A-LevelMathematicsTrigonometry9709/23 · Oct/Nov 2020 · Q710 Marks

A curve is defined by the parametric equations x = 3t – 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, the gradient of the curve is ⁵⁄₂.

(a)Show that the values of t at P and Q satisfy the equation 10 cos t – 8 sin t = 5.[3]
(b)Express 10 cos t – 8 sin t in the form R cos(t + α), where R > 0 and 0 < α < ½π. Give the exact value of R and the value of α correct to 3 significant figures.[3]
(c)Hence find the values of t at the points P and Q.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[B1] Obtain dx/dt = 3-2cos t and dy/dt = 5-4sin t
[M1] Equate expression for dy/dx to ⁵⁄₂
[A1] Obtain 10cos t - 8sin t = 5 AG – sufficient working to be shown.
[B1] State R = √164 or exact equivalent
[M1] Use appropriate trigonometry to find α
[A1] Obtain 0.675 with no errors seen AWRT
[M1] Carry out correct method to find one value of t Must be using the result from (b)
[A1] Obtain 0.495 AWRT
[M1] Carry out correct method to find second value of t
[A1] Obtain 4.44 AWRT

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