If $a_{1}, a_{2}, a_{3}, \ldots, a_{9}$ are in $AP$,then the value of $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is

If $\Delta_k=\left|\begin{array}{ccc}1 & 0 & 0 \\ 0 & k & k-1 \\ 0 & k-1 & k\end{array}\right|$,then $\Delta_1+\Delta_2+\ldots+\Delta_{20}$ is equal to

If $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right| = 10 k x y z \left(3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$,then $k = \text{ . . . . . . }$ (where $x, y, z \neq 0$ and $3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \neq 0$).

If $a_i, b_i, c_i \in \mathbb{R}$ for $i=1, 2, 3$ and $x \in \mathbb{R}$ and $\begin{vmatrix} a_1+b_1 x & a_1 x+b_1 & c_1 \\ a_2+b_2 x & a_2 x+b_2 & c_2 \\ a_3+b_3 x & a_3 x+b_3 & c_3 \end{vmatrix} = 0$, then:

If one of the roots of $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ is $-10$, then the other roots are

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: