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The points A, B, C lie on the circumference of the circle and [AC] is a diameter

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IB Questionbank Mathematics Higher Level 3rd edition 1

1. The diagram below shows a circle with centre O. The points A, B, C lie on the circumference of the circle and [AC] is a diameter.

Let OAaandOBb.

(a) Write down expressions for ABandCB in terms of the vectors a and b.

(2) (b) Hence prove that angle ABˆC is a right angle.

(3) (Total 5 marks)

2. Port A is defined to be the origin of a set of coordinate axes and port B is located at the point (70, 30), where distances are measured in kilometres. A ship S1 sails from port A at 10:00 in a straight line such that its position t hours after 10:00 is given by r = 

 

 20 t 10 .

A speedboat S2 is capable of three times the speed of S1 and is to meet S1 by travelling the shortest possible distance. What is the latest time that S2 can leave port B?

(Total 7 marks)

3. Given that a = 2 sin θ i + (1 – sin θ)j, find the value of the acute angle θ, so that a is perpendicular to the line x + y = 1.

(Total 5 marks)

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