# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#8

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Question In the diagram, AB is an arc of a circle with centre O and radius . The line XB is a tangent to the circle at B and A is the mid-point of OX.

i. Show that angle radians.

Express each of the following in terms of , and :

ii. the perimeter of the shaded region,

iii. the area of the shaded region.

Solution

i.

We are given that BX is tangent to the curve at point B, therefore; Now consider the .

Expression for trigonometric ratio in right-triangle is; Hence; We are given that and since A is the mid-point of  OX and , therefore; Hence;   ii.

It is evident from the diagram that; We are given that and since A is the mid-point of  OX and , therefore;    Now to find we consider the .

Expression for trigonometric ratio in right-triangle is; For the ; From (i), we have; We are also given that . Therefore;    Now we need length of arc AB.

Expression for length of a circular arc with radius and angle rad is; For the given case;   Hence;  iii.

It is evident from the diagram that; First we find area of .

Expression for the area of the triangle is; For ; We are given that and from (i) we have . Therefore;  Now we find area of sector OAB.

Expression for area of a circular sector with radius and angle rad is; For sector OAB. We are given that radius and from (i), we have .

Therefore;  Hence;  