# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#6

Question

a.   The function f, defined by for , is such that and .

i.       Find the values of the constants a and b.

ii.       Evaluate .

b.   The function g is defined by for for . The range of g is given by . Find the values of the constants c and d.

Solution

a.

i.

We are given the function as; We write it as; We are given that;  Therefore, we can write the given function as equations by substitution as follows;        Subtraction of second equation from the first yields;     Substitution of in the second equation;   Hence;  ii.

We are required to evaluate .

We are given the function as; We write it as; We have found in (a:i) that;  Therefore; Let’s first find .  Therefore when ;       b.

We are given the function as; We write it as; We are given that range of the function is; It is evident that the given function has maximum and minimum values -4 and 10, respectively.

We can also see that only variable in the function is which can have only following values; Therefore, when function has minimum value it is essential that ;   Similarly, when function has maximum value it is essential that ;   We can solve the following simultaneous equations obtained.  From first equation we can obtain and we can substitute it in the second equation.   Substitution of in the above second equation results;   Hence;  