The number of four letter words that can be formed using the letters of the word BARRACK is :

Suppose A is any 3$$ \times $$ 3 non-singular matrix and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I₃ and O = O₃. If $$\alpha $$A + $$\beta $$A^-1 = 4I, then $$\alpha $$ + $$\beta $$ is equal to :

Let f : A $$ \to $$ B be a function defined as f(x) = $${{x - 1} \over {x - 2}},$$ Where A = R $$-$$ {2} and B = R $$-$$ {1}. Then   f   is :

If f(x) is a quadratic expression such that f (1) + f (2) = 0, and $$-$$ 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :