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TMUA 2020 · Paper 1 · Question 5 of 20

TMUA 2020 Paper 1 Question 5

Algebra and functions — Quadratic · axis of symmetry and a repeated root. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Algebra and functionsQuadratic · axis of symmetry and a repeated root5 options
The curve S has equation y = px2 + 6x − q where p and q are constants.

S has a line of symmetry at x = −14 and touches the x-axis at exactly one point.

What is the value of p + 8q?

  1. A6
  2. B18
  3. C21
  4. D25
  5. E38
Show the answer and worked solution
answer · A
  1. A6
  2. B18
  3. C21
  4. D25
  5. E38
The axis of symmetry of px2 + 6x − q sits at x = −62p = −3p, so −3p = −14 and p = 12. Touching the x-axis once means a repeated root, so the discriminant vanishes: 62 − 4p(−q) = 36 + 4pq = 0. With p = 12 this gives 48q = −36, so q = −34. Then p + 8q = 12 − 6 = 6.