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← 2023 HSC Extension 1 paper Question 14(b)

Variation 3 · answer guide

Polynomial roots — Original curves, full chain

Question

Consider and , .

(i) Show that intersection -coordinates satisfy .

(ii) Hence, by imposing a repeated root of , find the exact value of for a single intersection point.

This practice question was inspired by Question 14(b) in the 2023 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: translate the requested symmetric expression into sums and products of roots from the coefficients.(i) Substitute and clear the denominator: .
  2. Finish the calculation, then check that the result meets the question’s conditions.(ii) (usable root). Set :
  3. State the final answer clearly in the form the question requested..

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

Verification: Complete both parts: polynomial setup then double-root condition.

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

← Back to 2023 HSC Extension 1 paper Question 14(b)