Determinants Te...

If A = $$\begin{bmatrix}
a & b\\
 c& d
\end{bmatrix}$$ (where $$b\neq c$$) and satisfies the equation $$A^{2}+kI=0$$, then 

If in a $$\displaystyle \Delta ABC; \frac{\cos A}{7}=\frac{\cos B}{19}=\frac{\cos C}{25}=k$$, then $$
\begin{vmatrix}-1/k & 25 & 19\\ 25 & -1/k & 7\\ 19 & 7 & -1/k\end{vmatrix}=$$

STATEMENT 1: In a $$\Delta ABC,\ a,\ b,\ c$$ denotes lengths of the sides and $$\begin{vmatrix}a & b & c\\ b & c & a\\ c & a & b\end{vmatrix}=0$$ then the triangle is equilateral triangle.

STATEMENT 2: Sum of three non-negative numbers $$=0\Rightarrow $$ each number is zero.

If f (x) = tan x and A, B, C are the angles of $$\Delta ABC$$, then $$\begin{vmatrix}
f(A) & f(\pi /4) & f(\pi /4)\\
f(\pi /4) &f(B) & f(\pi /4)\\
f(\pi /4) &f(\pi /4) & f(C)
\end{vmatrix}$$ 

If $$\mathrm{a}\neq b\neq \mathrm{c}$$ and if $$ax+by+\mathrm{c}=0\  bx+cy+\mathrm{a}=0$$ and $$cx+ay+b=0$$ are concurrent, 

If the lines $$2\mathrm{x}-\mathrm{a}\mathrm{y}+1 =0$$,$$\ 3\mathrm{x}-\mathrm{b}\mathrm{y}+1 =0$$,$$\ 4\mathrm{x}-\mathrm{c}\mathrm{y}+1 =0$$ are concurrent then $$a,b,c$$ are in ?

.

Value of the determinant $$\begin{vmatrix}x & y &z \\  p& q &r \\  yz& zx & xy\end{vmatrix}$$ is equal to

lf the lines $$3\mathrm{x}+2\mathrm{y}-5=0,\ 2\mathrm{x}-5\mathrm{y}+3=0,\ 5\mathrm{x}+\mathrm{b}\mathrm{y}+\mathrm{c}=0$$ are concurrent then $$\mathrm{b}+\mathrm{c}=$$

If the points $$A (1, 2), O (0, 0)$$ and $$C (a, b)$$ are collinear, then

Relation between $$x$$ and $$y$$, if the points $$(x, y), (1, 2)$$ and $$(7, 0)$$ are collinear is _____