Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2011  OctNov  (P29709/21)  Q#6
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Question
i. Verify by calculation that the cubic equation
has a root that lies between x=0.7 and x=0.8.
ii. Show that this root also satisfies an equation of the form
where the values of a and b are to be found.
iii. With these values of a and b, use the iterative formula
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
i.
We are required to verify by calculation that given cubic equation has a root that lies between x=0.7 and x=0.8.
We need to use signchange rule.
To use the signchange method we need to write the given equation as .
Therefore;
If the function is continuous in an interval of its domain, and if and have opposite signs, then has at least one root between and .
We can find the signs of at and as follows;
Since and have opposite signs for function , the function has root between and .
ii.
We are required to show that root of equation is also a root of the equation
If we can write the given equation and then transform it to , then both will have the same root.
Therefore, if the given equation can be rewritten as , it is evident that roots of both will be same.
Hence;
Since given equation can be rewritten as , the root of will also be root of .
iii.
If we can write the given equation and transform it to , then we can find the root of the equation by iteration method using sequence defined as.
We are given the iterative formula as;
As demonstrated in (iii) given equation can be rewritten as , therefore, iteration method can be used to find the root of the given equation using sequence defined by;
If the sequence given by the inductive definition , with some initial value , converges to a limit , then is the root of the equation .
Therefore, if , then is a root of .
We have already found in (i) through signchange rule that root of the given equation lies between
and .
Therefore, for iteration method we use;
We use as initial value.



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It is evident that .
Hence, is a root of .
The root given correct to 2 decimal places is 0.74.
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