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TMUA 2023 · Paper 1 · Question 2 of 20

TMUA 2023 Paper 1 Question 2

Algebra and functions — Discriminant · graphs that do not meet. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Algebra and functionsDiscriminant · graphs that do not meet6 options
The graphs of y = x2 + 5x + 6 and y = mx − 3, where m is a constant, are plotted on the same set of axes.

Given that the graphs do not meet, what is the complete range of possible values of m?

  1. A−1 < m < 11
  2. Bm < −1,  m > 11
  3. C−√11 < m < √11
  4. Dm < −√11,  m > √11
  5. E−11 < m < 1
  6. Fm < −11,  m > 1
Show the answer and worked solution
answer · A
  1. A−1 < m < 11
  2. Bm < −1,  m > 11
  3. C−√11 < m < √11
  4. Dm < −√11,  m > √11
  5. E−11 < m < 1
  6. Fm < −11,  m > 1
Setting them equal gives x2 + 5x + 6 = mx − 3, so x2 + (5−m)x + 9 = 0. The graphs miss each other exactly when this has no real root, so the discriminant is negative: (5−m)2 − 36 < 0. Hence |5−m| < 6, giving −6 < 5 − m < 6 and so −1 < m < 11.