Determinants

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

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?

The system of equations $x-2 y+3 z=5$, $2 x-2 y+z=0,-x+2 y-3 z=6$ has

If the point $P(\alpha, \beta, \gamma)$ lies on the plane $2 x+y+z=1$ and $[\alpha \beta \gamma]\left[\begin{array}{lll}1 & 9 & 1 \\ 8 & 2 & 1 \\ 7 & 3 & 1\end{array}\right]=\left[\begin{array}{lll}0 & 0 &…

Investigate the values of \(\lambda\) and \(\mu\) for the system \(x+2 y+3 z=6, x+3 y+5 z=9\), \(2 x+5 y+\lambda z=\mu\)$ and match the values in List - I with the items in List - II.

If $\alpha$ is a real root of the equation $x^3+6 x^2+5 x-42=0$, then the determinant of the matrix $\left[\begin{array}{ccc}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 &…

If the inverse of $\left[\begin{array}{ccc}-x & 14 x & 7 x \\ 0 & 1 & 0 \\ x & -4 x & -2 x\end{array}\right]$ is $\left[\begin{array}{ccc}2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1\end{array}\right]$, then…

A value of $\theta$ lying between 0 and $\frac{\pi}{2}$ and satisfying $|$\begin{array}{ccc} 1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta &…

$A=\left[\begin{array}{ccc}0 & k & k \\ k & -4 & -6 \\ k & -3 & -5\end{array}\right]$ is a singular matrix for

$\left|\begin{array}{ll}2 & 1 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1 & 1 / 3 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1 / 2 & 1 / 9 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1 / 4 & 1 / 27…

Match the following elements of $\left[\begin{array}{ccc}1 & -1 & 0 \\ 0 & 4 & 2 \\ 3 & -4 & 6\end{array}\right]$ with their co-factors and choose the correct answer. $\begin{array}{lcll} & \text { Element } & & \text {…

The values of $x$ for which the given matrix $\left[\begin{array}{ccc}-x & x & 2 \\ 2 & x & -x \\ x & -2 & -2\end{array}\right]$ will be non-singular are