Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2019  MayJun  (P29709/23)  Q#7
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Question
a.
i. Express in the form , where and .
ii. Hence find the smallest positive value of satisfying the equation
b.
Solve the equation
for , showing all necessary working and giving the answers correct to 2 decimal places.
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 find the smallest positive value of
As demonstrated in (a:i);
Therefore;
Using calculator we can find that;
iii.
We are required to solve the equation;
Let
Now we have two options.






Since,




Using calculator; 



Now we find all solutions in the interval
Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 


We utilize the periodicity/symmetry property of
Therefore;
For




For




Only following solutions (roots) are within the given interval
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