Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#4

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Question

The diagram shows a metal plate OABCDEF consisting of 3 sectors, each with centre O. The radius of sector COD is 2r and angle COD is  radians. The radius of each of the sectors BOA and FOE is  r, and AOED and CBOF are straight lines.

i.
Show that the area of the metal plate is .

ii.
Show that the perimeter of the metal plate is independent of .

Solution

i.

It is evident from the diagram that;

Now we find areas of all these individual sectors one-by-one.

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

For sector BOA; radius  and we need to find .

We know that vertically opposite angles are equal.

It is evident from the diagram that;

It is also evident from the diagram that;

Therefore;

Next we find area of sector COD.

For sector COD; radius  and angle . Therefore;

Now we find area of sector EOF.

For sector EOF; radius  and angle . Therefore;

Finally;

ii.

To show whether perimeter of the plate depends or not on the angle  we need expression for perimeter of the plate.

It is evident from the diagram that;

It is evident from the diagram that;

Now we find lengths of arcs.

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

First we find length of arc AB.

It is evident that  and as we have found in (i) . Therefore;

For arc CD, it is evident from the diagram that  and . Therefore;

For arc EF, it is evident from the diagram that  and  . Therefore;

Finally;

Hence, perimeter of the plate is independent of .