Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2019  OctNov  (P29709/23)  Q#5
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Question
It is given that
Where is a constant.
i. Show that
ii. Using the equation in part (i), show by calculation that 0.5 < a < 0.75.
iii. Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
Solution
i.
We are given that;
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
ii.
We are required to show by calculation that the xcoordinate lies between 0.5 <
a < 0.75.
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.
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 .
We use as initial value.



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It is evident that.
Hence, is a root of .
The root given correct to 3 significant figures is 0.651.
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