# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#3

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**Question**

In the diagram OCA and ODB are radii of a circle with centre O and radius cm. Angle adians. CD and AB are arcs of circles with centre O and radii cm and cm respectively. The perimeter of the shaded region ABDC is 4.4r cm.

** i. **Find the value of .

** ii. **It is given that the area of the shaded region is 30 cm2. Find the value of .

**Solution**

i.

It is evident from the diagram that;

We are given that;

Now we find lengths of arcs AB and CD.

Expression for length of a circular arc with radius and angle rad is;

Therefore, for length of arc CD;

Hence;

Next, for length of arc AB;

Hence;

Finally;

** ii.
**

It is evident from the diagram that;

We are given that;

Now we find areas of sectors AOB and COD.

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

Therefore, for area of sector COD;

Hence;

Next, for area of sector AOB;

Hence;

Finally;

We have found in (i) that ;

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