A-LevelMathematicsCircular measure9709/31 · May/June 2020 · Q68 Marks

The diagram shows a circle with centre O and radius r. The tangents to the circle at the points A and B meet at T, and angle AOB is 2x radians. The shaded region is bounded by the tangents AT and BT, and by the minor arc AB. The area of the shaded region is equal to the area of the circle. [Figure 6.1]

(a)Show that x satisfies the equation tan x = π + x.[3]
(b)This equation has one root in the interval 0 < x < ½π. Verify by calculation that this root lies between 1 and 1.4.[2]
(c)Use the iterative formula Xn+1 = tan¯¹(π + x) to determine the 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 (May/June 2020). The mark or mark type is shown in brackets.

[B1] (a) State or imply AT=r tan x or BT=r tan x
[M1] (a) Use correct area formula and form an equation in r and x
[A1] (a) Rearrange in the given form
[M1] (b) Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 1.4
[A1] (b) Complete the argument correctly with correct calculated values
[M1] (c) Use the iterative formula correctly at least once
[A1] (c) Obtain final answer 1.35
[A1] (c) Show sufficient iterations to 4 d.p. to justify 1.35 to 2 d.p. or show there is a sign change in the interval (1.345, 1.355)

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