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  4. Product of all real values of ' b ' such that there is no solution to the system of equations 2 x+5 y+ z=19,-4 x+ b y+6 z=-42,-3 y-b z=81 is

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Q. Product of all real values of ' $b$ ' such that there is no solution to the system of equations $2 x+5 y+ z=19,-4 x+ b y+6 z=-42,-3 y-b z=81$ is

AP EAMCETAP EAMCET 2020

Solution:

For no solution of given linear Equations value of given determinant is zero by Cramier's rule,
So, $D=\begin{vmatrix} 2 & 5 & 1 \\ -4 & b & 6 \\ 0 & -3 & -b \end{vmatrix}=0$
$\Rightarrow 2\left(-b^{2}+18\right)-5(4 b)+1(12)=0$
$\Rightarrow -2 b^{2}+36-20 b+12=0$
$\Rightarrow -2 b^{2}-20 b+48=0$
$\Rightarrow b^{2}+20 b-24=0$
Product of roots of above equations is,
Product $=\frac{c}{a}=\frac{-24}{1}=-24$