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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     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.



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 .


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 .



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.


Since given equation   can be rewritten as   , the root of will also be root of  .



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.











It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 0.74.

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