# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#3

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Question

i.
Express  in the form , where  and , giving exact  value of R and the value of  correct to 2 decimal places.

ii. Hence solve the equation

Giving all solutions in the interval .

Solution

i.

We are given that;

We are required to write it in the form;

If  and are positive, then;

can be written in the for

can be written in the for

where,

and , , with

Considering the given equation, we have following case at hand;

can be written in the for

Comparing it with given equation Therefore

Therefore;

Finally, we can find , utilizing the equation;

Using calculator we can find that;

Therefore;

ii.

We are required to solve the equation;

As demonstrated in (i), we can write;

Therefore, we need to solve;

Using calculator we can find that;

To find the other solution in the interval we utilize the odd/even property of

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We use odd/even property;

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots) over the interval , we utilize the periodic property of  for both these values of .

Therefore;

But it is evident that will yield values of outside the given interval .

However, we can utilize and the periodic property of to find the other possible values in the desired range.

Hence, the only possible values in the desired range  are;