Determinants

The equations $2 x+3 y+4=0 ; 3 x+4 y+6=0$ and $4 x+5 y+8=0$ are

The value of ' $a$ ' for which the system of equations $a x+y+z=0, x+a y+z=0, x+y+z=0$ possesses non-zero solutions are given by

The system $x+4 y-2 z=3,3 x+y+5 z=7$, $2 x+3 y+z=5$ has

If $\quad a_1 x+b_1 y+c_1 z=0, \quad a_2 x+b_2 y+c_2 z=0$, $a_3 x+b_3 y+c_3 z=0$ and $\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=0$, then the given system has

The existence of the unique solution of the system $\quad x+y+z=\lambda, \quad 5 x-y+\mu z=10$, $2 x+3 y-z=6$ depends on

Let $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$ be the roots of $a x^2+b x+c=0 \quad$ and $\quad p x^2+q x+r=0$ respectively. If the system of equations $\alpha_1 y+\alpha_2 z=0$ and $\beta_1 y+\beta_2 z=0$ has $a$…

The system of linear equations $x+y+z=0$, $2 x+y-z=0,3 x+2 y=0$ has

In the determinant $\left|\begin{array}{rrr}3 & x & -1 \\ 2 & -1 & 4 \\ 1 & y & -3\end{array}\right|$, the sum of the cofactors of $x$ and $y$ is

If the matrix $A=\left[\begin{array}{ccc}y+a & b & c \\ a & y+b & c \\ a & b & y+c\end{array}\right]$ has rank 3, then

Let $\lambda$ and $\alpha$ be real $\begin{array}{r} \lambda x+\sin \alpha \cdot y+\cos \alpha \cdot z=0 \\ x+\cos \alpha \cdot y+\sin \alpha \cdot z=0 \\ -x+\sin \alpha \cdot y-\cos \alpha \cdot z=0 \end{array}$ has a…

The values of $\lambda$ such that $(x, y, z) \neq(0,0,0)$ and $(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}) x+(3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}) y+(-4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}})…

The value of $\lambda$ and $\mu$ for which the system of equations $x+y+z=6, x+2 y+3 z=10$ and $x+2 y+\lambda z=\mu$ have no solution, are