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Date November 2008 Marks available 4 Reference code 08N.2.sl.TZ0.6
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 6 Adapted from N/A

Question

A ship leaves port A on a bearing of \(030^\circ \) . It sails a distance of \(25{\text{ km}}\) to point B. At B, the ship changes direction to a bearing of \(100^\circ \) . It sails a distance of \(40{\text{ km}}\) to reach point C. This information is shown in the diagram below.


A second ship leaves port A and sails directly to C.

Find the distance the second ship will travel.

[4]
a.

Find the bearing of the course taken by the second ship.

[3]
b.

Markscheme

finding \({\text{A}}\widehat {\rm{B}}{\rm{C}} = 110^\circ \) (\( = 1.92\) radians)     (A1)

evidence of choosing cosine rule     (M1)

e.g. \({\rm{A}}{{\rm{C}}^2} = {\rm{A}}{{\rm{B}}^2} + {\rm{B}}{{\rm{C}}^2} - 2({\rm{AB}})({\rm{BC}})\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\)

correct substitution     A1

e.g. \({\rm{A}}{{\rm{C}}^2} = {25^2} + {40^2} - 2(25)(40)\cos 110^\circ \)

\({\rm{A}}{{\rm{C}}^{}} = 53.9\) (km)     A1

a.

METHOD 1

correct substitution into the sine rule     A1

e.g.  \(\frac{{\sin {\rm{B}}\widehat {\rm{A}}{\rm{C}}}}{{40}} = \frac{{\sin 110^\circ }}{{53.9}}\)     A1

\({\rm{B}}\widehat {\rm{A}}{\rm{C}} = 44.2^\circ \)

bearing \( = 074^\circ \)     A1     N1

METHOD 2

correct substitution into the cosine rule     A1

e.g. \(\cos {\rm{B}}\widehat {\rm{A}}{\rm{C}} = \frac{{{{40}^2} - {{25}^2} - {{53.9}^2}}}{{ - 2(25)(53.9)}}\)     A1

\({\rm{B}}\widehat {\rm{A}}{\rm{C}} = 44.3^\circ \)

bearing \( = 074^\circ \)     A1     N1

[3 marks]

b.

Examiners report

A good number of candidates found this question very accessible, although some attempted to use Pythagoras' theorem to find AC.

a.

Often candidates correctly found \({\rm{B}}\widehat {\rm{A}}{\rm{C}}\) in part (b), but few added the \(30^\circ \) to obtain the required bearing. Some candidates calculated \({\rm{B}}\widehat {\rm{C}}{\rm{A}}\) , misinterpreting that the question required the course of the second ship.

b.

Syllabus sections

Topic 3 - Circular functions and trigonometry » 3.6 » Solution of triangles.
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