If $\left|\begin{array}{cc}1-i & i \\ 1+2 i & -i\end{array}\right|=x+i y$,then $x$ is equal to

$A$ block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined at an angle $\theta$ to the vertical. The minimum value of the coefficient of friction between the block and the surface for which the block will remain in equilibrium is:

The number of non-trivial solutions of the system $x-y+z=0, x+2y-z=0, 2x+y+3z=0$ is

If $\frac{x^3 - 6x^2 + 10x - 2}{x^2 - 5x + 6} = f(x) + \frac{A}{(x - 2)} + \frac{B}{(x - 3)}$,then $f(x) =$ ?

The equation of the straight line perpendicular to the straight line $3x + 2y = 0$ and passing through the point of intersection of the lines $x + 3y - 1 = 0$ and $x - 2y + 4 = 0$ is

$A$ block of weight $W$ is held against a vertical wall by applying a horizontal force $F$. The minimum value of $F$ needed to hold the block is