A-LevelMathematicsDifferentiation9709/22 · Feb/Mar 2023 · Q710 Marks

The diagram shows the curve with parametric equations x = k tan t, y = 3 sin 2t - 4 sin t, for 0 < t < ½π. It is given that k is a positive constant. The curve crosses the x-axis at the point P. [Figure X.X]

(a)Find the value of cos t at P, giving your answer as an exact fraction.[3]
(b)Express dy/dx in terms of k and cos t.[4]
(c)Given that the normal to the curve at P has gradient ⁹⁄₁₀, find the value of k, giving your answer as an exact fraction.[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] (a) Use identity sin2t=2sint cost
[M1] (a) Attempt solution of y=0 for cost
[A1] (a) Obtain cost = 2/3 Or exact equivalent.
[*M1] (b) Differentiate to obtain at least one of dx/dt and dy/dt correct
[A1] (b) Obtain dy/dt = 6cos2t-4cost and dx/dt = k sec² t
[DM1] (b) Attempt to express dy/dx in terms of cost Using correct identities.
[A1] (b) Obtain dy/dx = (6(2cos²t-1) cos² t - 4cos³ t) / k OE
[*M1] (c) Substitute value from part (a) in expression for dy/dx involving k and cost
[DM1] (c) Equate to -10/9 and solve for k
[A1] (c) Obtain k = 5/3

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