Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2004  OctNov  (P29709/02)  Q#8
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Question
i. Express in the form , where and , giving the exact values of R and .
ii. Hence show that
iii. By differentiating , show that if then .
iv. Using the results of parts (ii) and (iii), show that
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 show that;
As demonstrated in (i), we can write;
Therefore, we can write the given equation as;
Since provided that ;
Hence;
iii.
We are given that;
We are required to show that;
Since ;
Therefore;
If and are functions of , and if , then;
If , then;
Rule for differentiation of is;
Rule for differentiation of is;
We have the trigonometric identity;
Since provided that ;
iv.
We are required to show that;
As demonstrated in (ii), we can write;
Therefore;
Rule for integration of is:
Hence;
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