GATE EC 2015 Question Paper (31-Jan-2015) (Shift 2)

The bilateral Laplace transform of a function ≤ft( ) = ≤ft\{ {*20c {1if a ≤ ≤ }\\ {0otherwise} } .

The value of x for which all the Eigen–values of the matrix given below are real is

$\left[\begin{array}{ccc}10& 5+\mathrm{j}& 4\\ \mathrm{x}& 20& 2\\ 4& 2& -10\end{array}\right]$

Let $\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{az}+\mathrm{b}}{\mathrm{cz}+\mathrm{d}}.$ If $\mathrm{f}\left({\mathrm{z}}_1\right)=\mathrm{f}({\mathrm{z}}_2)$ for all ${\mathrm{z}}_1\ne {\mathrm{z}}_2,\mathrm{a}=2,\mathrm{b}=4\mathrm{and}\mathrm{c}=5$, then d should be equal to __________.

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1+\cos 2\mathrm{y}}{1-\cos 2\mathrm{x}}$ is

The magnitude and phase of the complex Fourier series coefficients ${\mathrm{a}}_{\mathrm{K}}$ of a periodic signal $\mathrm{x}\left(\mathrm{t}\right)$ are shown in the figure. Choose the correct statement form the four choices given. Notation: C is the set of complex numbers, R is the set of purely real numbers, and P is the set purely imaginary numbers.