The product of all real values of b such that | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For a system of linear equations to have no solution,the determinant of the coefficient matrix $D$ must be zero,and at least one of the Cramer's r

Vedclass · Accepted Answer

-$24$

Vedclass · Answer

(C) For a system of linear equations to have no solution,the determinant of the coefficient matrix $D$ must be zero,and at least one of the Cramer's rule determinants $(D_x, D_y, D_z)$ must be non-zero.First,we calculate the determinant $D$:$D = \begin{vmatrix} 2 & 5 & 1 \ -4 & b & 6 \ 0 & -3 & -b \end{vmatrix} = 0$$2(-b^2 + 18) - 5(4b - 0) + 1(12 - 0) = 0$$-2b^2 + 36 - 20b + 12 = 0$$-2b^2 - 20b + 48 = 0$$b^2 + 10b - 24 = 0$Factoring the quadratic equation: $(b + 12)(b - 2) = 0$. The roots are $b = -12$ and $b = 2$.Checking for $b = 2$: The system becomes $2x + 5y + z = 19$,$-4x + 2y + 6z = -42$,and $-3y - 2z = 81$. Solving this shows the system is inconsistent (no solution).Checking for $b = -12$: The system becomes $2x + 5y + z = 19$,$-4x - 12y + 6z = -42$,and $-3y + 12z = 81$. Solving this also shows the system is inconsistent (no solution).The product of these real values is $(-12) 	imes (2) = -24$.

The product of all real values of $b$ such that there is no solution to the system of equations $2x + 5y + z = 19$,$-4x + by + 6z = -42$,and $-3y - bz = 81$ is:

Similar Questions

$A, C$ are $3 \times 3$ matrices. $B, D$ are $3 \times 1$ matrices. If $AX=B$ has a unique solution and $CX=D$ has an infinite number of solutions, then:

If $\frac{5}{m}+\frac{2}{n}=9$ and $\frac{3}{m}+\frac{4}{n}=11$ and $mn \neq 0$,then the values of $m$ and $n$ are . . . . . . respectively.

The number of solutions of the equations $x + y - z = 0$,$3x - y - z = 0$,and $x - 3y + z = 0$ is

For the system $x-y+z=4, 2x+y-3z=0, x+y+z=2$,the values of $x, y, z$ respectively are given by

Solve the system of linear equations using the matrix method: $5x + 2y = 4$ and $7x + 3y = 5$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module