Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2011  OctNov  (P19709/12)  Q#5
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Question
i. Sketch, on the same diagram, the graphs of and for .
ii. Verify that is a root of the equation , and state the other root of this equation for which .
iii. Hence state the set of values of , for , for which
Solution
i.
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.






















Red curve is for while orange curve is for .
ii.
We have sketched both sides of the equation in (i) and we can see that both curves intersect at points. Therefore, there are two roots of the equation.
If we substitute in the given equation;
Using calculator we can find that;
Hence, both sides of the equation are equal for , therefore, is the root of the equation.
We utilize the symmetry property of to find another solution (root) of :


Symmetry 
or

Therefore;
If we substitute in the above equation;
Using calculator we can see that;
Hence other root for is .
We can also check this root for .
Using calculator we can see that;
Hence, roots of equation for are and .
iii.
We do not need to do tedious calculations for this part, we are required to “state”.
By closely observing the sketch graphed in (i) we can see that two points of intersection of the two curves are roots of the equation as found and discussed in (ii). Values of at these intersection points are are and .
By mere observation we can tell that curve remains below the curve of when and .
Therefore range of values of for the given condition is;
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