Hint
Exceptional parameter values must be analysed separately. Factor the determinant before dividing by parameter expressions.
Worked solution
(a)
\(\det A(t)=\left(t - 8\right)^{2}=0\)
Answer: \(8\).
Answer: \(8\)
(b)
\(c=2(10)=20\) because the third coefficient row is twice the first.
Answer: \(20\).
Answer: \(20\)
(c)
\(v=\left[\begin{matrix}8\\9\\11\end{matrix}\right]\) by elimination.
Answer: \(8\).
Answer: \(8\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Factor the determinant before dividing by parameter expressions.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not cancel a parameter factor and lose a singular case.
(b) · 2 marks
- 1 mark: Apply the same row dependency to the constants.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A singular system can have either no solution or infinitely many.
(c) · 2 marks
- 1 mark: Solve only after substituting the nonsingular parameter.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not cancel a parameter factor and lose a singular case.
Total: 6 marks.