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Date	May 2021	Marks available	4	Reference code	21M.2.SL.TZ1.8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that	Question number	8	Adapted from	N/A

Question

Two straight fences meet at point $A$ and a field lies between them.

A horse is tied to a post, $P$ , by a rope of length $r$ metres. Point $D$ is on one fence and point $E$ is on the other, such that $PD = PE = PA = r$ and $D \hat{P} E = θ$ radians. This is shown in the following diagram.

The length of the arc $DE$ shown in the diagram is $28 m$ .

A new fence is to be constructed between points $B$ and $C$ which will enclose the field, as shown in the following diagram.

Point $C$ is due west of $B$ and $AC = 800 m$ . The bearing of $B$ from $A$ is $195 °$ .

Write down an expression for $r$ in terms of $θ$ .

[1]

Show that the area of the field that the horse can reach is $\frac{392}{θ^{2}} (θ + \sin θ)$ .

[4]

The area of field that the horse can reach is $460 m^{2}$ . Find the value of $θ$ .

[2]

Hence, find the size of $D \hat{A} E$ .

[2]

Find the size of $A \hat{B} C$ .

[2]

e.i.

Find the length of new fence required.

[3]

e.ii.

Markscheme

$r = \frac{28}{θ}$ A1

[1 mark]

recognising sum of area of sector and area of triangle required (M1)

$\frac{1}{2} r^{2} θ + \frac{1}{2} r \times r \times \sin (π - θ) (= \frac{r^{2}}{2} (θ + \sin (π - θ))$ A1

$\sin (π - θ) = \sin θ$ (substitution seen anywhere) A1

$\frac{1}{2} {(\frac{28}{θ})}^{2} θ + \frac{1}{2} {(\frac{28}{θ})}^{2} \sin θ$ OR $\frac{1}{2} {(\frac{28}{θ})}^{2} (θ + \sin θ)$ A1

area $= \frac{392}{θ^{2}} (θ + \sin θ)$ AG

[4 marks]

$\frac{392}{θ^{2}} (θ + \sin θ) = 460$ (M1)

$θ = 1.43917 \dots$

$θ = 1.44$ A1

[2 marks]

$\frac{π - (π - θ)}{2}$ OR $\frac{θ}{2}$ (M1)

$D \hat{A} E = 0.719588 \dots$

$D \hat{A} E = 0.720$ A1

[2 marks]

$A \hat{B} C = 195 - 180 + 90$ (A1)

$= 105 °$ A1

[2 marks]

e.i.

choosing sine rule (M1)

$\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 105}$ OR $\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 1.83}$ A1

$BC = 546 (m)$ A1

[2 marks]

e.ii.

Examiners report

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area

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19M.1.SL.TZ1.T_8a:
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18M.1.SL.TZ1.T_8c:
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18N.2.AHL.TZ0.H_11c:
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19M.2.SL.TZ2.S_3b:
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17M.1.SL.TZ2.T_3b:
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19N.2.SL.TZ0.S_6:
The diagram below shows a triangular-based pyramid with base ${\text{ADC}}$ .
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19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.SL.TZ1.7b:
Given that $A \hat{V} B = 40 °$ , find $AB$ .
22M.2.SL.TZ2.7b:
Find the area of this piece of land.
22M.2.SL.TZ2.7c:
Find $CÂB$ .
22M.2.SL.TZ2.7d:
Find the distance from point $B$ to point $D$ .
20N.1.SL.TZ0.S_2a:
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20N.1.SL.TZ0.S_2b:
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20N.2.SL.TZ0.T_3b:
Show that angle $EDC = 48.0 °$ , correct to three significant figures.
21M.2.SL.TZ1.8e.ii:
Find the length of new fence required.
18M.1.SL.TZ2.T_8a:
Calculate the length of side AC in km.
SPM.1.SL.TZ0.1b:
Find the area of triangle ABC.
18N.2.SL.TZ0.S_7a:
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16N.2.SL.TZ0.T_5d:
the area of quadrilateral ABCD;
21M.2.SL.TZ1.8a:
Write down an expression for $r$ in terms of $θ$ .
21M.2.SL.TZ1.8e.i:
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21M.3.AHL.TZ1.2c:
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21M.3.AHL.TZ1.2d:
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17M.2.AHL.TZ2.H_4a:
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18M.1.SL.TZ1.T_8b:
Find the size of angle APB.
19N.1.SL.TZ0.S_9b:
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19M.1.AHL.TZ1.H_4a:
Show that ${\text{sin}}\,\theta = \frac{{\sqrt {15} }}{4}$ .
19N.1.AHL.TZ0.H_9c.i:
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19N.1.AHL.TZ0.H_9d:
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EXN.2.SL.TZ0.2b:
Show that $L = \sqrt{18 - 18 \cos θ}$ .
17N.2.SL.TZ0.S_1a:
Find BC.
SPM.2.SL.TZ0.7c:
Find ${\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{A}}$ .
21M.2.SL.TZ1.8d:
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SPM.2.SL.TZ0.7d:
Find the bearing that Jacob must take to point C.
EXN.1.AHL.TZ0.7:
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In $PQRS$ , $PQ = x$ , $SR = y$ , $R \hat{S} P = α$ and $Q \hat{R} S = β$ .

Find an expression for $PS$ in terms of $x, y, \sin β$ and $\sin (α + β)$ .
22M.2.SL.TZ2.1a:
Find the length of the chord $[AB]$ .
21N.2.SL.TZ0.5b:
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22M.2.SL.TZ2.7a:
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18M.2.SL.TZ1.T_1c:
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19M.2.SL.TZ1.T_2c:
Find the maximum number of complete panels that can be fitted to the whole roof.
17N.1.SL.TZ0.T_10b:
Write down the angle of elevation of B from E.
18M.2.SL.TZ2.T_5c:
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Find the angle of elevation of A from D.
17N.1.SL.TZ0.T_10c:
Find the vertical height of B above the ground.
19M.1.SL.TZ2.T_9b:
A second truck arrives whose ladder, when fully extended, is 30 metres long. The base of this ladder is also 4 metres above the ground. For safety reasons, the maximum angle of elevation that the ladder can make is 70º.

Find the maximum height on the wall that can be reached by the ladder on the second truck.
16N.2.SL.TZ0.T_5f:
the length of the fence, BP.
17M.2.SL.TZ1.T_2a.i:
Write down an equation for the area of ABCDE using the above information.
17N.1.SL.TZ0.T_10a:
Find the length of EB.
19M.1.SL.TZ1.T_10b:
Find the area, in km², of triangle ABC.
21N.2.SL.TZ0.5a:
Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .
19M.1.SL.TZ1.S_3a:
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19M.1.SL.TZ1.S_3b:
Find the value of ${\text{cos}}\,2\theta$ .
19M.2.AHL.TZ2.H_1:
In triangle ABC, AB = 5, BC = 14 and AC = 11.

Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.
18N.2.SL.TZ0.T_5b:
Calculate the area of triangle ABC.
19N.1.AHL.TZ0.H_9c.ii:
By considering triangle ${\text{CBD}}$ , find another expression for ${\text{B}}{{\text{D}}^2}$ in terms of $r$ and $q$ .
17N.1.SL.TZ0.S_4a:
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17M.1.SL.TZ2.T_3a:
Find the length of one of the straight pieces in the wireframe.
19M.1.SL.TZ1.S_3c:
Hence or otherwise, find ${\text{BC}}$ .
19N.1.AHL.TZ0.H_9a:
Given that ${\text{cos}}\,75^\circ = q$ , show that ${\text{cos}}\,105^\circ = - q$ .
19N.1.SL.TZ0.S_9c:
Calculate the area of triangle ${\text{AOC}}$ .
17M.1.SL.TZ1.S_3:
The following diagram shows triangle PQR.

Find PR.
19M.1.AHL.TZ1.H_4b:
Find the two possible values for the length of the third side.
17N.1.SL.TZ0.S_4b:
The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.
18M.2.SL.TZ2.S_2a:
Find DB.
18M.2.SL.TZ2.S_2b:
Find DC.
EXN.2.SL.TZ0.2a:
Show that arc $APB$ has length $6 π - 3 θ$ .
18N.2.SL.TZ0.S_7b:
Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.
21M.1.SL.TZ2.6:
The following diagram shows triangle $ABC$ , with $AB = 10$ , $BC = x$ and $AC = 2 x$ .

Given that $\cos \hat{C} = \frac{3}{4}$ , find the area of the triangle.

Give your answer in the form $\frac{p \sqrt{q}}{2}$ where $p, q \in ℤ^{+}$ .
21M.2.SL.TZ1.8c:
The area of field that the horse can reach is $460 m^{2}$ . Find the value of $θ$ .
21N.2.SL.TZ0.3:
Consider a triangle $ABC$ , where $AC = 12, CB = 7$ and $B \hat{A} C = 25 °$ .

Find the smallest possible perimeter of triangle $ABC$ .
17M.2.SL.TZ2.T_4e:
Calculate the total length of the track.
17M.1.SL.TZ1.T_13a:
Find the length of BC.
18N.2.SL.TZ0.T_5c:
Find the length of AC.
16N.2.SL.TZ0.T_5b:
the size of angle DAB;
19M.1.SL.TZ2.T_9a:
Find the angle of elevation from B to C.
17M.2.SL.TZ2.T_4b:
Show that angle DBC is 48.7°, correct to three significant figures.
17M.1.SL.TZ1.T_13b:
Find the area of the postage stamp.
20N.2.SL.TZ0.T_3d:
Pedro draws a circle, with centre at point $E$ , passing through point $C$ . Part of the circle is shown in the diagram.

Show that point $A$ lies outside this circle. Justify your reasoning.
17M.2.SL.TZ1.T_2c:
Show that angle ${\rm{B\hat AE}} = 67.4^\circ$ , correct to one decimal place.
17M.2.SL.TZ2.T_4d:
Find the length of the path CE.
16N.2.SL.TZ0.T_5a:
the length of BD;
19M.2.SL.TZ2.T_2d:
The length of path AD is 287 m.

Find the area of the region ABCD.
16N.1.SL.TZ0.T_11a:
Calculate CD, the height of the observation deck above the ground.
19M.1.SL.TZ1.T_10a:
Find the size, in degrees, of angle BÂC.
18M.2.SL.TZ1.T_1e:
Show that Farmer Brown is incorrect.
17M.2.SL.TZ1.T_2d:
Find the length of the perimeter of ABCDE.
18N.2.SL.TZ0.T_5a:
Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.
18M.2.SL.TZ2.T_5a.i:
Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.
18M.2.SL.TZ2.T_5a.ii:
Use Giovanni's diagram to calculate the length of AX.
18M.2.SL.TZ2.T_5a.iii:
Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.
18M.2.SL.TZ2.T_5b:
Find the percentage error on Giovanni’s diagram.
17N.2.SL.TZ0.T_3a:
Show that ${\text{BD}} = 93{\text{ m}}$ correct to the nearest metre.
17N.2.SL.TZ0.T_3b:
Calculate angle ${\rm{B\hat CD}}$ .
17N.2.SL.TZ0.T_3d.i:
Calculate Abdallah’s estimate for the area.
17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
19M.2.SL.TZ1.T_2a:
Find the length ${\text{AB}}$ , giving your answer to four significant figures.
19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
19M.2.SL.TZ1.T_2e:
Olivia investigates arranging the panels, such that the longer edge of each panel is parallel to ${\text{AB}}$ or ${\text{AC}}$ .

State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
19M.2.SL.TZ2.T_2a:
Find the distance from A to C.
19M.2.SL.TZ2.T_2b.i:
Find the size of angle ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ .
19M.2.SL.TZ2.T_2b.ii:
Find the size of angle ${\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}}$ .
19M.2.SL.TZ2.T_2c:
Find the size of angle ${\text{A}}\mathop {\text{C}}\limits^ \wedge {\text{D}}$ .
18M.2.SL.TZ1.T_1a:
Calculate the area of triangle EAD.
18M.2.SL.TZ1.T_1b:
Calculate the total volume of the barn.
18M.2.SL.TZ1.T_1f:
Calculate the total length of metal required for one support.
17M.2.SL.TZ1.T_2a.ii:
Show that the equation in part (a)(i) simplifies to $3{x^2} + 19x - 414 = 0$ .
17M.2.SL.TZ1.T_2b:
Find the length of CD.
17M.2.SL.TZ1.T_2e:
Calculate the length of CF.
18N.2.SL.TZ0.T_5d:
A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.
17M.2.SL.TZ2.T_4a:
Find the length of the path BD.
17M.2.SL.TZ2.T_4c:
Find the area of the park.
16N.2.SL.TZ0.T_5c:
the area of triangle ABD;
16N.2.SL.TZ0.T_5e:
the length of AP;
20N.2.SL.TZ0.T_3a:
Calculate the length of $BD$ .
20N.2.SL.TZ0.T_3c:
Calculate the area of triangle $BDC$ .

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