A-LevelMathematicsCoordinate geometry9709/12 · Oct/Nov 2018 · Q109 Marks

The equation of a curve is y = 2x + 12/x and the equation of a line is y + x = k, where k is a constant.

(i)Find the set of values of k for which the line does not meet the curve.[3]
(ii)In the case where k = 15, the curve intersects the line at points A and B. Find the coordinates of A and B.[3]
(iii)Find the equation of the perpendicular bisector of the line joining A and B.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[*M1] Attempt to eliminate y (or x) to form a 3 term quadratic. Expect 3x² - kx + 12 or 3y² – 5ky + (2k² + 12) (= 0)
[DM1] Using the discriminant, allow <, = 0; expect k² - 144 < 0
[A1] -12 < k < 12
[M1] Using k = 15 in their 3 term quadratic
[A1] Either pair of x or y values correct..
[A1] Both pairs of coordinates
[B1FT] Gradient of AB = -1 → Perpendicular gradient = +1
[M1] Finding their midpoint using their (1, 14) and (4, 11)
[A1] Equation: y – 12½ = (x – 2½) [y = x + 10]

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Topic: Coordinate geometry, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2018 Paper 1 Variant 2, question 10.

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