Question

The value of a for which the system of equations

$a^{3}x + (a + 1)^{3}y + (a + 2)^{3}z = 0,$

$ax + (a + 1)y + (a + 2)z = 0,x + y + z = 0,$

has a non zero solution is.

• A. – 1
• B. 0
• C. 1
• D. None of these

Answer 1

Answer: (A)

– 1

Answer 2

The system will have a non-zero solution, if

a^{3} & (a + 1)^{3} & (a + 2)^{3} \\ a & a + 1 & a + 2 \\ 1 & 1 & 1 \end{matrix} \right| = 0$$ $$\Rightarrow \left| \begin{matrix} a^{3} & 3a^{2} + 3a + 1 & 3(a + 1)^{2} + 3(a + 1) + 1 \\ a^{2} & 1 & 1 \\ 1 & 0 & 0 \end{matrix} \right| = 0$$ by $C_{2} \rightarrow C_{2} - C_{1} $ $$C_{3} \rightarrow C_{3} - C_{2}$ $ ⇒ $3a^{2} + 3a + 1 - \{ 3(a + 1)^{2} + 3(a + 1) + 1\}$ (expanding along $R_{3}$) ⇒ $- 6(a + 1) = 0 \Rightarrow a = - 1$.