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TMUA 2020 · Paper 1 · Question 17 of 20

TMUA 2020 Paper 1 Question 17

Algebra and functions — Line meeting a square-root curve twice. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Algebra and functionsLine meeting a square-root curve twice6 options
Find the complete set of values of m in terms of c such that the graphs of y = mx + c and y = √x have two points of intersection.
  1. A0 < m < 14c
  2. B0 < m < 4c2
  3. Cm > 14c
  4. Dm < 14c
  5. Em > 4c2
  6. Fm < 4c2
Show the answer and worked solution
answer · A
  1. A0 < m < 14c
  2. B0 < m < 4c2
  3. Cm > 14c
  4. Dm < 14c
  5. Em > 4c2
  6. Fm < 4c2
Substituting t = √ x, where t ≥ 0, turns √ x = mx + c into mt2 − t + c = 0, and each root t ≥ 0 gives exactly one intersection point. Two intersections therefore need two distinct non-negative roots. The discriminant condition is 1 − 4mc > 0, the sum of roots 1m must be positive, forcing m > 0, and the product cm must be non-negative, forcing c ≥ 0. With m > 0 and c > 0, 1 − 4mc > 0 rearranges to m < 14c, so the answer is 0 < m < 14c. Dropping the requirement m > 0 gives the tempting option D, but a line with negative gradient cuts y = √ x at most once.