Let a, lambda, mu in mathbb{R}. Consider the | vedclass

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

Question

(C) The given system of equations is:$ax + 2y = \lambda$$3x - 2y = \mu$The determinant of the coefficient matrix is:$\Delta = \begin{vmatrix} a & 2 \

Vedclass · Accepted Answer

$B, C, D$

Vedclass · Answer

(C) The given system of equations is:$ax + 2y = \lambda$$3x - 2y = \mu$The determinant of the coefficient matrix is:$\Delta = \begin{vmatrix} a & 2 \ 3 & -2 \end{vmatrix} = -2a - 6 = -2(a + 3)$.For a unique solution,$\Delta 
eq 0$,which implies $a 
eq -3$. Thus,statement $(B)$ is correct.If $a = -3$,then $\Delta = 0$. The system will have either no solution or infinitely many solutions.We calculate $\Delta_1 = \begin{vmatrix} \lambda & 2 \ \mu & -2 \end{vmatrix} = -2\lambda - 2\mu = -2(\lambda + \mu)$.If $\lambda + \mu = 0$,then $\Delta_1 = 0$. Since $\Delta = 0$ and $\Delta_1 = 0$,the system has infinitely many solutions. Thus,statement $(C)$ is correct.If $\lambda + \mu 
eq 0$,then $\Delta_1 
eq 0$. Since $\Delta = 0$ and $\Delta_1 
eq 0$,the system has no solution. Thus,statement $(D)$ is correct.Therefore,statements $(B)$,$(C)$,and $(D)$ are correct.

Similar Questions

For the equations $x+2y+3z=1$,$2x+y+3z=2$,and $5x+5y+9z=4$,which of the following is true?

If the set of equations $x+2y+3z=6$,$x+3y+5z=9$,and $2x+5y+az=b$ has a unique solution,then:

Solve the system of equations $2x + 5y = 1$ and $3x + 2y = 7$.

If the system of simultaneous linear equations $x+\lambda y-2 z=1$,$x-y+\lambda z=2$,and $x-2 y+3 z=3$ is inconsistent for $\lambda=\lambda_1$ and $\lambda_2$,then $\lambda_1+\lambda_2=$

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?

Vedclass Test Series

Exam Paper Generator

Online Exam Module