Determinants

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

For $\alpha\,,\beta\,\in\,\mathbb{R}$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha\,z=8$ $3x-y+4z=\beta\,$ has infinitely many solutions. Then $\alpha\,$ and $\beta\,$ are the roots of

If the system of linear equations $2x+3y-z=-2$ $x+y+z=4$ $x-y+\left|\lambda\,\right|z=4\lambda\,-4$where$\lambda\,\in\,\mathbb{R}$, has no solution, then

If the system of linear equations. $8x+y+4z=-2$ $x+y+z=0$ $\lambda\,x-3y=\mu\,$ has infinitely many solutions, then the distance of the point$\left(\lambda\,,\mu\,,-\frac{1}{2}\right)$from the plane$8x+y+4z+2=0$is:

The value of the determinant $\begin{vmatrix} a & b & c \\ l & m & n \\ p & q & r \end{vmatrix}$ is equal to

A train covers the first 5 km of its journey at a speed of 30 $\mathrm{km} / \mathrm{hr}$ and the next 15 km at a speed of $45 \mathrm{~km} / \mathrm{hr}$. What is the average speed of the train?

If $A$ is a square matrix of order $n \gt 1$, then which one of the following is correct?

If $\left|\begin{array}{ccc}8 & -5 & 1 \\ 5 & x & 1 \\ 6 & 3 & 1\end{array}\right|=2$, then what is the value of $x ?$

The determinant of an odd order skew symmetric matrix is always:

What is the value of the minor of the element 9 in the determinant $\left|\begin{array}{ccc}10 & 19 & 2 \\ 0 & 13 & 1 \\ 9 & 24 & 2\end{array}\right|$ ?

If any two adjacent rows or columns of a determinant are interchanged in position, the value of the determinant:

Let A be an $n \times n$ matrix. If $\operatorname{det}(\lambda \mathrm{A})=\lambda^s \operatorname{det}(\mathrm{~A})$, what is the value of $s$ ?