Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be $$\left( {{{10} \over 3},{7 \over 3}} \right)$$. If $$\alpha$$, $$\beta$$ are the roots of the equation $$a{x^2} + bx + 1 = 0$$, then the value of $${\alpha ^2} + {\beta ^2} - \alpha \beta $$ is :

Let a, b$$ \in $$R. If the mirror image of the point P(a, 6, 9) with respect to the line

$${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$$ is (20, b, $$-$$a$$-$$9), then | a + b |, is equal to :

For the system of linear equations:

$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,

consider the following statements :

(A) The system has unique solution if $$k \ne 2,k \ne - 2$$.

(B) The system has unique solution if k = $$-$$2

(C) The system has unique solution if k = 2

(D) The system has no solution if k = 2

(E) The system has infinite number of solutions if k $$ \ne $$ $$-$$2.

Which of the following statements are correct?

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $$-$$ x) for all x$$ \in $$ (0, 2), f(0) = 1 and f(2) = e². Then the value of $$\int\limits_0^2 {f(x)} dx$$ is :