Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M^TM is seven, is ________.

Let B_i (i = 1, 2, 3) be three independent events in a sample space. The probability that only B₁ occur is $$\alpha $$, only B₂ occurs is $$\beta $$ and only B₃ occurs is $$\gamma $$. Let p be the probability that none of the events B_i occurs and these 4 probabilities satisfy the equations $$\left( {\alpha - 2\beta } \right)p = \alpha \beta $$ and $$\left( {\beta - 3\gamma } \right)p = 2\beta \gamma $$ (All the probabilities are assumed to lie in the interval (0, 1)).
Then $${{P\left( {{B_1}} \right)} \over {P\left( {{B_3}} \right)}}$$ is equal to ________.

A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is $$-$$Q. The electric field at the centre of cube is :

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillations will be :