Complete combustion of 3g of ethane gives x $$\times$$ 10²² molecules of water. The value of x is __________. (Round off to the Nearest Integer). [Use : N_A = 6.023 $$\times$$ 10²³; Atomic masses in u : C : 12.0; O : 16.0; H : 1.0]

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

The solutions of the equation $$\left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr {{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr {4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr } } \right| = 0,(0 < x < \pi )$$, are

Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system of equations (in u, v, w) given by $$\alpha$$u + $$\beta$$v + $$\gamma$$w = 0, $$\beta$$u + $$\gamma$$v + $$\alpha$$w = 0; $$\gamma$$u + $$\alpha$$v + $$\beta$$w = 0 has non-trivial solution, then the value of $${{{a^2}} \over b}$$ is