# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#2

Question

i.
Express the equation  in the form , where k is a  constant.

ii.       Hence solve the equation for .

Solution

i.

We are given that;

We know that therefore;

Comparison with given  yields;

ii.

We are required to solve  for .

From (i) we know that  can be written in the form;

Therefore, we solve  for .

To solve this equation for , we can substitute . Hence,

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

Multiplying the entire inequality with 2;

Since ;

Hence the given interval for  is .

To solve  equation for interval ,

Using calculator we can find the value of .

Properties of  are listed below.

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We utilize the periodic property of  to find other solutions (roots) of .

Therefore;

For;

Only following solutions (roots) are within the given interval ;

Since ;

Therefore;

Hence, all the solutions of the equation  within the interval for  are;