# 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;  