Determinants Te...

If $$A = \begin{bmatrix} \dfrac {-1 + i\sqrt {3}}{2i}& \dfrac {-1 - i\sqrt {3}}{2i}\\ \dfrac {1 + i\sqrt {3}}{2i} & \dfrac {1 - i\sqrt {3}}{2i}\end{bmatrix}, i = \sqrt {-i}$$ and $$f(x) = x^{2} + 2$$, then $$f(A)$$ is equal to

$$\begin{vmatrix} { \left( { a }^{ x }+{ a }^{ -x } \right)  }^{ 2 } & { \left( { a }^{ x }-{ a }^{ -x } \right)  }^{ 2 } & 1 \\ { \left( b^{ x }+{ b }^{ -x } \right)  }^{ 2 } & { \left( { b }^{ x }-{ b }^{ -x } \right)  }^{ 2 } & 1 \\ { \left( { c }^{ x }+{ c }^{ -x } \right)  }^{ 2 } & { \left( { c }^{ x }-{ c }^{ -x } \right)  }^{ 2 } & 1 \end{vmatrix}$$ is equal to

If $$x=cy+bz,y=az+cx,z=bx+ay$$, where $$x,y,z$$ are not all zero, then the value of $${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+2abc$$

If the vectors $$\vec {a}, \vec {b}, \vec {c}$$ are coplanar, then the value of $$\begin{vmatrix}\vec {a}& \vec {b} & \vec {c}\\ \vec {a}.\vec {a} & \vec {a}.\vec {b} & \vec {a}.\vec {c}\\ \vec {b}.\vec {a} & \vec {b}.\vec {b} & \vec {b} . \vec {c}\end{vmatrix} =$$

Let $$A^{-1}\begin{bmatrix} 1 & 2017 & 2\\ 1 & 2017 & 4 \\ 1 & 2018 & 8\end {bmatrix}$$. Then $$|2A|-|2A^{-1}|$$ is equal to.

If $$A, B, C$$ are collinear points such that $$A(3, 4), C(11, 10)$$ and $$AB = 2.5$$ then point $$B$$ is

Let $$z = \begin{vmatrix} 1& 1 + 2i & -5i\\ 1 - 2i & -3 & 5 + 3i\\ 5i & 5 - 3i & 7\end{vmatrix}$$, then

If $$A = \begin{bmatrix} 2& -3\\ 4 & 1\end{bmatrix}$$, then adjoint of matrix $$A$$ is _______.

Three distinct points A, B and C are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point $$(1, 0)$$ to the distance from the point $$(-1, 0)$$ is equal to $$\displaystyle \frac{1}{3}$$. Then the circumcentre of the triangle ABC is at the point:

The number of values of k for which the system of linear equations, $$(2k+1)x+5ky=k+2$$ and $$kx+(k+2)y=2$$ has no solution, is: