Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | 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;      