The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetric positive definite matrix is ____________.

Consider the following system of equations:
3x + 2y = 1
4x + 7z = 1
x + y + z =3
x - 2y + 7z = 0
The number of solutions for this system is ______________________

Let the function
$$f\left( \theta \right) = \left| {\matrix{ {\sin \,\theta } & {\cos \,\theta } & {\tan \,\theta } \cr {\sin \left( {{\pi \over 6}} \right)} & {\cos \left( {{\pi \over 6}} \right)} & {\tan \left( {{\pi \over 6}} \right)} \cr {\sin \left( {{\pi \over 3}} \right)} & {\cos \left( {{\pi \over 3}} \right)} & {\tan \left( {{\pi \over 3}} \right)} \cr } } \right|$$

Where $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ and $$f\left( \theta \right)$$ denote the derivative of $$f$$ with repect to $$\theta $$. Which of the following statements is/are TRUE?

$${\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$= 0$$.
$${\rm I}{\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$ \ne 0$$.

A function $$f(x)$$ is continuous in the interval $$\left[ {0,2} \right]$$. It is known that $$f(0)$$ $$=$$ $$f(2)$$ $$= -1$$ and $$f(1)$$ $$ = 1$$. Which one of the following statements must be true?