If $A = \begin{bmatrix} 5x & 10 \\ 8 & 7 \end{bmatrix}$ and $|A| = 25$,then $x = $ . . . . . . .

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:

If $\omega$ is an imaginary root of unity,then the value of $\left| \begin{array}{ccc} a & b\omega^2 & a\omega \\ b\omega & c & b\omega^2 \\ c\omega^2 & a\omega & c \end{array} \right|$ is

If the system of homogeneous equations $\begin{aligned} & t x+(t+1) y+(t-1) z=0 \\ & (t+1) x+t y+(t+2) z=0 \\ & (t-1) x+(t+2) y+t z=0\end{aligned}$ in $x, y, z$ has a non-trivial solution,then $t$ is a root of the equation

If $\left| \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} \right| = k(a + b + c)(a^2 + b^2 + c^2 - bc - ca - ab)$,then $k =$

Let $\theta \in \left(0, \frac{\pi}{2}\right)$. If the system of linear equations
$(1+\cos^2 \theta) x + \sin^2 \theta y + 4 \sin 3\theta z = 0$
$\cos^2 \theta x + (1+\sin^2 \theta) y + 4 \sin 3\theta z = 0$
$\cos^2 \theta x + \sin^2 \theta y + (1+4 \sin 3\theta) z = 0$
has a non-trivial solution,then the value of $\theta$ is: