Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | May-Jun | (P2-9709/02) | Q#8

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Question

The constant , where , is such that


    i.       Find an equation satisfied by , and show that it can be written in the form


ii.      
Verify, by calculation, that the equation  ) has a root between 3 and 3.5.

   iii.       Use the iterative formula

with a1 = 3.2, to calculate the value of a correct to 2 decimal places. Give the result of each iteration  to 4 decimal places.

Solution

     i.
 

We are given;

Rule for integration of  is:

Rule for integration of  is:

For ;

Since , , therefore; 

Since , therefore; 

 

   ii.
 

We are required to verify by calculation that the root of equation  lies between 3  and 3.5. 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 .

 

  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.

As demonstrated in (i) 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 (ii) 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

It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 3.26.

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