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← 2020 HSC Extension 1 paper Question 5

Variation 1 · answer guide

Polynomial graphs — Irreducible quadratic factor

Question

A monic quartic is divisible by and has a double real zero at . Which statement is true?

  1. It touches the -axis only at and both ends go to
  2. It crosses at and both ends go to
  3. It has four distinct real zeros
  4. It is undefined 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.Form is monic of degree .
  2. Simplify the previous line carefully, keeping exact values where possible. always, so the only real zero is the double root (touch, no cross).
  3. Finish the calculation, then check that the result meets the question’s conditions.Even degree and positive leading coefficient: both ends .
  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: Expand or substitute to confirm and .

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

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