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 the system of linear equations $x + ay + z = 3$,$x + 2y + 2z = 6$,and $x + 5y + 3z = b$ has no solution,then:

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4 \end{bmatrix}$,$B = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$ and $X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$ such that $AX = B$,then the value of $x_1 + x_2 + x_3$ is:

Let $A = \begin{bmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{bmatrix}$ and $B = \begin{bmatrix} a \\ -b \\ -c \end{bmatrix}$. If $A$ and $[A: B]$ have the same rank, then:

Let $A=\begin{bmatrix} 0 \\ -6 \\ 8 \end{bmatrix}$,$B=\begin{bmatrix} 3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0 \end{bmatrix}$ and $X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $D=[\alpha, \beta, \gamma]^{T}$ is the solution of $X^{T} B^{T}=A^{T}$,then $D^{T} A=$

If the system of linear equations $x-2y+z=-4$; $2x+\alpha y+3z=5$; $3x-y+\beta z=3$ has infinitely many solutions,then $12\alpha+13\beta$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module