Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2018  MayJun  (P29709/21)  Q#7
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Question
i. Express in the form , where and . Give the value of correct to 4 decimal places.
ii. Using your answer from part (i), solve the equation
for .
Solution
i.
We are given the expression;
We are required to write it in the form;
If and are positive, then;
can be written in the form
can be written in the form
where,
and , , with
Considering the given equation, we have following case at hand;
can be written in the form
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;
provided that
provided that
As demonstrated in (i), we can write;
Therefore;
Hence, we need to solve;
Using calculator we can find that;
To find the other solution of 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;
Therefore;




For

For



For















Hence all the solutions (roots) of the equation for are;


iii.
We are required to find;
As demonstrated in (i), we can write;
Therefore;
Hence;
provided that
Rule for integration of is:
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