Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2020  FebMar  (P19709/12)  Q#11
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Question
i. Solve the equation for .
ii. Find the set of values of for which the equation has no solution.
iii. For the equation , state the value of for which there are three solutions in the interval , and find these solutions.
Solution
i.
We have the equation;
Let ;
Now we have two options.





Since ;




Using calculator; 



Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 


We utilize the periodicity property of to find other solutions (roots) of : .
Therefore;
For;
for








for







We are given that for .
Only following solutions (roots) are within the given interval ;


ii.
We are given the equation;
It is evident that given equation is a quadratic one.
Standard form of quadratic equation is;
Expression for discriminant of a quadratic equation is;
If ; Quadratic equation has two real roots.
If ; Quadratic equation has no real roots.
If ; Quadratic equation has one real root/two equal roots.
Therefore, if the given equation has no solution, then for it;
As can be seen;
Hence;
iii.
We are given the equation;
It is evident that given equation is a quadratic one.
We are given that this equation has three solutions.
We know that only cubic equation can have three solutions but not the quadratic equation.
The solutions to a quadratic equation are the places where it crosses the xaxis. Since the graph of a quadratic equation is a parabola, it is impossible for it to cross the axis in more than two places.
Therefore, we need to find a situation where this equation can yield three solutions.
By inspection it is evident that if , the equation will yield more than two solutions.
Let ;
Now we have two solutions.




Using calculator; 



Using calculator; 


Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 


We utilize the periodicity property of to find other solutions (roots) of : .
Therefore;
For;
for








for







We are given that for .
Only following solutions (roots) are within the given interval ;



Hence, the equation for has 03 solutions when .
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