Let\(\rm v_1=\begin{bmatrix}1\\ 2\\\ 0\end{bmatrix}\) and\(\rm v_2=\begin{bmatrix}2\\ 1\\\ 3\end{bmatrix}\) be two vectors. The value of the coefficient 𝛼 in the expression v₁= αv₂+ e, which minimizes the length of the error vector 𝒆, is

The rate of increase, of a scalar field 𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧, in the direction 𝒗 = (2, 1, 2) at a point (0, 2, 1) is

Let 𝑤⁴ = 16𝑗. Which of the following cannot be a value of 𝑤?

The value of the contour integral,\(\rm \oint_c\left(\frac{z+2}{z^2+2z+2}\right)dz\),where the contour C is\(\rm \left\{z:\left|z+1-\frac{3}{2}j\right|=1\right\}\)taken in the counter clockwise direction, is

Let the sets of eigenvalues and eigenvectors of a matrix 𝐵 be {𝜆_𝑘 |1 ≤ 𝑘 ≤ 𝑛} and {𝒗_𝑘 |1 ≤ 𝑘 ≤ 𝑛}, respectively. For any invertible matrix 𝑃, the sets of eigenvalues and eigenvectors of the matrix 𝐴, where 𝐵 = 𝑃⁻¹𝐴𝑃, respectively, are

In a semiconductor, if the Fermi energy level lies in the conduction band, then the semiconductor is known as

For an intrinsic semiconductor at temperature 𝑇 = 0 𝐾, which of the following statement is true?

A series 𝑅𝐿𝐶 circuit has a quality factor 𝑄 of 1000 at a center frequency of 10⁶ rad/s. The possible values of 𝑅, 𝐿 and C are

For a MOS capacitor, V_fb and V_t are the flat-band voltage and the threshold voltage, respectively. The variation of the depletion width (W_dep) for varying gate voltage (V_g) is best represented by

Consider a narrow band signal, propagating in a lossless dielectric medium (𝜀_𝑟 = 4, 𝜇_𝑟 = 1), with phase velocity 𝑣_𝑝 and group velocity 𝑣_𝑔. Which of the following statement is true? (𝑐 is the velocity of light in vacuum.)

In the circuit shown below, V₁ and V₂ are bias voltages. Based on input and output impedances, the circuit behaves as a

A cascade of common-source amplifiers in a unity gain feedback configuration oscillates when

In the circuit shown below, P and Q are the inputs. The logical function realized by the circuit shown below is

The synchronous sequential circuit shown below works at a clock frequency of 1 GHz. The throughput, in Mbits/s, and the latency, in ns, respectively, are

The open loop transfer function of a unity negative feedback system is\(\rm G(s)=\frac{k}{s(1+sT_1)(1+sT_2)}\),where 𝑘, 𝑇₁ and 𝑇₂ are positive constants. The phase crossover frequency, in rad/s, is

Consider a system with input 𝑥(𝑡) and output 𝑦(𝑡) = 𝑥(𝑒^𝑡). The system is

Let 𝑚(𝑡) be a strictly band-limited signal with bandwidth 𝐵 and energy 𝐸. Assuming 𝜔₀ = 10𝐵, the energy in the signal 𝑚(𝑡) cos 𝜔₀𝑡 is

The Fourier transform 𝑋(𝜔) of\(\rm x(t)=e^{-t^2}\) is

Note:\(\rm \int_{-\infty}^{\infty}e^{-y^2}dy=\sqrt{\pi}\)

In the table shown below, match the signal type with its spectral characteristics.

For a real signal, which of the following is/are valid power spectral density/densities?

​The signal-to-noise ratio (SNR) of an ADC with a full-scale sinusoidal input is given to be 61.96 dB. The resolution of the ADC is ________ bits (rounded off to the nearest integer).

In the circuit shown below, the current i flowing through 200 Ω resistor is ______ mA (rounded off to two decimal places).

For the two port network shown below, the [Y]-parameters is given as

\(\rm [Y]=\frac{1}{100}\begin{bmatrix}2&-1\\\ -1&4/3\end{bmatrix}S\)

The value of load impedance Z_L, in Ω, for maximum power transfer will be _______ (rounded off to the nearest integer).

For the circuit shown below, the propagation delay of each NAND gate is 1 ns. The critical path delay, in ns, is __________ (rounded off to the nearest integer).

In the circuit shown below, switch S was closed for a long time. If the switch is opened at 𝑡 = 0, the maximum magnitude of the voltage V_R, in volts, is _________ (rounded off to the nearest integer).

A random variable X, distributed normally as 𝑁(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of 𝑌 is

(In the options given below, 𝑎, 𝑏, 𝑐 are non-zero constants and 𝑔(𝑦) is piece-wise continuous function)

The value of the line integral\(\rm \int_P^Q(z^2dx+3y^2dy+2xzdz)\) along the straight line joining the points 𝑃 (1, 1, 2) and 𝑄 (2, 3, 1) is

Let 𝒙 be an 𝑛 × 1 real column vector with length \(\rm l=\sqrt{x^Tx}\). The trace of the matrix 𝑃 = 𝒙𝒙^𝑇 is

The\(\rm \frac{V_{OUT}}{V_{IN}}\)of the circuit shown below is

In the circuit shown below, D₁ and D₂ are silicon diodes with cut-in voltage of 0.7 V. V_IN and V_OUT are input and output voltages in volts. The transfer characteristic is

A closed loop system is shown in the figure where k > 0 and α > 0. The steady state error due to a ramp input (R(s) = α/s²) is given by

In the following block diagram, R(s) and D(s) are two inputs. The output Y(s) is expressed as Y(s) = G₁(s)R(s) + G₂(s)D(s).

G₁(s) and G₂(s) are given by

The state equation of a second order system is

Suppose λ₁ and λ₂ are two distinct eigenvalues of A and 𝒗₁ and 𝒗₂ are the corresponding eigenvectors. For constants 𝛼₁ and 𝛼₂, the solution, 𝒙(𝑡), of the state equation is

The switch S₁ was closed and S₂ was open for a long time. At 𝑡 = 0, switch S₁ is opened and S₂ is closed, simultaneously. The value of i_c(0⁺), in amperes, is

Let a frequency modulated (FM) signal

\(\rm x(t)=A\cos(\omega_ct+k_f\int_{-\infty}^{t}m(\lambda)d\lambda)\), where 𝑚(𝑡) is a message signal of bandwidth W. It is passed through a non-linear system with output 𝑦(𝑡) = 2𝑥(𝑡) + 5(𝑥(𝑡))² . Let 𝐵_𝑇 denote the FM bandwidth. The minimum value of 𝜔_𝑐 required to recover 𝑥(𝑡) from 𝑦(𝑡) is

The h-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain\(\rm \frac{V_{out}}{V_s}\)is

Consider a discrete-time periodic signal with period 𝑁 = 5. Let the discrete-time Fourier series (DTFS) representation be\(\rm x[n]=\Sigma_{k=0}^4a_ke^{\frac{jk2\pi n}{s}}\), where 𝑎₀ = 1, 𝑎₁ = 3𝑗, 𝑎₂ = 2𝑗, 𝑎₃ = −2𝑗 and 𝑎₄ = −3𝑗. The value of the sum\(\rm \Sigma_{n=0}^4x[n]\sin\frac{4\pi n}{5}\)is

Let an input 𝑥[𝑛] having discrete time Fourier transform

𝑋(𝑒^𝑗𝛺) = 1 − 𝑒^−𝑗𝛺 + 2𝑒^−3𝑗𝛺 be passed through an LTI system. The frequency response of the LTI system is \(\rm H(e^{j\Omega)}=1-\frac{1}{2}e^{-j2\Omega}\). The output 𝑦[𝑛] of the system is

Let 𝑥(𝑡) = 10 cos(10.5 𝑊𝑡) be passed through an LTI system having impulse response\(\rm h(t)=\pi \left(\frac{\sin Wt}{\pi t}\right)^2\cos 10Wt\).The output of the system is

Let x₁(t) and x₂(t) be two band-limited signals having bandwidth 𝐵 = 4𝜋 × 10³ rad/s each. In the figure below, the Nyquist sampling frequency, in rad/s, required to sample y(t), is

The S-parameters of a two port network is given as

\(\rm [S]=\begin{bmatrix}S_{11}&S_{12}\\\ S_{21}&S_{22}\end{bmatrix}\)

with reference to 𝑍₀. Two lossless transmission line sections of electrical lengths 𝜃₁ = 𝛽𝑙₁ and 𝜃₂ = 𝛽𝑙₂ are added to the input and output ports for measurement purposes, respectively. The S-parameters [𝑆′] of the resultant two port network is

The standing wave ratio on a 50 Ω lossless transmission line terminated in an unknown load impedance is found to be 2.0. The distance between successive voltage minima is 30 cm and the first minimum is located at 10 cm from the load. 𝑍_𝐿 can be replaced by an equivalent length 𝑙_𝑚 and terminating resistance 𝑅_𝑚 of the same line. The value of 𝑅_𝑚 and 𝑙_𝑚, respectively, are

The electric field of a plane electromagnetic wave is

𝑬 = 𝒂_𝒙𝐶_1𝑥 cos(𝜔𝑡 − 𝛽𝑧) + 𝒂_𝒚𝐶_1𝑦 cos(𝜔𝑡 − 𝛽𝑧 + 𝜃) V/m.

Which of the following combination(s) will give rise to a left handed elliptically polarized (LHEP) wave?

The following circuit(s) representing a lumped element equivalent of an infinitesimal section of a transmission line is/are

The value of the integral\(\rm \iint_R xydxdy\)over the region R, given in the figure, is _______ (rounded off to the nearest integer).

In an extrinsic semiconductor, the hole concentration is given to be 1.5𝑛_𝑖 where 𝑛_𝑖 is the intrinsic carrier concentration of 1 × 10¹⁰ 𝑐𝑚⁻³ . The ratio of electron to hole mobility for equal hole and electron drift current is given as __________ (rounded off to two decimal places).

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is\(\rm (s)=\frac{k(s+z)^a}{s^b(s+p)^c}\),where 𝑘, 𝑧, 𝑝, 𝑎, 𝑏 and 𝑐 are positive constants. The value of (𝑎 + 𝑏 + 𝑐) is __________(rounded off to the nearest integer).

Let x₁(t) = u(t + 1.5) − u(t − 1.5) and x₂(t) is shown in the figure below. For y(t) = x₁(t) ∗ x₂(t), the\(\rm \int_{-\infty}^{\infty}y(t)dt\) is ________ (rounded off to the nearest integer).

Let 𝑋(𝑡) be a white Gaussian noise with power spectral density\(\frac{1}{2}\) W/Hz. If 𝑋(𝑡) is input to an LTI system with impulse response 𝑒^−𝑡𝑢(𝑡). The average power of the system output is ___________ W (rounded off to two decimal places).

A transparent dielectric coating is applied to glass (𝜀_𝑟 = 4, 𝜇_𝑟 = 1) to eliminate the reflection of red light (𝜆₀ = 0.75 μm). The minimum thickness of the dielectric coating, in μm, that can be used is _________ (rounded off to two decimal places).

In a semiconductor device, the Fermi-energy level is 0.35 eV above the valence band energy. The effective density of states in the valence band at T = 300 K is 1 × 10¹⁹ cm⁻³ . The thermal equilibrium hole concentration in silicon at 400 K is ____________ × 10¹³ cm⁻³ (rounded off to two decimal places).

Given kT at 300 K is 0.026 eV

A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of 1 μs. The time for the sampling switch to stay closed to charge a capacitor adequately to a full scale voltage of 1 V with 12-bit accuracy is _______ μs (rounded off to two decimal places).

In a given sequential circuit, initial states are Q₁ = 1 and Q₂ = 0. For a clock frequency of 1 MHz, the frequency of signal Q₂ in kHz, is ___________ (rounded off to the nearest integer).

In the circuit below, the voltage V_L is ____________ V (rounded off to two decimal places).

The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of “yes/no” questions which are assumed to be truthfully answered. The average number of questions when asked in the most efficient sequence, to determine the chosen symbol, is _______ (rounded off to two decimal places).