Consider the system of linear equations:-x+y+ | vedclass

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

Question

(C) The determinant of the coefficient matrix is $\Delta = \begin{vmatrix} -1 & 1 & 2 \ 3 & -a & 5 \ 2 & -2 & -a \end{vmatrix}$.Expanding along the

Vedclass · Accepted Answer

$n(S_{1})=2, n(S_{2})=0$

Vedclass · Answer

(C) The determinant of the coefficient matrix is $\Delta = \begin{vmatrix} -1 & 1 & 2 \ 3 & -a & 5 \ 2 & -2 & -a \end{vmatrix}$.Expanding along the first row:$\Delta = -1(a^2 + 10) - 1(-3a - 10) + 2(-6 + 2a)$$= -a^2 - 10 + 3a + 10 - 12 + 4a = -a^2 + 7a - 12 = -(a-3)(a-4)$.For the system to be inconsistent or have infinitely many solutions,we must have $\Delta = 0$,which gives $a = 3$ or $a = 4$.Now,calculate $\Delta_1 = \begin{vmatrix} 0 & 1 & 2 \ 1 & -a & 5 \ 7 & -2 & -a \end{vmatrix}$.$\Delta_1 = 0(a^2 + 10) - 1(-a - 35) + 2(-2 + 7a) = a + 35 - 4 + 14a = 15a + 31$.For $a = 3$,$\Delta_1 = 15(3) + 31 = 76 
eq 0$.For $a = 4$,$\Delta_1 = 15(4) + 31 = 91 
eq 0$.Since $\Delta = 0$ and $\Delta_1 
eq 0$ for both $a=3$ and $a=4$,the system is inconsistent for these values. Thus,$S_1 = \{3, 4\}$ and $n(S_1) = 2$.For infinitely many solutions,we require $\Delta = 0$ and $\Delta_1 = \Delta_2 = \Delta_3 = 0$. Since $\Delta_1 
eq 0$ for $a=3$ and $a=4$,there are no values of $a$ for which the system has infinitely many solutions. Thus,$S_2 = \emptyset$ and $n(S_2) = 0$.

Similar Questions

If $\begin{bmatrix} 3 & 1 \\ 4 & 1 \end{bmatrix} X = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$,then $X =$

Solve the following system of equations by matrix method: $3x - 2y + 3z = 8$,$2x + y - z = 1$,$4x - 3y + 2z = 4$.

The system of equations $3x + 2y + z = 6$,$3x + 4y + 3z = 14$ and $6x + 10y + 8z = a$ has an infinite number of solutions if $a$ is equal to

The number of values of $\alpha$ for which the system of equations: $x+y+z=\alpha$,$\alpha x+2 \alpha y+3 z=-1$,and $x+3 \alpha y+5 z=4$ is inconsistent,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module