A scalar valued function is defined as $$f\left( x \right){x^T}Ax + {b^T}x + c,$$ where $$A$$ is a symmetric positive definite matrix with dimension $$n \times n;$$ $$b$$ and $$x$$ are vectors of dimension $$n \times 1$$. The minimum value of $$f(x)$$ will occur when $$x$$ equals.

For the matrix $$A$$ satisfying the equation given below, the eigen values are $$$\left[ A \right]\left[ {\matrix{ 1 & 2 & 3 \cr 7 & 8 & 9 \cr 4 & 5 & 6 \cr } } \right] = \left[ {\matrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 8 & 9 \cr } } \right]$$$

Given $$x\left( t \right) = 3\,\sin \,\left( {1000\pi t} \right)\,\,$$ and $$\,\,y\left( t \right) = 5\cos \,\left( {1000\pi t{\pi \over t}} \right)$$
The $$x$$-yplot will be

$$A$$ vector is defined as $$f = y\widehat i + x\widehat j + z\widehat k\,\,$$. Where $$\widehat i,\widehat j,$$ and $$\widehat k$$ are unit vectors in cartesian $$(x, y, z)$$ coordinate system. The surface integral - -over the closed surface $$S$$ of a cube with vertices having the following coordinates: $$(0,0,0), (1, 0, 0), (0, 1, 0), (0,0,1), (1, 0, 1), (1,1,1), (0, 1, 1), (1, 1, 0)$$ is _______.

Given that $$x$$ is a random variable in the range $$\left[ {0,\infty } \right]$$ with a probability density function $${{{e^{ - {x \over 2}}}} \over K},$$ the value of the constant $$K$$ is _______

The figure shown the schematic of a production process with machines $$A, B$$ and $$C.$$ An input job needs to be pre-processed either by $$A$$ or by $$B$$ before it is fed to $$C,$$ from which the final finished product comes out. The probabilities of failure of the machines are given as:
$${P_{\rm A}} = 0.15,\,\,{P_{\rm B}} = 0.05\,\,\& \,\,{P_C} = 0.1$$

Assuming independence of failures of the machines, the probability that a given job is successfully processed (up to the third decimal place) is _____.

The figure shows the plot of $$y$$ as a function of $$x$$

The function shown in the solution of the differential equation (assuming all initial conditions to be zero) is