A-LevelMathematicsTrigonometry9709/22 · May/June 2018 · Q710 Marks

Show that 2 cosec² 2x(1 – cos 2x) = sec²x.

(i)Show that 2 cosec² 2x(1 – cos 2x) = sec²x.[3]
(ii)Solve the equation 2 cosec² 2x(1 – cos 2x) = tan x + 21 for 0 < x < π, giving your answers correct to 3 significant figures.[4]
(iii)Find ∫ [2 cosec²(4y + 2) – 2 cosec²(4y + 2) cos(4y + 2)] dy.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[B1] Express cosec² 2x as 1/(4sin²x cos²x)
[M1] Must be using correct working for M1 Attempt to express LHS in terms of sin x and cos x only
[A1] AG; necessary detail needed Obtain 2sin²x / (4sin²x cos²x) or equivalent and hence sec²x
[B1] Express equation as 1 + tan² x = tan x + 21
[M1] Solve 3-term quadratic equation for tan x
[A1] Or greater accuracy 1.3734... Obtain tan x = 5 and hence x = 1.37
[A1] Or greater accuracy 1.8157... Obtain tan x = -4 and hence x = 1.82
[B1] Use x = 2y+1
[M1] Condone absence of or error with dy Identify integral as of form ∫ sec²(ay + b) dy
[A1] Obtain ½ tan(2y + 1) + c

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