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← 2022 HSC Extension 1 paper Question 10

Variation 3 · answer guide

Differential equation — Reject a decreasing sketch

Question

A proposed solution graph of falls from to as increases. This is

  1. impossible, because on
  2. possible near an equilibrium
  3. required by uniqueness
  4. possible if is negative

This practice question was inspired by Question 10 in the 2022 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: separate or integrate the differential equation, include the constant, then use the initial condition.On , so ; must increase.
  2. Finish the calculation, then check that the result meets the question’s conditions.A decrease is forbidden.
  3. 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: Evaluate the sign of the RHS on the -interval shown.

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

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