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← 2021 HSC Extension 1 paper Question 12(a)

Variation 1 · answer guide

Differential equation — Through a node on y'=x

Question

On the direction field for , sketch the solution through . Describe the curve.

Direction field for $\dfrac{dy}{dx}=x$ with the initial point $P(0,1)$.
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 are horizontal on the -axis and increase with .
  2. State the final answer clearly in the form the question requested.Through the solution is , an upward parabola.

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

Verification: Must be tangent to horizontal mark at and open upward.

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

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