GATE EC 2021 Question Paper (07-Feb-2021) (Shift 2)

A real 2 × 2 non-singular matrix A with repeated eigenvalue is given as

A = ≤ft[ {*20c x& - 3.0\\ 3.0&4.0 } ]

where x is a real positive number. The value of x (rounded off to one decimal place) is _______

For a vector field D = ρ cos² ϕ a${}_{\rho }$ + z² sin² ϕ a${}_{\text{ϕ}}$ in a cylindrical coordinate system (ρ, ϕ, z) with unit vectors a${}_{\rho }$, a${}_{\text{ϕ}}$ and a${}_z$, the net flux of D leaving the closed surface of the cylinder (ρ = 3, 0 ≤ z ≤ 2) (rounded off to two decimal places) is ______.

In the circuit shown in the figure, the switch is closed at time t = 0, while the capacitor is initially charged to - 5 V (i.e. vc(0) = -5V)



The time after which the voltage across the capacitor becomes zero (rounded off to three decimal places) is ______ ms.

The exponential Fourier series representation of a continuous-time periodic signal x(t) is defined as:

$x(t)=\sum _{k=-\infty }^{\infty }{a}_k{e}^{jk{\omega }_{0}t}$

Where ω₀ is the fundamental angular frequency of x(t) and the coefficients of the series are aₖ. The following information is given about x(t) and aₖ.

I. x(t) is real and even having a fundamental period of 6

II. The average value of x(t) is 2

III. $a_k=\{\begin{array}{c}k,1\le k\le 3\\ 0,k>3\end{array}$

The average power of the signal x(t) (rounded off to one decimal place) is _______.

For a unit step input u[n], a discrete-time LTI system produces an output signal (2δ[n + 1] + δ[n] + δ[n - 1]). Let y[n] be the output of the system for an input $\left({\left(\frac{1}{2}\right)}^{n}u[n]\right)$. The value of y[0] is _____