GATE IN 2011 Question Paper (13-Feb-2011) (Shift-2)

The matrix $M=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$ has eigenvalues -3, -3, 5. An eigenvector corresponding to the eigenvalue 5 is $[12-1{]}^T$. One of the eigenvectors of the matrix $M^3$ is

The contour integral
$${\oint }_C{e}^{1/z}dz$$
with C as the counter-clockwise unit circle in the z-plane is equal to

Consider the signal
$$x(t)=\{\begin{array}{ll}{e}^{-t}& ,t\ge 0\\ 0& ,t<0\end{array}$$
Let $X(\omega )$ denote the Fourier transform of this signal. The integral
$$\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )d\omega \text{ is }$$

The continuous-time signal $x(t)=\sin {\omega }_{0}t$ is a periodic signal. However, for its discrete-time counterpart $x[n]=\sin {\omega }_{0}n$ to be periodic, the necessary condition is

Consider a periodic signal $x(t)$ as shown below



It has a Fourier series representation $x(t)={\sum }_{k=-\infty }^{\infty }{a}_k{e}^{j(2\pi /T)kt}$ Which one of the following statements is TRUE?