Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/12)  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;


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