Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2006  OctNov  (P29709/02)  Q#5
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Question
The diagram shows a chord joining two points, A and B, on the circumference of a circle with centre O and radius r. The angle AOB is radians, where . The area of the shaded segment is one sixth of the area of the circle.
i. Show that satisfies the equation
ii. Verify by calculation that lies between and .
iii. Use the iterative formula
with initial value x_{1 }= 2, to determine correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
i.
We are given that area of the shaded segment is one sixth of the area of the circle.
Expression for area of a circle with radius is;
From the given diagram it is evident that;
Expression for area of a circular sector with radius and angle rad is;
Expression for the area of a triangle for which two sides (a and b) and the included angle (C ) is given;
According to given condition;
Therefore;
Hence if ;
ii.
We are required to verify by calculation that that lies between and .
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.
We are given the 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 (ii) 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 1.97.
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