Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2012  OctNov  (P29709/21)  Q#5
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Question
The diagram shows the curve , for . A rectangle OABC is drawn, where B is the point on the curve with xcoordinate , and A and C are on the axes, as shown. The shaded region R is bounded by the curve and by the lines x = and y = 0.
The shaded region R is bounded by the curve and the lines y = 1 and x = p, where p is a constant.
i. Find the area of R in terms of .
ii. The area of the rectangle OABC is equal to the area of R. Show that
iii. Use the iterative formula
, with initial value , to determine the value of correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
i.
We are required to find area of shaded region R.
It is evident that area of shaded region R is area under the curve as x varies from to .
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
For the given case;
Therefore;
Rule for integration of is:
Hence, area of shaded region R is;
ii.
We are given that;
Area of Shaded Region R = Area of Rectangle OABC
We have found in (i) that;
Now we find area of rectangle.
Expression for the area of the rectangle is;
Therefore;
Therefore;
iii.
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 .
Therefore, iterative formula 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 .
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