GATE EC 2014 Question Paper (15-Feb-2014) (Shift 1)

The Taylor series expansion of 3 sin x + 2 cos x is

For a function g(t), it is given that $\underset{-\infty }{\overset{+\infty }{\int }}g\left(t\right)e^{-j\omega t}dt=\omega e^{-2{\omega }^2}$ for any real value $\omega$.
If, $y\left(t\right)=\underset{-\infty }{\overset{t}{\int }}g\left(\tau \right)d\tau$, then $\underset{-\infty }{\overset{\infty }{\int }}y\left(t\right)dt$ is:

The volume under the surface $\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}+\mathrm{y}$ and above the triangle in the x-y plane defined by $\{0\le \mathrm{y}\le \mathrm{x}\mathrm{and}0\le \mathrm{x}\le 12$ } is _________.

Consider the matrix

$J_6=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right]$

Which is obtained by reversing the order of columns of the Identity matrix I₆.

Let P = I₆ + α J₆, where α is a non-negative real number. The value of α for which det (P) = 0 is _________.

A Y-network has a resistances of 10 Ω each in two of its arms, while the third arm has a resistance of 11 Ω. In the equivalent Δ-network, the lowest value (in Ω) among the three resistance is_______.