Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2012  OctNov  (P29709/22)  Q#4
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Question
The diagram shows the part of the curve for .
i. Use the trapezium rule with 2 intervals to estimate the value of
giving your answer correct to 2 decimal places.
ii. The line y=x intersects the curve at point P. Use the iterative formula
to determine the xcoordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
i.
We are required to apply Trapezium Rule to evaluate;
The trapezium rule with intervals states that;
If the graph is bending downwards over the whole interval from to , then trapezium rule will give an underestimate of the true area.
We are given that there are two intervals, .
We are also given that and .
Hence;




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Therefore;
The value correct to 2 decimal places is .
ii.
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 are given the part of the curve for and since line and the curve intersect, the point of intersection must lie between and .
Hence, 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 1.06.
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