# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#4

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**Question**

In the diagram, D lies on the side AB of triangle ABC and CD is an arc of a circle with centre A and radius 2 cm. The line BC is of length 2√3 cm and is perpendicular to AC. Find the area of the shaded region BDC, giving your answer in terms of and .

**Solution**

It is evident from the diagram that;

First we find the area of the triangle ABC.

Expression for the area of the triangle is;

For the given case;

Now we find area of circular sector ADC.

Expression for area of a circular sector with radius and angle rad is;

We are given that radius of this circular section is 2.

We need to find the area of circular sector ACD.

We know that is a right angled triangle with .

Expression for trigonometric ratio in right-triangle is;

For we take and therefore;

Hence, area of circular sector ACD;

Finally, we can find area of shaded region;

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