# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | 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 sign-change rule.

To use the sign-change 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 sign-change rule that root of the given equation lies between and .  Therefore, for iteration method we use; We use as initial value.   1  2  3  4  5  6  7  8  9  10  It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 0.74.