A-LevelMathematicsQuadratics9709/12 · Oct/Nov 2017 · Q78 Marks

Points A and B lie on the curve y = x² – 4x + 7. Point A has coordinates (4, 7) and B is the stationary point of the curve. The equation of a line L is y = mx – 2, where m is a constant.

(i)In the case where L passes through the mid-point of AB, find the value of m.[4]
(ii)Find the set of values of m for which L does not meet the curve.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

dy/dx = 2x - 4 = 0 (Can use completing the square.) (M1)
→ x = 2, y = 3 (B1B1)
Midpoint of AB is (3, 5) (B1 FT) FT on (their 2, their 3) with (4,7)
m = 7/3 (or 2.33) (B1)
Simultaneous equations → x² - 4x - mx + 9 (= 0) (*M1) Equates and sets to 0 must contain m
Use of b² - 4ac → (m + 4)² - 36 (DM1) Any use of b²-4ac on equation set to 0 must contain m
Solves = 0 → -10 or 2 (A1) Correct end-points.
-10 < m < 2 (A1) Don't condone ≤ at either or both end(s). Accept -10 < m, m < 2.

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