A-LevelMathematicsTrigonometry9709/22 · Feb/Mar 2019 · Q711 Marks
The parametric equations of a curve are x = 2t - sin 2t, y = 5t + cos 2t, for 0 ≤ t ≤ ½π. At the point P on the curve, the gradient of the curve is 2.
- (i)Show that the value of the parameter at P satisfies the equation 2 sin 2t – 4 cos 2t = 1.[4]
- (ii)By first expressing 2 sin 2t – 4 cos 2t in the form R sin(2t – α), where R > 0 and 0 < α < ½π, find the coordinates of P. Give each coordinate correct to 3 significant figures.[7]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.
- [B1] Obtain dx/dt = 2 – 2 cos 2t
- [B1] Obtain dy/dt = 5 – 2 sin 2t
- [M1] Equate attempt at dy/dx to 2 and rearrange
- [A1] Confirm equation 2 sin 2t – 4 cos 2t = 1
- [B1] State R = √20 or 4.47
- [M1] Use appropriate trigonometry to find α
- [A1] Obtain α = 1.107 with no errors seen
- [M1] Carry out correct method to find value of t
- [A1] Obtain t = 0.666
- [M1] Substitute value of t between 0 and ½π into expressions for x and y
- [A1] Obtain x = 0.361, y = 3.57
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