Vector motion — Show D increasing under angle bound

Question

With and , show that if then is increasing for all .

This practice question was inspired by Question 14(d) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: differentiate position to get velocity and acceleration, then translate the angle condition into a dot product.Work with ; .
Finish the calculation, then check that the result meets the question’s conditions.Discriminant of the quadratic is proportional to ; negative when .
State the final answer clearly in the form the question requested.Answer: under the given angle bound, for , so is increasing.

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

Verification: Sign link: with .

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