Determinants Te...

Let $$\Delta =$$$$\begin{vmatrix} sin\theta cos \phi & sin\theta sin\phi & cos\theta \\ cos\theta cos\phi & cos\theta sin\phi & -sin\theta \\ -sin\theta sin\phi & sin\theta cos\phi & 0\end{vmatrix}$$, then

$$\begin{vmatrix} 1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b) \end{vmatrix}=$$

If the system of equations of $$3x-2y+z=0, kx-14y+15z=0,x+2y+3z=0$$ has non trivial solution then $$k=$$

$$A=\left[ \begin{matrix} 5 & 5\alpha  & \alpha  \\ 0 & \alpha  & 5\alpha  \\ 0 & 0 & 5 \end{matrix} \right] $$; If $$\left| { A }^{ 2 } \right| =25$$, then $$\left| \alpha  \right| =$$

If $$f(x) = \left|
\begin{array}{111}
x-3 & 2x^2 -18 & 3x^3 -81\\
x-5 & 2x^2-50 & 4x^3-500\\
1 & 2 & 3 \\
\end {array}
\right|
$$ then $$f(1). f(3) + f(3) .f(5) + f(5) .f(1)$$ is equal to-

The points $$({X}_{1},{Y}_{1})$$, $$({X}_{2},{Y}_{2})$$, $$({X}_{1},{Y}_{2})$$ and $$({X}_{2},{Y}_{1})$$ are always

$$\begin{vmatrix}\dfrac{1}{a} &bc &a^2 \\ \dfrac{1}{b} &ca & b^2\\ \dfrac{1}{c} & ab & c^2\end{vmatrix}$$ is equal to -

$$\begin{vmatrix} log e & log e^2 & log e^3\\ log e^2 & log e^3 & log e^4\\ log e^3 & log e^4 & log e^5\end{vmatrix}=$$?

If $$ \begin{vmatrix} \lambda^2 + 3 \lambda & \lambda -1 & \lambda +3 \\ \lambda + 1 & 2- \lambda & \lambda - 4 \\ \lambda-3 & \lambda + 4 & 3 \lambda \end{vmatrix} = p \lambda^4 + q \lambda^3 + r \lambda^2 + s \lambda + t $$ then $$ t = $$

$$\begin{vmatrix} a + x& a - x & a - x\\ a - x & a + x & a - x\\ a - x & a - x & a + x\end{vmatrix} = 0$$ then $$x$$ is