC = \angle ABD = 40^\circ$ (alternate angles) [1] for reasoning, [1] for answer.

(c) $\angle ADC = \angle ADB + \angle BDC$. Need $\angle ADB$. In $\triangle ABD$: $\angle ADB = 180^\circ - 75^\circ - 40^\circ = 65^\circ$ [1] $\angle ADC = 65^\circ + 40^\circ = 105^\circ$ [1]

(d) $\angle DBC = \angle DAC$ (angles in same segment) $\angle DAC = \angle DAB - \angle CAB$? Not given. Alternative: $\angle DBC = 180^\circ - \angle BCD - \angle BDC = 180^\circ - 105^\circ - 40^\circ = 35^\circ$ [1] for method, [1] for answer.

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19. (a) $AB = \sqrt{(8-2)^2 + (5-1)^2} = \sqrt{36 + 16} = \sqrt{52} = 7.21$ units (3 s.f.) [1]

(b) $BC = \sqrt{(4-8)^2 + (9-5)^2} = \sqrt{16 + 16} = \sqrt{32} = 5.66$ units (3 s.f.) [1]

(c) $AC = \sqrt{(4-2)^2 + (9-1)^2} = \sqrt{4 + 64} = \sqrt{68} = 8.25$ units (3 s.f.) [1]

(d) Area $= \frac{1}{2}|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|$ [1] $= \frac{1}{2}|2(5-9) + 8(9-1) + 4(1-5)|$ $= \frac{1}{2}|2(-4) + 8(8) + 4(-4)|$ [1] $= \frac{1}{2}|-8 + 64 - 16| = \frac{1}{2}|40| = 20$ square units [1]

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20. (a) $EF^2 = 9^2 + 13^2 - 2(9)(13)\cos 50^\circ$ [1] $= 81 + 169 - 234(0.6428) = 250 - 150.4 = 99.6$ $EF = \sqrt{99.6} = 9.98$ cm (3 s.f.) [1]

(b) Area of $\triangle DEF = \frac{1}{2} \times 9 \times 13 \times \sin 50^\circ$ [1] $= 58.5 \times 0.7660 = 44.8$ cm$^2$ (3 s.f.) [1]

(c) Area of $\triangle DEG = \frac{DG}{DF} \times$ Area of $\triangle DEF$ (same height from $E$ to $DF$) [1] $= \frac{5}{13} \times 44.8 = 17.2$ cm$^2$ (3 s.f.) [1]

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END OF ANSWER KEY