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;

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