Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  MayJun  (P19709/11)  Q#9
Question
The function f is defined by for , .
i. Find an expression for .
The function g is defined by for , where a is a constant.
ii. Find the value of a for which .
iii. Find the possible values of a given that the equation has two equal roots.
Solution
i.
We have;
We write it as;
To find the inverse of a given function we need to write it in terms of rather than in terms of . Therefore;
Interchanging ‘x’ with ‘y’;
ii.
We have;
We can write these functions as;
Therefore, for ;
Hence, for ;
We are given that;
Therefore;
iii.
We are given the equation;
We have found in (i) that;
Let’s find from the given function;
We write it as;
To find the inverse of a given function we need to write it in terms of rather than in terms of . Therefore;
Interchanging ‘x’ with ‘y’;
Therefore given equation can be written as;
We
are also given that equation has
two equal roots.
Therefore also has two equal roots.
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.
We can see that is a quadratic equation and has two equal roots. Therefore, in this case;
Now we have two options.






Comments