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TMUA 2020 · Paper 2 · Question 1 of 20

TMUA 2020 Paper 2 Question 1

Algebra and functions — Discriminant · line meeting a parabola. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Algebra and functionsDiscriminant · line meeting a parabola6 options
Find the complete set of values of k for which the line y = x − 2 crosses or touches the curve y = x2 + kx + 2.
  1. A−1 ≤ k ≤ 3
  2. B−3 ≤ k ≤ 5
  3. C−4 ≤ k ≤ 4
  4. Dk ≤ −1 or k ≥ 3
  5. Ek ≤ −3 or k ≥ 5
  6. Fk ≤ −4 or k ≥ 4
Show the answer and worked solution
answer · E
  1. A−1 ≤ k ≤ 3
  2. B−3 ≤ k ≤ 5
  3. C−4 ≤ k ≤ 4
  4. Dk ≤ −1 or k ≥ 3
  5. Ek ≤ −3 or k ≥ 5
  6. Fk ≤ −4 or k ≥ 4
"Crosses or touches" means the equation for the intersections has at least one real root. Setting x2 + kx + 2 = x − 2 gives x2 + (k−1)x + 4 = 0, so the condition is that the discriminant is not negative: (k−1)2 − 16 ≥ 0. That factorises as (k−5)(k+3) ≥ 0, giving k ≤ −3 or k ≥ 5. The trap is to solve (k−1)2 < 16 instead and pick the interval between the roots.