# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#10

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Question

i.       Solve the equation  for .

ii.       Sketch, on the same diagram, the graphs of  and for  .

iii.       Use your answers to parts (i) and (ii) to find the set of values of x for   for which .

Solution

i.

We have the equation;

We know that ;

Using calculator we can find;

 Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry

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

Therefore;

For;

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

ii.

We are required to sketch and  for .

First we sketch  for .

We can sketch the graph of  for  as follows.

We can find the points of the graph as follows.

Now we sketch  for

We can sketch the graph of for as follows.

We can find the points of the graph as follows.

Now we can sketch both the curve on same graph.

iii.

We have found in (i) that solutions of the equation for  is;

It shows that and have equal but opposite values at  and so that their sum is equal to zero.

Also we have sketched in (ii) and ;

As is evident from the graph that and have equal values at
and .

If we invert to make it , the above graph will look like;

Hence,  for;