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← 2020 HSC Extension 1 paper Question 5

Variation 2 · answer guide

Polynomial graphs — Opposite end behaviour trap

Question

A monic quartic has a double zero at and is divisible by (no real zeros). Which graph feature is forced?

  1. Touches at only; both ends
  2. Crosses at ; both ends
  3. Two simple real zeros only
  4. A vertical asymptote at

This practice question was inspired by Question 5 in the 2020 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

  1. Set up the solution: use degree, leading coefficient, root multiplicities and any forced factors to decide shape.Monic quartic both ends .
  2. Simplify the previous line carefully, keeping exact values where possible.Irreducible quadratic factor contributes no real zeros.
  3. Finish the calculation, then check that the result meets the question’s conditions.Double zero touch-and-turn at .
  4. Match the result to the choices and state the correct option.Therefore option A is correct.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Discriminant of is .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.

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