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Date	May 2018	Marks available	3	Reference code	18M.2.AHL.TZ2.H_4
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	H_4	Adapted from	N/A

Question

Consider the following diagram.

The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].

Find AM.

[3]

a.i.

Find $A \land M P$ ${\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}$ in radians.

[2]

a.ii.

Find the area of the shaded region.

[3]

b.

Markscheme

METHOD 1

PC $= \frac{\sqrt{3}}{2}$ $= \frac{{\sqrt 3 }}{2}$ or 0.8660 (M1)

PM $= \frac{1}{2}$ $= \frac{1}{2}$ PC $= \frac{\sqrt{3}}{4}$ $= \frac{{\sqrt 3 }}{4}$ or 0.4330 (A1)

AM $= \sqrt{\frac{1}{4} + \frac{3}{16}}$ $= \sqrt {\frac{1}{4} + \frac{3}{{16}}}$

$= \frac{\sqrt{7}}{4}$ $= \frac{{\sqrt 7 }}{4}$ or 0.661 (m) A1

METHOD 2

using the cosine rule

AM² $= 1^{2} + {(\frac{\sqrt{3}}{4})}^{2} - 2 \times \frac{\sqrt{3}}{4} \times cos (30^{\circ})$ $= {1^2} + {\left( {\frac{{\sqrt 3 }}{4}} \right)^2} - 2 \times \frac{{\sqrt 3 }}{4} \times {\text{cos}}\left( {30^\circ } \right)$ M1A1

AM $= \frac{\sqrt{7}}{4}$ $= \frac{{\sqrt 7 }}{4}$ or 0.661 (m) A1

[3 marks]

a.i.

tan ( $A \land M P$ ${\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}$ ) $= \frac{2}{\sqrt{3}}$ $= \frac{2}{{\sqrt 3 }}$ or equivalent (M1)

= 0.857 A1

[2 marks]

a.ii.

EITHER

$\frac{1}{2} A M^{2} (2 A \land M P - sin (2 A \land M P))$ $\frac{1}{2}{\text{A}}{{\text{M}}^2}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - {\text{sin}}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}} \right)} \right)$ (M1)A1

OR

$\frac{1}{2} A M^{2} \times 2 A \land M P - = \frac{\sqrt{3}}{8}$ $\frac{1}{2}{\text{A}}{{\text{M}}^2} \times 2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - = \frac{{\sqrt 3 }}{8}$ (M1)A1

= 0.158(m²) A1

Note: Award M1 for attempting to calculate area of a sector minus area of a triangle.

[3 marks]

b.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.4—Circle: radians, arcs, sectors

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