A-LevelMathematicsDifferentiation9709/12 · Feb/Mar 2017 · Q911 Marks

The point A (2, 2) lies on the curve y = x² – 2x + 2.

(i)Find the equation of the tangent to the curve at A.[3]
(ii)The normal to the curve at A intersects the curve again at B. Find the coordinates of B.[4]
(iii)The tangents at A and B intersect each other at C. Find the coordinates of C.[4]

Mark scheme

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

[B1B1] Numerical m dy/dx = 2x-2. At x = 2, m = 2
[B1] Expect y = 2x – 2 Equation of tangent is y-2=2(x-2)
[M1] Through (2, 2) with gradient = -1/m . Expect y = -1/2x + 3 Equation of normal y-2=-1/2(x-2)
[M1] Equate and simplify to 3-term quadratic x² - 2x + 2 = -1/2x + 3 → 2x² - 3x - 2 = 0
[A1A1] Ignore answer of (2, 2) x = -1/2, y = 13/4
[B1] Ft their -1/2. At x = −½, grad=2(−½)−2=−3
[M1] Through their B with grad their –3 (not m₁ or m₂). Expect y = -3x + 7/4 Equation of tangent is y-3¼ = -3(x+½)
[DM1] Equate their tangents or attempt to solve simultaneous equations 2x-2 = -3x+7/4
[A1] Both required. x = 3/4, y = -1/2

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