If the system of equations x+2y-3z=2,2x+lambd | vedclass

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

Question

(D) For a system of linear equations to have infinitely many solutions,the determinant $D$ must be $0$,and $D_1 = D_2 = D_3 = 0$.$D = \begin{vmatrix}

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(D) For a system of linear equations to have infinitely many solutions,the determinant $D$ must be $0$,and $D_1 = D_2 = D_3 = 0$.$D = \begin{vmatrix} 1 & 2 & -3 \ 2 & \lambda & 5 \ 14 & 3 & \mu \end{vmatrix} = 1(\lambda\mu - 15) - 2(2\mu - 70) - 3(6 - 14\lambda) = 0$$\lambda\mu - 15 - 4\mu + 140 - 18 + 42\lambda = 0 \Rightarrow \lambda\mu + 42\lambda - 4\mu + 107 = 0$.Now,consider $D_3 = \begin{vmatrix} 1 & 2 & 2 \ 2 & \lambda & 5 \ 14 & 3 & 33 \end{vmatrix} = 1(33\lambda - 15) - 2(66 - 70) + 2(6 - 14\lambda) = 0$$33\lambda - 15 + 8 + 12 - 28\lambda = 0 \Rightarrow 5\lambda + 5 = 0 \Rightarrow \lambda = -1$.Substitute $\lambda = -1$ into the equation for $D = 0$:$(-1)\mu + 42(-1) - 4\mu + 107 = 0 \Rightarrow -5\mu + 65 = 0 \Rightarrow \mu = 13$.Thus,$\lambda + \mu = -1 + 13 = 12$.

If the system of equations $x+2y-3z=2$,$2x+\lambda y+5z=5$,$14x+3y+\mu z=33$ has infinitely many solutions,then $\lambda+\mu$ is equal to:

Similar Questions

If the system of equations $x+y+2z=3$,$x+2y+3z=4$ and $x+y+cz=5$ is inconsistent,then:

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$,and a system of linear equations
$x+y+z=5$
$x+2y+3z=\mu$
$x+3y+\lambda z=1$
is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution,then:

The system of equations ${x_1} + 2{x_2} + 3{x_3} = a$,$2{x_1} + 3{x_2} + {x_3} = b$,and $3{x_1} + {x_2} + 2{x_3} = c$ has:

Find the matrix $X$ such that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module