Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/21) | 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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