2023 HSC Mathematics Extension 1
Question 3:
Differential equation
The diagram shows the direction field of a differential equation. A particular solution to the differential equation passes through .
Where does the solution that passes through cross the -axis?
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
- Start at the marked point and follow the slope marks of the direction field to the right toward .
- The field lines indicate a gradual rise: the solution curve increases as increases from to .
- Tracing carefully (not a straight-line extrapolation from a single slope) yields a -intercept near .
- Options A and B underestimate the cumulative rise; D overestimates it.
- Answer: C. (NESA multiple-choice key: C.)
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
On a direction field for , the slope mark at is
Question 2
Using one Euler step of size for starting at , the estimate of is
Question 3
For with and , the solution curves are