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← 2021 HSC Extension 1 paper Question 12(a)

Variation 2 · answer guide

Differential equation — Through isocline

Question

For , describe the solution through on a direction field.

Direction field for $\dfrac{dy}{dx}=y$ with the initial point $P(0,2)$.
Direction field for with the initial point .

This practice question was inspired by Question 12(a) in the 2021 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.Slopes proportional to height: steeper further from the -axis.
  2. State the final answer clearly in the form the question requested.Solution is increasing and concave up for .

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

Verification: Never crosses ; exponential growth from .

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

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