If $a, b, c$ are respectively the $p^{th}, q^{th}, r^{th}$ terms of an $A.P.$,then $\left| \begin{array}{ccc} a & p & 1 \\ b & q & 1 \\ c & r & 1 \end{array} \right| = $

Find the value of $x$,if $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2x & 5\end{array}\right|$.

The number of real values of $t$ such that the system of homogeneous equations
$\begin{aligned}
t x+(t+1) y+(t-1) z &=0 \\
(t+1) x+t y+(t+2) z &=0 \\
(t-1) x+(t+2) y+t z &=0
\end{aligned}$
has non-trivial solutions is

The number of values of $\lambda$ for which the points $(\lambda + 1, 1)$,$(2\lambda + 1, 3)$,and $(2\lambda + 2, 2\lambda)$ are collinear is:

Evaluate $\left|\begin{array}{rr}2 & 4 \\ -1 & 2\end{array}\right|$

Let $A = \begin{bmatrix} [x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4] \end{bmatrix}$,where $[t]$ denotes the greatest integer less than or equal to $t$. If $\operatorname{det}(A) = 192$,then the set of values of $x$ is the interval: