Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2005  OctNov  (P29709/02)  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;


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