Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  MayJun  (P19709/13)  Q#7
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Question
a.
i. Express in the form , where and are constants to be found.
ii. Hence, or otherwise, and showing all necessary working, solve the equation
For .
b.
The diagram shows the graphs of and for . The graphs intersect at the points A and B.
i. Find the xcoordinate of A.
ii. Find the ycoordinate of B.
Solution
a.
i.
We are given the expression;
Since ;
Since ;
We can write as;
Therefore;
ii.
We are required to solve the equation;
As demonstrated in (i);
Therefore;
We have two options;




Using calculator; 



We utilize the symmetry property of to find other solution(s) (root) of :
Properties of 

Domain 

Range 

Odd/Even 

Periodicity 



Translation/ Symmetry 






Hence;
For 
For 




It is evident these above solutions and any further solutions obtained through periodicity property of of will be out of given range .
Hence, only solution within desired range is;
b.
i.
We are required to find the xcoordinate of point A which is the intersection point of graphs of and .
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).
Equation of the first curve is;
Equation of the second curve is;
Equating both equations;
Since ;
Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 


We utilize the periodicity property of to find the other solutions within the given range .
Therefore, for ;







It is evident that only values within range are;


Two values of x indicate that there are two intersection points.
However, it is evident from the diagram that point A is on the positive side of xaxis.
Hence, xcoordinates of point A is 1.11.
ii.
As demonstrated in (B(i)) the points of intersection of graphs of and are;


It is also demonstrated in (B(i)) that xcoordinate of point A is .
It is evident then that xcoordinate of point B is .
Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.
We choose;
Therefore;
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