Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2014  MayJun  (P29709/22)  Q#7
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Question
It is given that
i. Show that
ii. Show by calculation that the value of a lies between 1.0 and 1.5.
iii. Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Solution
i.
We are given that;
Rule for integration of is:
Rule for integration of , or ;
Rule for integration of is:
Taking logarithm of both sides;
Since for any ;
ii.
We are required to show by calculation that the xcoordinate of a lies between 1.0 and 1.5.
We need to use signchange rule.
To use the signchange method we need to write the given equation as .
From (ii) we have;
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.
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 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 1.343.
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