Let $$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ and $$B = \left[ {\matrix{ \alpha \cr \beta \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr } } \right]$$ such that AB = B and a + d = 2021, then the value of ad $$-$$ bc is equal to ___________.

If 1, log₁₀(4^x $$-$$ 2) and log₁₀$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real number x, then the value of the determinant $$\left| {\matrix{ {2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} \cr 1 & 0 & x \cr x & 1 & 0 \cr } } \right|$$ is equal to :

If $$A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]$$, then the value of det(A⁴) + det(A¹⁰ $$-$$ (Adj(2A))¹⁰) is equal to _____________.

Let $$A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$$ and $$B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]$$ be two 2 $$\times$$ 1 matrices with real entries such that A = XB, where

$$X = {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & { - 1} \cr 1 & k \cr } } \right]$$, and k$$\in$$R.

If $$a_1^2$$ + $$a_2^2$$ = $${2 \over 3}$$(b$$_1^2$$ + b$$_2^2$$) and (k² + 1) b$$_2^2$$ $$\ne$$ $$-$$2b₁b₂, then the value of k is __________.