2024 HSC Mathematics Extension 1
Question 12(a):
Vectors
The vectors and are perpendicular.
Find the possible values of .
How to recognise this question
Question type: vectors
- The command and notation point to vectors.
- The requested response is a worked solution.
How to handle it: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Perpendicular dot product zero: .
- Expand: .
- Test : , so is a factor.
- Factor as (by division or Vieta: remaining sum , product ).
- . Roots: , , .
- NESA 12(a): 3 marks correct; 2 marks for obtaining (or equivalent merit); 1 mark for the cubic equation .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks 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
The vectors and are perpendicular. Find all possible values of .
Question 2
Find so that is perpendicular to .
Question 3
Show that if , then , and solve this cubic.