A-LevelMathematicsNumerical methods9709/31 · May/June 2017 · Q58 Marks
The diagram shows a semicircle with centre O, radius r and diameter AB. The point P on its circumference is such that the area of the minor segment on AP is equal to half the area of the minor segment on BP. The angle AOP is x radians. [Figure X.X]
- (i)Show that x satisfies the equation x = ½(π + sin x).[3]
- (ii)Verify by calculation that x lies between 1 and 1.5.[2]
- (iii)Use an iterative formula based on the equation in part (i) to determine x correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]
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Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- Use correct sector formula at least once and form an equation in r and x M1
- Obtain a correct equation in any form A1
- Rearrange in the given form A1
- Calculate values of a relevant expression or expressions at x = 1 and x = 1.5 M1
- Complete the argument correctly with correct calculated values A1
- Use the iterative formula correctly at least once M1
- Obtain final answer 1.374 A1
- Show sufficient iterations to 5 d.p. to justify 1.374 to 3 d.p., or show there is a sign change in the interval (1.3745, 1.3755) A1
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