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

The correct option is C $-1$

The system has a non-zero system solutions, if

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

Applying $C_2\to C_2-C_1$ and $C_3\to C_3-C_2$,we get

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

$3a^2+3a+1-3a^2-9a-7=0$$

$-6a-6=0$

$a=-1$