# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#7

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Question

a.

i.       Express  in the form , where and  .

ii.       Hence find the smallest positive value of  satisfying the equation

b.
Solve the equation

for , showing all necessary working and giving the answers correct to 2  decimal places.

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 find the smallest positive value of satisfying the equation

As demonstrated in (a:i);

Therefore;

Using calculator we can find that;

iii.

We are required to solve the equation;

except where  or undefined

Let , then;

Now we have two options.

Since, ;

 Using calculator;

Now we find all solutions in the interval .

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

We utilize the periodicity/symmetry property of   to find other solutions (roots) of

Therefore;

For ;

For ;

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