A-LevelMathematicsDifferential equations9709/32 · May/June 2018 · Q36 Marks
In the diagram, the tangent to a curve at the point P with coordinates (x, y) meets the x-axis at T. The point N is the foot of the perpendicular from P to the x-axis. The curve is such that, for all values of x, the gradient of the curve is positive and TN = 2. [Figure X.X]
- (i)Show that the differential equation satisfied by x and y is dy/dx = 1/2y.[1]
- (ii)The point with coordinates (4, 3) lies on the curve. Solve the differential equation to obtain the equation of the curve, expressing y in terms of x.[5]
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 2018). The mark or mark type is shown in brackets.
- [B1] Fully justify the given statement
- [B1] Separate variables and attempt integration of at least one side
- [B1] Obtain terms ln y and 1/2 x
- [M1] Use x = 4, y = 3 to evaluate a constant or as limits in a solution with terms a ln y and bx, where ab ≠ 0
- [A1] Obtain correct solution in any form
- [A1] Obtain answer y = 3e^(x/2 - 2) or equivalent
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