# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#7

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Question

a)   Find the possible values of  for which , giving your answers correct to 3 decimal places.

b)  Solve the equation  for  giving  in terms of  in your answers.

Solution

a)

We have;

We can rearrange the equation as;

We know that;

Therefore we can write above equation as;

b)

To solve the equation  for , let

Since given interval is  , for    interval can be found as follows;

Multiplying the entire inequality with 2;

Since  ;

Hence the given interval for  is .

Therefore we can write the given equation as;

Using calculator we can find the value of .

We utilize the symmetry property of   to find another solution (root) of :

 Symmetry Property or

Hence;

For ;

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots) of the  we utilize the periodic property of .

 Periodic Property or

Hence

 For For

For

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

Since

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