GATE EC 2016 Question Paper (30-Jan-2016) (Shift 1)

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor

(C) elements, connected all in series or all in parallel, is excited with a source of the form

$\sum _{\mathrm{k}=1}^3{\mathrm{a}}_{\mathrm{k}}\cos \left(\mathrm{k}{\omega }_0\mathrm{t}\right),$ where ${\mathrm{a}}_{\mathrm{k}}\ne 0,{\omega }_{\mathrm{o}}\ne 0$

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for t ≥ 0:

X: Impulse P: 1 − 𝑒${}^{-\text{t}/\text{T}}$

Y: Unit step Q: t − T(1 − 𝑒${}^{-\text{t}/\text{T}}$)

Z: Ramp R: 𝑒${}^{-\text{t}/\text{T}}$

An AC voltage source V = 10 sin(t) volts is applied to the following network. Assume that R₁ = 3 kΩ, R₂= 6 kΩ and R₃= 9 kΩ, and that the diode is ideal.



RMS current I${}_{rms}$ (in mA) through the diode is ________

In the circuit shown in the figure, the maximum power (in watt) delivered to the resistor R is __________

Consider the signal

$\mathrm{x}[\mathrm{n}]=6\delta [\mathrm{n}+2]+3\delta [\mathrm{n}+1]+8\delta [\mathrm{n}]+7\delta [\mathrm{n}-1]+4\delta [\mathrm{n}-2]$ .

If $\mathrm{X}({\mathrm{e}}^{\mathrm{j}\omega })$ is the discrete-time Fourier transform of $\mathrm{x}[\mathrm{n}]$,

then $\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}\mathrm{X}\left({\mathrm{e}}^{\mathrm{j}\omega }\right){\sin }^2\left(2\omega \right)\mathrm{d}\omega$ is equal to _________.