Matrices & Determinants
System of equations with non-trivial solution — minimum value
MJAT_TS5_P2
Grade 12
Question:
**Paragraph:** Let $\alpha,\beta,\gamma$ be real roots of $x^3+ax^2+bx+c=0$ ($a,b,c\in\mathbb{R}$, $a,b\neq 0$). If the system $\alpha u+\beta v+\gamma w=0$, $\beta u+\gamma v+\alpha w=0$, $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the minimum value of $x(3x+2a)+b+1$ is:
A) $5$\quad B) $3$\quad C) $1$\quad D) $0$
Step-by-Step Solution
Key Concept: Non-trivial solution $\Leftrightarrow$ determinant of the circulant matrix $=0$: $(\alpha+\beta+\gamma)(\alpha+\omega\beta+\omega^2\gamma)(\alpha+\omega^2\beta+\omega\gamma)=0$ where $\omega=e^{2\pi i/3}$. This gives $\alpha+\beta+\gamma=0$ or roots are in geometric progression with ratio $\omega$.
Answer: **C** ($\min = 1$).
Correct Answer: C