A-LevelMathematicsQuadratics9709/12 · May/June 2022 · Q59 Marks
The equation of a curve is y = 4x² - kx + (1/2)k² and the equation of a line is y = x - a, where k and a are constants.
- (a)Given that the curve and the line intersect at the points with x-coordinates 0 and ¾, find the values of k and a.[4]
- (b)Given instead that a = -3/2, find the values of k for which the line is a tangent to the curve.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.
- [M1] 4x(0)² - k(0) + (1/2)k² = 0 - a
- [M1] 4(3/4)² - k(3/4) + (1/2)k² = 3/4 - a
- [A1 A1] k = 2, a = -2
- [M1] 4x² - kx + (1/2)k² = x + 7/2 leading to 4x² - (k+1)x + (1/2)k² - 7/2 = 0
- [DM1] (k+1)² - 4(4)((1/2)k² - 7/2) = 0
- [A1] 7k² - 2k - 57 = 0
- [DM1] (k-3)(7k+19) = 0 or other valid method
- [A1] k = 3, k = -19/7
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