# 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;  