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2020 HSC Mathematics Extension 1

Question 7:
Differential equation

1 markdifferential equations

Which of the following best represents the direction field for the differential equation ?

The four direction-field options supplied for dy/dx = -x/(4y).
The four direction-field options supplied for dy/dx = -x/(4y).
  1. Signs in Q1/Q3 and in Q2/Q4; steep near -axis; horizontal on -axis
  2. Similar signs but slopes flatten near the -axis
  3. Incorrect sign in at least one quadrant
  4. Positive slopes in every quadrant

How to recognise this question

Question type: differential equations

  • The command and notation point to differential equations.
  • The requested response is a multiple choice.

How to handle it: separate or integrate the differential equation, include the constant, then use the initial condition.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. The slope is . On the -axis, (and ) so : horizontal marks.
  2. Sign of equals sign of : negative when and have the same sign (Q1, Q3); positive when they have opposite signs (Q2, Q4).
  3. As with , : marks become vertical near the -axis (singular line of the DE).
  4. Only option A matches the sign pattern and the steep behaviour near .
  5. Answer: A.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

For , which statement is true in the first quadrant ()?

  1. Slopes are positive
  2. Slopes are negative
  3. Slopes are zero
  4. Slopes are undefined everywhere in Q1
See the step-by-step answer guide →

Question 2

For , where are the slope marks horizontal (away from )?

  1. On the -axis ()
  2. On the -axis ()
  3. On the line
  4. Nowhere
See the step-by-step answer guide →

Question 3

For , the isocline where the slope equals is

See the step-by-step answer guide →