Given the following statements about a function \(f:R\rightarrow R\), select the right option:

P: If \(f(x)\) is continuous at \(x=x_0\), then it is also differentiable at \(x=x_0\)

Q: If \(f(x)\) is continuous at \(x=x_0\), then it may not be differentiable at \(x=x_0\)

R: If \(f(x)\) is differentiable at \(x=x_0\), then it is also continuous at \(x=x_0\)

Consider the plot of f(x) versus x as shown below.

Consider a two-port network with the transmission matrix: \({\rm{T}} = \left( {\begin{array}{*{20}{c}} {\rm{A}}&{\rm{B}}\\ {\rm{C}}&{\rm{D}} \end{array}} \right).\) If the network is reciprocal, then

A small percentage of impurity is added to an intrinsic semiconductor at 300 K. Which one of the following statements is true for the energy band diagram shown in the following figure?

Consider the following statements for a metal oxide semiconductor field effect transistor

(MOSFET):

P: As channel length reduces, OFF-state current increases.

Q: As channel length reduces, output resistance increases.

R: As channel length reduces, threshold voltage remains constant.

S: As channel length reduces, ON current increases.

Which of the above statements are INCORRECT?

Consider the constant current source shown in the figure below. Let 𝛽 represent the current gain of the transistor.

The load current 𝐼₀ through R_L is

The following signal V_i of peak voltage 8 V is applied to the non-inverting terminal of an ideal opamp. The transistor has V_BE = 0.7 V, β = 100; V_LED = 1.5 V, V_CC = 10 V and −V_CC = −10 V.

The number of times the LED glows is ________

Consider the oscillator circuit shown in the figure. The function of the network (shown in dotted lines) consisting of the 100 kΩ resistor in series with the two diodes connected back-to-back is to:

The block diagram of a frequency synthesizer consisting of a Phase Locked Loop (PLL) and a divide-by-𝑁 counter (comprising ÷ 2 ,÷ 4, ÷ 8, ÷ 16 outputs) is sketched below. The synthesizer is excited with a 5 kHz signal (Input 1). The free-running frequency of the PLL is set to 20 kHz. Assume that the commutator switch makes contacts repeatedly in the order 1-2-3-4.

The corresponding frequencies synthesized are:

The output of the combinational circuit given below is

What is the voltage V_out in the following circuit?

Match the inferences X, Y, and Z, about a system, to the corresponding properties of the elements of first column in Routh’s Table of the system characteristic equation.

X: The system is stable …

Y: The system is unstable …

Z: The test breaks down …

P: … when all elements are positive

Q: … when any one element is zero

R: … when there is a change in sign of coefficients

A super heterodyne receiver operates in the frequency range of 58 MHz − 68 MHz. The

The integral \(\frac{1}{{2\pi }}\mathop \int\!\!\!\int \limits_{D\;}^\; \left( {x + y + 10} \right)dxdy\), where D denotes the disc 𝑥² + 𝑦² ≤ 4, evaluates to__________.

A sequence x[n] is specified as \(\left[ {\begin{array}{*{20}{c}} {{\rm{x}}\left[ {\rm{n}} \right]}\\ {{\rm{x}}\left[ {{\rm{n}} - 1} \right]} \end{array}} \right] = {\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&0 \end{array}} \right]^{\rm{n}}}\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right],{\rm{\;for\;n}} \ge 2.\)

In the following integral, the contour C encloses the points 2πj and −2πj

\(- \frac{1}{{2{\rm{\pi }}}}\mathop \oint \limits_{\rm{c}} \frac{{\sin {\rm{z}}}}{{{{\left( {{\rm{z}} - 2{\rm{\pi j}}} \right)}^3}}}{\rm{dz}}\)

The Laplace transform of the causal periodic square wave of period T shown in the figure below is

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

\(\mathop \sum \limits_{{\rm{k}} = 1}^3 {{\rm{a}}_{\rm{k}}}\cos \left( {{\rm{k}}{{\rm{\omega }}_0}{\rm{t}}} \right),\) where \(\rm a_k ≠ 0, ω_o ≠ 0\)

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^−𝑡/𝑇

Y: Unit step     Q: t − T(1 − 𝑒^−𝑡/𝑇)

Z: Ramp          R: 𝑒^−𝑡/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

\(\rm x[n] = 6 \delta[n + 2] + 3\delta[n + 1] + 8\delta[n] + 7\delta[n − 1] + 4\delta[n − 2]\) .

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

then \(\frac{1}{{\rm{\pi }}}\mathop \smallint \limits_{ - {\rm{\pi }}}^{\rm{\pi }} {\rm{X}}\left( {{{\rm{e}}^{{\rm{j\omega }}}}} \right){\sin ^2}\left( {2{\rm{\omega }}} \right){\rm{d\omega \;}}\) is equal to _________.

Consider a silicon p-n junction with a uniform acceptor doping concentration of 10¹⁷ cm⁻³ on the p side and a uniform donor doping concentration of 10¹⁶ cm−3 on the n-side. No external voltage is applied to the diode. Given: kT/_q = 26 mV, n_i =1.5 ×10¹⁰ cm⁻³, ε_Si = 12ε₀, ε₀ = 8.85 × 10⁻¹⁴ F/m, and q = 1.6 ×10⁻¹⁹ C. The charge per unit junction area (nC cm⁻²) in the depletion region on the p-side is ___________.

The figure below shows the doping distribution in a p-type semiconductor in log scale.

The magnitude of the electric field (in kV/cm) in the semiconductor due to non-uniform

doping is _________.

Consider a silicon sample at T = 300 K, with a uniform donor density 𝑁_𝑑 = 5 × 10¹⁶

cm⁻³, illuminated uniformly such that the optical generation rate is 𝐺_𝑜𝑝𝑡= 1.5 × 10²⁰ cm⁻³𝑠⁻¹ throughout the sample. The incident radiation is turned off at 𝑡 = 0. Assume low-level injection to be valid and ignore surface effects. The carrier lifetimes are 𝜏_𝑝0= 0.1 μs and 𝜏_𝑛0 = 0.5 μs.

The hole concentration at 𝑡 = 0 and the hole concentration at 𝑡 = 0.3 μs, respectively, are

An ideal opamp has voltage sources V₁, V₃, V₅, …, V_N-1 connected to the non-inverting input and V₂, V₄, V₆, …, V_N connected to the inverting input as shown in the figure below (+V_CC = 15 volt, −V_CC = −15 volt). The voltages V₁, V₂, V₃, V₄, V₅, V₆,… are 1, − 1/2, 1/3, −1/4, 1/5, −1/6, … volt, respectively. As N approaches infinity, the output voltage (in volt) is ___________

A p-i-n photodiode of responsivity 0.8A/W is connected to the inverting input of an ideal opamp as shown in the figure, +V_cc = 15 V, −V_cc = −15V, Load resistor R_L = 10 kΩ. If 10 μW of power is incident on the photodiode, then the value of the photocurrent (in μA) through the load is ________

Identify the circuit below.

The functionality implemented by the circuit below is

The open-loop transfer function of a unity-feedback control system is

\({\rm{G}}\left( {\rm{S}} \right) = \frac{{\rm{K}}}{{{{\rm{s}}^2} + 5{\rm{s}} + 5}}\)

The open-loop transfer function of a unity-feedback control system is given by

\({\rm{G}}\left( {\rm{s}} \right) = \frac{{\rm{K}}}{{{\rm{s}}\left( {{\rm{s}} + 2} \right)}}\)

The transfer function of a linear time invariant system is given by

\(\rm H(s) = 2s^4 − 5s^3 + 5s +2\)

The current density in a medium is given by

\({\rm{\vec J}} = \frac{{400{\rm{sin\theta }}}}{{2{\rm{\pi }}\left( {{{\rm{r}}^2} + 4} \right)}}{{\rm{\hat a}}_{\rm{r}}}{\rm{A}}{{\rm{m}}^{ - 2}}\)

The total current and the average current density flowing through the portion of a spherical surface \({\rm{r}} = 0.8{\rm{\;m}},\frac{{\rm{\pi }}}{{12}}{\rm{\;}} \le {\rm{\;\theta \;}} \le \frac{{\rm{\pi }}}{4},{\rm{\;}}0{\rm{\;}} \le {\rm{\;}}\phi {\rm{\;}} \le {\rm{\;}}2{\rm{\pi }}\) are given, respectively, by

The electric field of a uniform plane wave traveling along the negative z-direction is given by the following equation:

\({\rm{\vec E}}_{\rm{w}}^{\rm{i}} = \left( {{{{\rm{\hat a}}}_{\rm{x}}} + {\rm{j}}{{{\rm{\hat a}}}_{\rm{y}}}} \right){{\rm{E}}_0}{{\rm{e}}^{{\rm{jkz}}}}\)

This wave is incident upon a receiving antenna placed at the origin and whose radiated electric field towards the incident wave is given by the following equation:

\({\rm{\vec E}}_{\rm{a}}^{\rm{i}} = \left( {{{{\rm{\hat a}}}_{\rm{x}}} + {\rm{j}}{{{\rm{\hat a}}}_{\rm{y}}}} \right){{\rm{E}}_{\rm{I}}}\frac{1}{{\rm{r}}}{\rm{\;}}{{\rm{e}}^{{\rm{-jkr}}}}\)

The polarization of the incident wave, the polarization of the antenna and losses due to the polarization mismatch are, respectively,

The far-zone power density radiated by a helical antenna is approximated as:

\({{\rm{\vec W}}_{{\rm{rad}}}} = {{\rm{\vec W}}_{{\rm{average}}}} \approx \widehat {{{\rm{a}}_{\rm{r}}}}{{\rm{C}}_0}\frac{1}{{{{\rm{r}}^2}}}{\cos ^4}{\rm{\theta }}\)

The radiated power density is symmetrical with respect to φ and exists only in the upper

hemisphere: 0 ≤ 𝜃 ≤ 𝜋/2 ; 0 ≤ 𝜙 ≤ 2𝜋; 𝐶₀ is a constant. The power radiated by the antenna (in watts) and the maximum directivity of the antenna, respectively, are