# 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

Hence;

We are given that

Hence;

ii.

It is evident from the diagram that;

We are given that

Now to find

Expression for

For the

From (i), we have;

We are also given that

Now we need length of arc AB.

Expression for length of a circular arc with radius

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

Now we find area of sector OAB.

Expression for area of a circular sector with radius

For sector OAB.

We are given that radius

Therefore;

Hence;

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