Let lambda be a real number for which the sys | vedclass

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

Question

(A) For a system of linear equations to have infinitely many solutions,the determinant of the coefficient matrix $D$ must be $0$,and the augmented mat

Vedclass · Accepted Answer

$\lambda^2 - \lambda - 6 = 0$

Vedclass · Answer

(A) For a system of linear equations to have infinitely many solutions,the determinant of the coefficient matrix $D$ must be $0$,and the augmented matrix must satisfy the condition for consistency.The coefficient matrix is:$D = \begin{vmatrix} 1 & 1 & 1 \ 4 & \lambda & -\lambda \ 3 & 2 & -4 \end{vmatrix}$Expanding along the first row:$D = 1(-4\lambda - (-2\lambda)) - 1(-16 - (-3\lambda)) + 1(8 - 3\lambda) = 0$$D = (-4\lambda + 2\lambda) - (-16 + 3\lambda) + (8 - 3\lambda) = 0$$-2\lambda + 16 - 3\lambda + 8 - 3\lambda = 0$$-8\lambda + 24 = 0 \Rightarrow \lambda = 3$Now,check the options to see which quadratic equation has $\lambda = 3$ as a root:For option $A$: $\lambda^2 - \lambda - 6 = 0 \Rightarrow (\lambda - 3)(\lambda + 2) = 0$. Here,$\lambda = 3$ is a root.Thus,the correct option is $A$.

Let $\lambda$ be a real number for which the system of linear equations $x + y + z = 6$,$4x + \lambda y - \lambda z = \lambda - 2$,and $3x + 2y - 4z = -5$ has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation:

Similar Questions

If the system of equations $2x - y + z = 4$,$5x + \lambda y + 3z = 12$,and $100x - 47y + \mu z = 212$ has infinitely many solutions,then $\mu - 2\lambda$ is equal to

Consider the following system of equations in matrix form $\begin{bmatrix} 1 \\ 2 \\ \lambda \end{bmatrix} \begin{bmatrix} 1 & 2 & \lambda \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = 0$. Then which one of the following statements is true?

If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$,where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$,then $|x + y + z|$ is equal to:

If the system of linear equations
$7x + 11y + \alpha z = 13$
$5x + 4y + 7z = \beta$
$175x + 194y + 57z = 361$
has infinitely many solutions,then $\alpha + \beta + 2$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module