The set of all values of K > $$-$$1, for which the equation $${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$$ has real roots, is :

Let $$\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} $$ and $$\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} $$. If $$8{x^2} + bx + c = 0$$ is a quadratic equation whose roots are $$\alpha$$^1/5 and $$\beta$$^1/5, then the value of c $$-$$ b is equal to :

Let $$\alpha$$, $$\beta$$ be two roots of the

equation x² + (20)^1/4x + (5)^1/2 = 0. Then $$\alpha$$⁸ + $$\beta$$⁸ is equal to

If [x] be the greatest integer less than or equal to x,

then $$\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]} $$ is equal to :