A-LevelMathematicsNumerical methods9709/32 · Feb/Mar 2023 · Q79 Marks

The diagram shows a circle with centre O and radius r. The angle of the minor sector AOB of the circle is x radians. The area of the major sector of the circle is 3 times the area of the shaded region. [Figure 7.1]

(a)Show that x = ¾ sin x + ½π.[4]
(b)Show by calculation that the root of the equation in (a) lies between 2 and 2.5.[2]
(c)Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

[B1] State or imply area of major sector = ½r²(2π – x) OE
[B1] State or imply area of shaded segment = ½r²x - ½r² sin x OE
[M1] State ½r²(2π – x) = 3(½r²x - ½r² sin x) OE
[A1] Obtain the given answer x = ¾ sin x + ½π after full and correct working AG Allow rectified slip if before penultimate line.
[M1] Calculate the values of a relevant expression or pair of expressions at x = 2 and x = 2.5
[A1] Complete the argument correctly with correct calculated values Degrees award 0/2
[M1] Use the iterative formula correctly at least twice
[A1] Obtain final answer 2.18
[A1] Show sufficient iterations to 4 d.p. to justify 2.18 to 2 d.p. or show there is a sign change in the interval (2.175, 2.185) Degrees award 0/3

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