Question

The value of ${}^{\prime }{a}^{\prime }$ for which the system of equations $(a+1{)}^{3}x+(a+2{)}^{3}y=(a+3{)}^3,(a+1)x+(a+2)y=a+3,x+y=1$ is consistent, is

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-2$

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

By applying $C_1\to C_1-C_2$ and $C_2\to C_2-C_3,$ we get

$\Rightarrow \left|\begin{array}{ccc}-(3a^2+9a+7)& -(3a^2+15a+19)& (a+3{)}^3\\ -1& -1& a+3\\ 0& 0& 1\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}(3a^2+9a+7)& (3a^2+15a+19)& (a+3{)}^3\\ 1& 1& a+3\\ 0& 0& 1\end{array}\right|=0$

By applying $C_1\to C_1-C_2,$ we get

$\Rightarrow \left|\begin{array}{ccc}-6a-12& (3a^2+15a+19)& (a+3{)}^3\\ 0& 1& a+3\\ 0& 0& 1\end{array}\right|=0$

$\Rightarrow 6a+12=0$

$\Rightarrow a=-2$