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TMUA 2022 · Paper 1 · Question 20 of 20

TMUA 2022 Paper 1 Question 20

Graphs and transformations — Intersections of two modulus graphs. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Graphs and transformationsIntersections of two modulus graphs5 optionshard
n is the number of points of intersection of the graphs y = |x2 − a2|  and  y = a2|x − 1| where a is a real number.

What is the smallest value of n that is not possible?

  1. An = 1
  2. Bn = 2
  3. Cn = 3
  4. Dn = 4
  5. En = 5
Show the answer and worked solution
answer · B
  1. An = 1
  2. Bn = 2
  3. Cn = 3
  4. Dn = 4
  5. En = 5
Taking a = 0 collapses both graphs to y = x2 and y = 0, meeting only at the origin, so n = 1 occurs. Taking a2 = 1 gives |x2−1| = |x−1|, that is |x−1|(|x+1| − 1) = 0, with solutions x = 1,  0,  −2 — so n = 3 occurs, and larger counts arise for other a. The value that never occurs is n = 2.