The vector function F(r) = -x î + yĵ is defined over a circular are C shown in the figure.

The line integral of ∫_C F(r) ⋅ dr is 

Consider the differential equation given below:

\(\dfrac{dy}{dx} + \dfrac{x}{1-x^2} y= x\sqrt{y}\)

The integrating factor of the differential equation is:

Two continuous random variables X and Y are related as

Y = 2X + 3

Let \(\sigma_X^2\) and \(\sigma_Y^2\) denote the variances of X and Y, respectively. The variances are related as

Consider a real-value based-band signal x(t), band limited to 10 kHz. The Nyquist rate for the signal y(t) = x(t) \(x(1+ \frac{t}{2})\) is

Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of x[n] and h[n]. The value(s) of k for which z[k] = y[k] is/are

A bar of silicon is doped with a boron concentration of 10¹⁶ cm^-3 and assumed to be fully ionized. It is exposed to light such that electron-hole pairs are generated throughout the volume of the bar at the rate of 10²⁰ cm^-3s^-1. If the recombination lifetime is 100 μs, the intrinsic carrier concentration of silicon is 10¹⁰ cm^-3 and assuming 100% ionization of boron, then the approximate product of steady-state electron and hole concentrations due to this light exposure is

The energy band diagram of a p-type semiconductor bar of length L under equilibrium condition (i.e., the Fermi energy level E_F is constant) is shown in the figure. The valance band E_V is sloped since doping is non-uniform along the bar. The difference between the energy levels of the valence band at the two edges of the bar is Δ.

If the charge of an electron is q, then the magnitude of the electric field developed inside this semiconductor bar is

In a circuit shown in the figure, the transistors M₁ and M₂ are operating in saturation. The channel length modulation coefficients of both the transistors are non-zero. The transconductance of the MOSFETs M1 and M2 are g_m1 and g_m2, respectively, and the internal resistance of the MOSFETs M1 and M2 are r₀₁ and r₀₂, respectively.

Ignoring the body effect, the ac small-signal voltage gain (∂V_out / ∂V_in) of the circuit is:

For the circuit with an ideal OPAMP shown in the figure. V_REF is fixed.

If V_OUT = 1 for V_IN = 0.1 volt and V_OUT = 6 volt for V_IN = 1 volt, where V_OUT is measured across R_L connected at the output of this OPAMP, the value of R_F / R_IN is

Consider the circuit with and ideal OPAMP shown in the figure.

Assuming |V_IN| << |V_CC| and |V_REF| << |V_CC|, the condition at which V_OUT equals to zero is

If (1235)_x = (3033)_y, where x and y indicate the bases of the corresponding numbers, then 

Addressing of a 32K × 16 memory is realized using a single decoder. The minimum number of AND gates required for the decoder is

The block diagram of a feedback control system is shown in the figure.

The transfer function \(\frac{Y(s)}{X(s)}\) of the system is

The complete Nyquist plot of the open-loop transfer function G(s) H(s) of a feedback control system is shown in the figure.

If G(s) H(s) has one zero in the right-half of the s-plane. the number of poles that the closed-loop system will have in the right-half of the s-plane is

Consider a rectangular coordinate system (x, y, z) with unit vectors a_x, a_y, and a_z. A plane wave traveling in the region z ≥ 0 with electric field vector E = 10 cos (2 × 10⁸ t + βz) a_y is incident normally on the plane at z = 0, where β is the phase constant. The region z ≥ 0 is in free space and the region z < 0 is filled with a lossless medium (permittivity ε = ε₀, permeability μ = 4μ₀, where ε₀ = 8.85 × 10^-12 F/m  and μ₀ = 4π × 10^-7 H/m). The value of the reflection coefficient is  

If the vectors (1.0, -1.0, 2.0), (7.0, 3.0, x) and (2.0, 3.0, 1.0) in R³ are linearly dependent, the value of x is ______

Consider the vector field F = ax (4y - c1z) + ay (4x + 2z) +az (2y +z) in rectangular coordinated system (x, y, z) with unit vectors ax, ay, and az, If the filed F is irrotational (conservative), then the constant c₁ (in integer) is _____

Consider the circuit shown in the figure.

The current I flowing through the 7 Ω resistor between P and Q (rounded off to one decimal place) is _______ A.

Consider the circuit shown in the figure.

The value of v₀ (rounded off to one decimal place) is ______ V.

An 8-bit unipolar (all analog output values are positive) digital-to-analog converter (DAC) has a full-scale voltage range from 0 V to 7.68 V. If the digital input code is 10010110 (the leftmost bit is MSB), then the analog output voltage of DAC (rounded off to one decimal place) is _______ V.

The auto correlation function RX(τ) of a wide-sense stationary random process X(t) is shown in the figure.

The average power of X(t) is _______

Consider a carrier signal which is amplitude modulated by a single-tone sinusoidal message signal with a modulation index of 50%. If the carrier and one of the sidebands are suppressed in the modulated signal, the percentage of power saved (rounded off to one decimal place) is _______.

A speech signal, band-limited to 4 kHz, is sampled at 1.25 times the Nyquist rate. The speech samples assumed to be statistically independent and uniformly distributed in the range -5 V to +5 V, are subsequently quantized in an 8-bit uniform quantizer and then transmitted over a voice-grade AWGN telephone channel. If the ratio of the transmitted signal power to channel noise power is 26 dB, the minimum channel bandwidth required to ensure reliable transmission of the signal with an arbitrarily small probability of transmission error (rounded off to two decimal places) is _______ kHz.

A 4 kHz sinusoidal message signal having amplitude 4 V is fed to a delta modulator (DM) operating at a sampling rate of 32 kHz. The minimum step size required to avoid slope overload noise in the DM (rounded off to two decimal places) is _______ V.

The refractive indices of the core and cladding of an optical fiber are 1.50 and 1.48, respectively. The critical propagation angle, which is defined as the maximum angle that the light beam makes with the axis of the optical fiber to achieve the total internal reflection, (rounded off to two decimal places) is ______ degree.

Consider the integral \(\displaystyle\oint_C \dfrac{\sin (x)}{x^2 (x^2 + 4)}dx\)

Where C is a counter-clockwise oriented circle defined as |x - i| = 2. The value of the integral is

A box contains the following three coins.

I. A fair coin with head on one face and tail on the other face.

II. A coin with heads on both the faces.

III. A coin with tails on both the faces.

A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the probability of getting a head in the second toss is

The switch in the circuit in the figure is in position P for long time and then moved to position Q at time t = 0.

The value of \(\frac{{dv\left( t \right)}}{{dt}}\) at t = 0⁺ is

Consider the two-port network shown in the figure.

The admittance parameters, in siemens, are

For an n-channel silicon MOSFET with 10 nm gate oxide thickness, the substrate sensitivity (∂V_T/∂|V_BS|) is found to be 50 mV/V at a substance voltage |V_BS| = 2 V, where V_T is the threshold voltage of the MOSFET. Assume that, |V_BS| >> 2ϕ_B, where qϕ_B is the separation between the Fermi energy level E_F and the intrinsic level E_i in the bulk. Parameters given are

Electron charge (q) = 1.6 × 10^-19 C

Vacuum permittivity (ε₀) = 8.85 × 10^-12 F/m

Relative permittivity of silicon (ε_Si) = 12

Relative permittivity of oxide (ε_ox) = 4

The doping concentration of the substrate is      

The propagation delays of the XOR gate, AND gate, and multiplexer (MUX) in the circuit shown in the figure are 4 ns, 2 ns, and 1 ns, respectively.

If all the inputs P, Q, R, S, and T are applied simultaneously and held constant the maximum propagation delay of the circuit is

The content of the registers are R₁ = 25H, R₂ = 30H and R₃ = 40H. The following machine instructions are executed.

PUSH{R1}

PUSH{R2}

PUSH{R3}

POP{R₁}

POP{R2}

POP{R3}

After execution, the content of registers R1, R2, R3 are

The electrical system shown in the figure converts input source current i_s(t) to output voltage v₀(t).

Current i_L(t) in the inductor and voltage v_c(t) acros the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., i_L(0) = 0 and v_c(0) = 0. Then system is

A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (m_i ; c_i), where c_i is the codeword corresponding to the i^th message m_i, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?

The impedance matching network shown in the figure is to match a lossless line having characteristic impedance Z₀ = 50 Ω with a load impedance Z_L. A quarter-wave line having a characteristic impedance Z₁ = 75 Ω is connected to Z_L. Two stubs having characteristic impedance of 75 Ω each are connected to this quarter-wave line. One is a short-circuited (S.C.) stub of length 0.25 λ connected across PQ and the other one is an open-circuited (O.C.) stub of length 0.5 λ connected across RS.

The impedance matching is achieved when the real part of Z_L is

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

\(A = \left[ {\begin{array}{*{20}{c}} x&{ - 3.0}\\ {3.0}&{4.0} \end{array}} \right]\)

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_ρ + z² sin² ϕ a_ϕ in a cylindrical coordinate system (ρ, ϕ, z) with unit vectors a_ρ, a_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:

\(x(t)=\displaystyle\sum_{k=-\infty}^\infty a_k e^{jkω_0 t}\)

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

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

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

III. \(a_k = \left\lbrace \begin{matrix} k, \ \ 1 \le k \le 3 \\\ 0, \ \ \ k > 3 \end{matrix} \right.\)

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 \(\left(\left(\dfrac{1}{2}\right)^n u[n]\right)\). The value of y[0] is _____   

Consider the signals x[n] = 2^n-1 u[-n + 2] and y[n] = 2^-n+2 u[n + 1], where u[n] is the unit step sequence. Let X(e^jω) and Y(e^jω) be the discrete-time Fourier transform of x[n] and y[n], respectively. The value of the integral

\(\dfrac{1}{2\pi}\displaystyle\int_0^{2\pi} X(e^{j\omega}) Y(e^{-j\omega})d\omega\)

(rounded off to one decimal place) is _____  

A silicon P-N junction is shown in the figure. The doping in the P region is 5 × 10¹⁶ cm^-3 and doping in the N region is 10 × 10¹⁶ cm^-3. The parameters given are

Built-in voltage (Φ_bi) = 0.8 V

Electron charge (q) = 1.6 × 10^-19 C

Vacuum permittivity (ε₀) = 8.85 × 10^-12 F / m

Relative permittivity of silicon (ε_Si) = 12

The magnitude of reverse bias voltage that would completely deplete one of the two regions (P or N) prior to the other (rounded off to one decimal place) is _______ V.

An asymmetrical periodic pulse train v_in of 10 V amplitude with on-time T_ON = 1 ms and off-time T_OFF = 1 μs is applied to the circuit shown in the figure. The diode D₁ is ideal. 

The difference between the maximum voltage and minimum voltage of the output waveform v₀ (in integer) is _____V

For the transistor M` in the circuit shown in the figure, μn Cox = 100μA/V^2 and (W/L) = 10, where μn is the mobility of electron, Cox is the oxide capacitance per unit area, W is the width and L is the length.

The channel length modulation coefficient is ignored. If the gate-to-source voltage VGS is 1 V to keep the transistor at the edge of saturation, then the threshold voltage of the transistor (rounded off to one decimal place) is _____ V.

A circuit with an ideal OPAMP is shown in the figure. A pulse V_IN of 20 ms duration is applied to the input. The capacitors are initially uncharged,

The output voltage V_OUT of this circuit t = 0⁺ (in integer) is ______V.

The propagation delay of the exclusive-OR (XOR) gate in the circuit in the figure is 3 ns. The propagation delay of all the flip-flops is assumed to be zero. The clock (Clk) frequency provided to the circuit is 500 MHz.

Starting from the initial value of the flip-flop outputs Q₂ Q₁ Q₀ = 1 1 1 with D2 = 1, then the minimum number of triggering clock edges after which the flip-flop outputs Q₂ Q₁ Q₀ becomes 1 0 0 (in integer) is ______

The circuit in the figure contains a current source driving a load having an inductor and a resistor in series, with a shunt capacitor across the load. The ammeter is assumed to have zero resistance. The switch is closed at time t = 0.

Initially, when the switch is open, the capacitor is discharged and the ammeter reads zero ampere. After the switch is closed, the ammeter reading keeps fluctuating for some time till it settles to a final steady value. The maximum ammeter reading that one will observe after the switch is closed (rounded off to two decimal places) is _______ A.

A unity feedback system that uses proportional-integral (PI) control is shown in the figure.

The stability of the overall system is controlled by tuning the PI control parameters K_P and K_I. The maximum value of K_I that can be chosen so as to keep  the overall system stable or, in the worst case, marginally stable (rounded off to three decimal places) is ________.

A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to a phase modulator with phase deviation constant 2 rad / volt. If the carrier signal is c(t) = 2cos(2π106t). the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is _______ Hz.

Consider a superheterodyne receiver tuned to 600 kHz. If the local oscillator feeds a 1000 kHz signal to the mixer, the image frequency (in integer) is _______ kHz.

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal places) is _______ bits.

A message signal having peak-to-peak value of 2 V, root mean square value of 0.1 V and bandwidth of 5 kHz is sampled and fed to a pulse code modulation (PCM) system that uses a uniform quantizer. The PCM output is transmitted over a channel that can support a maximum transmission rate of 50 kbps. Assuming that the quantization error is uniformly distributed, the maximum signal to quantization nose ratio that can be obtained by the PCM system (rounded off to two decimal places) is _____

Consider a polar non-return to zero (NRZ) waveform, using +2 V and -2 V for representing binary '1' and '0' respectively, is transmitted in the presence of additive zero-mean white Gaussian noise with variance 0.4 V². If the a priori probability of transmission of a binary '1' is 0.4, the optimum threshold voltage for a maximum a posteriori (MAP) receiver (rounded off to two decimal places) is _____ V.

A standard air-filled rectangular waveguide with dimensions a = 8 cm, b = 4cm, operates at 3.4 GHz, For the dominant mode of wave propagation, the phase velocity of the signal is vp. The value (rounded off two decimal places) of vp / C, where C denotes the velocity of light, is _____

An antenna with a directive gain of 6 dB is radiating a total power of 16 kW. The amplitude of the electric field in free space at a distance of 8 km from the antenna in the direction of 6 dB gain (rounded off to three decimal places) is _______ V / m.