GATE EC 2018 Question Paper (10-Feb-2018) (Shift 1)

Let Mbe a real 4 × 4 matrix. Consider the following statements:

S1: M has 4 linearly independent eigenvectors.

S2:M has 4 distinct eigenvalues.

S3:Mis non-singular (invertible).

Which one among the following is TRUE?

Let $\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{a}{\mathrm{x}}^2+\mathrm{b}{\mathrm{y}}^2}{\mathrm{xy}}$, where a and b are constants. If $\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\frac{\partial \mathrm{f}}{\partial \mathrm{y}}$ at x = 1 and y = 2, then the relation between a and b is

A discrete-time all-pass system has two of its poles at 0.25∠0° and 2∠30°. Which one of the following statement about the system is TRUE?

Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by aₖ, that is:

$x\left(t\right)=\sum _{k=-\infty }^{\infty }{a}_k{e}^{jk\frac{2\pi }{T}t}$

The same function x(t) can also be considered as a periodic function with period T′ = 40. Let $b_k$ be the Fourier series coefficients when the period is taken as T’. If $\sum _{\mathrm{k}=-\infty }^{\infty }\left|{\mathrm{a}}_{\mathrm{k}}\right|=16$, then $\sum _{\mathrm{k}=-\infty }^{\infty }\left|{\mathrm{b}}_{\mathrm{k}}\right|$ is equal to

Consider the following amplitude modulated signal: s(𝑡) = cos(2000 𝜋𝑡) + 4 cos(2400 𝜋𝑡) + cos(2800 𝜋𝑡).The ratio (accurate to three decimal places) of the power of the message signal to the power of the carrier signal is ________.