A-LevelMathematicsDifferentiation9709/22 · May/June 2017 · Q811 Marks
The diagram shows the curve with parametric equations x = 2 - cos 2t, y = 2 sin³ t + 3 cos³t + 1 for 0 ≤ t ≤ ½π. The end-points of the curve are (1, 4) and (3, 3). [Figure 8.1]
- (i)Show that dy/dx = ¾ sin t - ¾ cos t.[5]
- (ii)Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.[3]
- (iii)Find the exact gradient of the normal to the curve at the point for which x = 2.[3]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- B1 Obtain dx/dt = 2sin2t
- B1 Obtain dy/dt = 6sin²t cost - 9 cos² tsint
- M1 Use dy/dx = (dy/dt) / (dx/dt) for their first derivatives
- B1 Use identity sin 2t = 2sint cost
- A1 Simplify to obtain ¾ sint-¾ cost with necessary detail present
- M1 Equate dy/dx to zero and obtain tant =k
- A1 Obtain tant = 1/3 or equivalent
- A1 Substitute value of t to obtain coordinates (2.38, 2.66)
- B1 Identify t = ¼π
- M1 Substitute to obtain exact value for gradient of the normal
- A1 Obtain gradient 8√2/3 or similarly simplified exact equivalent
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