Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2008  MayJun  (P29709/02)  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 a_{1 }= 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 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 .
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 signchange rule that root of the given equation lies between and .
Therefore, for iteration method we use;
We use as initial value.



1 


2 


3 


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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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