Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2013  OctNov  (P29709/22)  Q#7
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Question
The diagram shows the curve . The shaded region R is bounded by the curve and by the lines x = 0, y = 0 and x = a, where a is positive. The area of R is equal to .
i. Find an equation satisfied by a, and show that the equation can be written in the form
ii. Verify by calculation that the equation has a root between 0.2 and 0.3.
iii. Use the iterative formula to determine this root correct to 2 decimal places. Give
the result of each iteration to 4 decimal places.
Solution
i.
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
It is evident from the diagram that area R of shaded region is area under the curve along xaxis from x=0 to x=a.
Therefore;
Rule for integration of is:
Rule for integration of is:
Rule for integration of , or ;
Therefore;
We are given that;
Therefore;
ii.
We are required to verify by calculation that has a root that lies between a=0.2 and a=0.3.
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 iterative formula;
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 signchange rule that root of the given equation lies between
and .
Therefore, for iteration method we use;
We use as initial value.



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It is evident that .
Hence, is a root of .
The root given correct to 2 decimal places is 0.29.
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