Question

The value of a for which the system of equations

x + y + z = 0

ax + (a + 1)y + (a + 2)z = 0 a³x + (a + 1)³y + (a + 2)³z = 0

has a non-zero solution is –

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

Answer 1

Answer: (C)

–1

Answer 2

The system of equations will have a non-zero solution if and

only if

D = $\left| \begin{matrix} 1 & 1 & 1 \\ a & a + 1 & a + 2 \\ a^{3} & (a + 1)^{3} & (a + 2)^{3} \end{matrix} \right|$ = 0

Using C₃ ® C₃ – C₂, C₂ ® C₂ – C₁, we get

D = $\left| \begin{matrix} 1 & 0 & 0 \\ a & 1 & 1 \\ a^{3} & 3a^{2} + 3a + 1 & 3a^{2} + 9a + 7 \end{matrix} \right|$= 0

Ž 6a + 6 = 0 or a + 1 = 0

Or a = –1.