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A-LevelMathematicsQuadratics9709/12 · Oct/Nov 2010 · Q67 Marks

A curve has equation y = kx² + 1 and a line has equation y = kx, where k is a non-zero constant.

  1. (i)Find the set of values of k for which the curve and the line have no common points.[3]
  2. (ii)State the value of k for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.

  1. [M1] kx²-kx + 1 = 0 y eliminated
  2. [M1] k²-4k < 0 Applying b² - 4ac < 0 or = or < or ≥
  3. [A1] 0<k<4 co
  4. [B1√] k = 4 only ft from their k² – 4k = 0. (Not k = 0)
  5. [M1] (2x - 1)² = 0 ft from their k
  6. [A1, A1] x=1/2, y=2 or (1/2, 2)

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About this question

Topic
Quadratics, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source
A-Level Mathematics, Oct/Nov 2010 Paper 1 Variant 2, question 6.

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