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;
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Therefore, we can write the given function as equations by substitution as follows;
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Subtraction of second equation from the first yields;
Substitution of in the second equation;
Hence;
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ii.
We are required to evaluate .
We are given the function as;
We write it as;
We have found in (a:i) that;
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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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