Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is

The state equation of a second order system is

$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.

Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and $$v_1$$ and $$v_2$$ are the corresponding eigenvectors. For constants $$\alpha_1$$ and $$\alpha_2$$, the solution, $$x(t)$$, of the state equation is

The value of the integral $$\int\limits\!\!\!\int_R {xy\,dx\,dy} $$ over the regioin R, given in the figure, is _________ (rounded off to the nearest integer).

A series $$RLC$$ circuit has a quality factor Q of 1000 at a center frequency of $$10^6$$ rad/s. The possible values of R, L and C are