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Date	November 2019	Marks available	1	Reference code	19N.1.AHL.TZ0.H_9
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Show that	Question number	H_9	Adapted from	N/A

Question

In the following diagram, the points $A$ ${\text{A}}$ , $B$ ${\text{B}}$ , $C$ ${\text{C}}$ and $D$ ${\text{D}}$ are on the circumference of a circle with centre $O$ ${\text{O}}$ and radius $r$ $r$ . $[AC]$ $\left[ {{\text{AC}}} \right]$ is a diameter of the circle. $BC = r$ ${\text{BC}} = r$ , $AD = CD$ ${\text{AD}} = {\text{CD}}$ and $A \land B C = A \land D C = 90^{\circ}$ ${\text{A}}\mathop {\text{B}}\limits^ \wedge {\text{C}} = {\text{A}}\mathop {\text{D}}\limits^ \wedge {\text{C}} = 90^\circ$ .

Given that $cos 75^{\circ} = q$ ${\text{cos}}\,75^\circ = q$ , show that $cos 105^{\circ} = - q$ ${\text{cos}}\,105^\circ = - q$ .

[1]

a.

Show that $B \land A D = 75^{\circ}$ ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 75^\circ$ .

[3]

b.

By considering triangle $ABD$ ${\text{ABD}}$ , show that $B D^{2} = 5 r^{2} - 2 r^{2} q \sqrt{6}$ ${\text{B}}{{\text{D}}^2} = 5{r^2} - 2{r^2}q\sqrt 6$ .

[4]

c.i.

By considering triangle $CBD$ ${\text{CBD}}$ , find another expression for $B D^{2}$ ${\text{B}}{{\text{D}}^2}$ in terms of $r$ $r$ and $q$ $q$ .

[3]

c.ii.

Use your answers to part (c) to show that $cos 75^{\circ} = \frac{1}{\sqrt{6} + \sqrt{2}}$ ${\text{cos}}\,75^\circ = \frac{1}{{\sqrt 6 + \sqrt 2 }}$ .

[3]

d.

Markscheme

$cos 105^{\circ} = cos (180^{\circ} - 75^{\circ}) = - cos 75^{\circ}$ ${\text{cos}}\,105^\circ = {\text{cos}}\left( {180^\circ - 75^\circ } \right) = - {\text{cos}}\,75^\circ$ R1

$= - q$ $= - q$ AG

Note: Accept arguments using the unit circle or graphical/diagrammatical considerations.

[1 mark]

a.

$AD = CD \Rightarrow C \land A D = 45^{\circ}$ ${\text{AD}} = {\text{CD}} \Rightarrow {\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 45^\circ$ A1

valid method to find $B \land A C$ ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ (M1)

for example: $BC = r \Rightarrow B \land C A = 60^{\circ}$ ${\text{BC}} = r \Rightarrow {\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{A}} = 60^\circ$

$\Rightarrow B \land A C = 30^{\circ}$ $\Rightarrow {\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}} = 30^\circ$ A1

hence $B \land A D = 45^{\circ} + 30^{\circ} = 75^{\circ}$ ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 45^\circ + 30^\circ = 75^\circ$ AG

[3 marks]

b.

$AB = r \sqrt{3}$ ${\text{AB}} = r\sqrt 3$ , $AD = (CD) = r \sqrt{2}$ ${\text{AD}} = \left( {{\text{CD}}} \right) = r\sqrt 2$ A1A1

applying cosine rule (M1)

$B D^{2} = {(r \sqrt{3})}^{2} + {(r \sqrt{2})}^{2} - 2 (r \sqrt{3}) (r \sqrt{2}) cos 75^{\circ}$ ${\text{B}}{{\text{D}}^2} = {\left( {r\sqrt 3 } \right)^2} + {\left( {r\sqrt 2 } \right)^2} - 2\left( {r\sqrt 3 } \right)\left( {r\sqrt 2 } \right){\text{cos}}\,75^\circ$ A1

$= 3 r^{2} + 2 r^{2} - 2 r^{2} \sqrt{6} cos 75^{\circ}$ $= 3{r^2} + 2{r^2} - 2{r^2}\sqrt 6 \,{\text{cos}}\,75^\circ$

$5{r^2} - 2{r^2}q\sqrt 6$ AG

[4 marks]

c.i.

${\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{D}} = 105^\circ$ (A1)

attempt to use cosine rule on $\Delta {\text{BCD}}$ (M1)

${\text{B}}{{\text{D}}^2} = {r^2} + {\left( {r\sqrt 2 } \right)^2} - 2r\left( {r\sqrt 2 } \right){\text{cos}}\,105^\circ$

$= 3{r^2} + 2{r^2}q\sqrt 2$ A1

[3 marks]

c.ii.

$5{r^2} - 2{r^2}q\sqrt 6 = 3{r^2} + 2{r^2}q\sqrt 2$ (M1)(A1)

$2{r^2} = 2{r^2}q\left( {\sqrt 6 + \sqrt 2 } \right)$ A1

Note: Award A1 for any correct intermediate step seen using only two terms.

$q = \frac{1}{{\sqrt 6 + \sqrt 2 }}$ AG

Note: Do not award the final A1 if follow through is being applied.

[3 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

Syllabus sections

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

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19N.2.SL.TZ0.S_6:
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19M.1.SL.TZ1.S_3b:
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19M.2.AHL.TZ2.H_1:
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18N.2.SL.TZ0.T_5b:
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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:
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19M.1.SL.TZ1.S_3c:
Hence or otherwise, find ${\text{BC}}$ .
19N.1.SL.TZ0.S_9c:
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17M.1.SL.TZ1.S_3:
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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:
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EXN.2.SL.TZ0.2a:
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18N.2.SL.TZ0.S_7b:
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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:
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21N.2.SL.TZ0.3:
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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:
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17M.2.SL.TZ1.T_2d:
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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
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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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