Determinants Te...

If [x] stands greatest integer $$\leq x$$ then the value of
$$\begin{vmatrix}
\left [ e \right ] & \left [ \pi  \right ] & \left [ \pi ^{2}-6 \right ]\\
\left [ \pi  \right ] & \left [ \pi ^{2}-6 \right ] & \left [ e \right ]\\
\left [ \pi ^{2}-6 \right ] & \left [ e \right ] & \left [ \pi  \right ]
\end{vmatrix}$$ equals to=?

If $$D_k = \begin{vmatrix}1 & n & n\\ 2k & n^2 + n + 1 & n^2 + n\\ 2k-1 & n^2 & n^2 + n + 1\end{vmatrix} $$ and $$\displaystyle \sum_{k = 1}^n D_k = 56 $$ then n equals

If $$\Delta =\begin{vmatrix}
x+1 & x+2 & x+a\\
x+2 & x+3 & x+b\\
x+3 & x+4 & x+c
\end{vmatrix}=0$$, then
the family of lines $$ax+by+c=0$$ passes through

Consider the points $$P=(-\sin (\beta -\alpha ), -\cos \beta )$$, $$Q=(\cos (\beta -\alpha ), \sin \beta )$$ and $$R=(\cos (\beta -\alpha +\theta ), \sin (\beta -\theta ))$$, where $$0< \alpha , \beta < \dfrac{\pi }{4}$$ then 

If the points $$(a, 1), (1, b)$$ and $$(a -1, b -1)$$ are collinear, $$\alpha ,\beta $$ are respectively the arithmetic and geometric means of $$a$$ and $$b $$, then $$4\alpha -\beta^{2}$$ is equal to

If $$A$$ and $$B$$ are two similar matrices, then

If $$\Delta =\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$ and $$c_{ij}=\left ( -1 \right )^{i+j}$$ (determinant obtained by deleting ith row and jth column), then $$\begin{vmatrix} c_{11} & c_{12} & c_{13}\\ c_{21} & c_{22} & c_{23}\\ c_{31} & c_{32} & c_{33} \end{vmatrix}=\Delta ^{2}$$

If $$\begin{vmatrix} 1 & x & x^{ 2 } \\ x & x^{ 2 } & 1 \\ x^{ 2 } & 1 & x \end{vmatrix}=7$$ and $$\Delta =\begin{vmatrix}
x^{3}-1 & 0 & x-x^{4}\\
0 & x-x^{4} & x^{3}-1\\
x-x^{4} & x^{3}-1 & 0
\end{vmatrix}$$, then

The number of distinct real roots of $$\begin{vmatrix} \sin\, x&\cos\, x&\cos \,x \\ \cos\, x &\sin\, x&\cos\, x \\ \cos\, x&\cos\, x&\sin \, x\end{vmatrix}=0$$ in the interval $$-\dfrac{\pi}{4} < x \le \dfrac{\pi}{4}$$ is

If the points $$(-1,3), (2,p)$$ and $$(5,-1)$$ are collinear, the value of $$p$$ is

$$(1,6), (3.-2)$$ and $$(-2,K)$$ are collinear points. What is $$K$$?