Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#3
Hits: 190
Question
i. Express in the form
, where
and
, giving exact value of R and the value of
correct to 2 decimal places.
ii. Hence solve the equation
Giving all solutions in the interval .
Solution
i.
We are given that;
We are required to write it in the form;
If and
are positive, then;
can be written in the for
can be written in the for
where,
and
,
, with
Considering the given equation, we have following case at hand;
can be written in the for
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;
As demonstrated in (i), we can write;
Therefore, we need to solve;
Using calculator we can find that;
To find the other solution in the interval 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;
But it is evident that will yield values of
outside the given interval
.
However, we can utilize and the periodic property of
to find the other possible values in the desired range.
Hence, the only possible values in the desired range are;
|
|
Comments