In the given circuit, the silicon transistor has $$\beta $$ $$=75$$ and a collector voltage $${V_C} = 9\,V.$$ Then the ratio of $${R_B}$$ and $${R_C}$$ is ___________.

Consider the circuit shown in the figure. In this circuit $$R = 1\,\,k\Omega ,$$ and $$C = 1\,\,\mu F.$$ The input voltage is sinusoidal with a frequency of $$50$$ $$Hz,$$ represented as a phasor with magnitude $${V_i}$$ and phase angle $$0$$ radian as shown in the figure. The output voltage is represented as a phasor with magnitude $${V_0}$$ and phase angle $$\delta $$ radian. What is the value of the output phase angle $$\delta $$ (in radian) relative to the phase angle of the input voltage?

Of the four characteristics given below, which are the major requirements for an instrumentation amplifier?
P. $$\,\,\,\,$$High common mode rejection ratio
Q. $$\,\,\,\,$$High input impedance
R. $$\,\,\,\,$$High linearity
S. $$\,\,\,\,$$High output impedance

The op-amp shown in the figure has a finite gain $$A = 1000$$ and an infinite input resistance. A step voltage $${V_i} = 1\,\,mV$$ is applied at the input at time $$t = 0$$ as shown. Assuming that the operational amplifier is not saturated, the time constant (in millisecond) of the output voltage $${V_o}$$ is

Find the transfer function $${{Y\left( s \right)} \over {X\left( s \right)}}$$ of the system given below:

For the signal-flow graph shown in the figure, which one of the following expressions is equal to the transfer function $${\left. {{{Y\left( s \right)} \over {{X_2}\left( s \right)}}} \right|_{{x_1}\left( s \right) = 0}}?$$

A Bode magnitude plot for the transfer function $$𝐺(𝑠)$$ of a plant is shown in the figure. Which one of the following transfer functions best describes the plant?

The transfer function of a second order real system with a perfectly flat magnitude response of unity has a pole at $$\left( {2 - j3} \right).$$ List all the poles and zeros.

In the signal flow diagram given in the figure, $${u_1}$$ and $${u_2}$$ are possible inputs whereas $${y_1}$$ and $${y_2}$$ are possible outputs. When would the $$SISO$$ system derived from this diagram be controllable and observable?

$$f\left( {A,\,B,\,C,\,D} \right) = \Pi M\left( {0,1,3,4,5,7,9,11,12,13,14,15} \right)$$ is a Maxterm representation of a Boolean function $$f(A,B,C,D)$$ where $$A$$ is the $$MSB$$ and $$D$$ is the $$LSB$$. The equivalent minimized representation of this function is

In the $$4 \times 1$$ multiplexer, the output $$F$$ is given by $$F = A \oplus B.$$ Find the required input $${{\rm I}_3}{{\rm I}_2}{{\rm I}_1}{{\rm I}_0}.$$

The figure shows a digital circuit constructed using negative edge triggered $$J-K$$ flip flops. Assume a starting state of $${Q_2}\,{Q_1}\,{Q_0} = 000.$$ This state $${Q_2}\,{Q_1}\,{Q_0} = 000$$ will repeat after _____ number of cycles of the clock $$CLK.$$

An $$8$$-bit, unipolar Successive Approximation Register type $$ADC$$ is used to convert $$3.5$$ $$V$$ to digital equivalent output. The reference voltage is $$+5 V.$$ The output of the $$ADC,$$ at the end of $$3$$rd clock pulse after the start of conversion, is

In the given circuit, the parameter k is positive, and the power dissipated in the 2 Ω resistor is 12.5 W. The value of k is ________.

For the given circuit the Thevenin equivalent is to be determined. The Thevenin voltage, V_Th (in volt), seen from terminal AB is _________.

An inductor is connected in parallel with a capacitor as shown in the figure. As the frequency of current i is increased, the impedance (Z) of the network varies as

The circuit shown in the figure has two sources connected in series. The instantaneous voltage of the AC source (in volt) is given by v(t) = 12sint. If the circuit is in steady-state, then the rms value of the current (in Ampere) flowing in the circuit is ______.

In a linear two-port network, when 10 V is applied to Port 1, a current of 4 A flows through Port 2 when it is short-circuited. When 5V is applied to Port1, a current of 1.25 A flows through a 1 Ω resistance connected across Port 2. When 3V is applied to Port 1, then current (in Ampere) through a 2 Ω resistance connected across Port 2 is _________.

The voltages developed across the 3Ω and 2Ω resistors shown in the figure are 6V and 2V respectively, with the polarity as marked. What is the power (in Watt) delivered by the 5V voltage source?

A ($$0-50$$ $$A$$) moving coil ammeter has a voltage drop of $$0.1$$ $$V$$ across its terminals at full scale deflection. The external shunt resistance (in $$milliohms$$) needed to extend its range to ($$0 – 500$$ $$A$$) is ______.

When the Wheatstone bridge shown in the figure is used to find the value of resistor $${R_x},$$ the galvanometer $$G$$ indicates zero current when $${R_1} = 50\,\Omega ,\,\,{R_2} = 65\,\Omega $$ and $$\,{R_3} = 100\,\Omega ,\,\,$$ If $${R_3}$$ is known with $$\, \pm 5\% \,$$ tolerance on its nominal value of $$100\,\Omega .$$ What is the range of $${R_x}$$ in Ohms?

An unbalanced $$DC$$ Wheatstone bridge is shown in the figure. At what value of $$p$$ will the magnitude of $${V_0}$$ be maximum?

A separately excited DC generator has an armature resistance of 0.1 $$\Omega$$ and negligible armature inductance. At rated field current and rated rotor speed, its open-circuit voltage is 200 V. When this generator is operated at half the rated speed, with half the rated field current, an un-charged 1000 μF capacitor is suddenly connected across the armature terminals. Assume that the speed remains unchanged during the transient. At what time (in microsecond) after the capacitor is connected will the voltage across it reach 25 V?

A separately excited DC motor runs at 1000 rpm on no load when its armature terminals are connected to a 200 V DC source and the rated voltage is applied to the field winding. The armature resistance of this motor is 1 Ω. The no-load armature current is negligible. With the motor developing its full load torque, the armature voltage is set so that the rotor speed is 500 rpm. When the load torque is reduced to 50% of the full load value under the same armature voltage conditions, the speed rises to 520 rpm. Neglecting the rotational losses, the full load armature current (in Ampere) is _______.

A DC motor has the following specifications: 10 Hp, 37.5 A, 230 V; flux/pole = 0.01 Wb, number of poles = 4, number of conductors = 666, number of parallel paths = 2. Armature resistance = 0.267 Ω. The armature reaction is negligible and rotational losses are 600 W. The motor operates from a 230 V DC supply. If the motor runs at 1000 rpm, the output torque produced (in Nm) is ________.

The self inductance of the primary winding of a single phase, 50 Hz, transformer is 800 mH, and that of the secondary winding is 600 mH. The mutual inductance between these two windings is 480 mH. The secondary winding of this transformer is short circuited and the primary winding is connected to a 50 Hz, single phase, sinusoidal voltage source. The current flowing in both the winding is less than their respective rated currents. The resistance of both windings can be neglected. In this connection, what is the effective inductance (in mH) seen by the source?

Two single-phase transformers T₁ and T₂ each rated at 500 kVA are operated in parallel. Percentage impedances of T₁ and T₂ are (1 + j6) and (0.8 + j4.8), respectively. To share a load of 1000 kVA at 0.8 lagging power factor, the contribution of T₂ (in kVA) is ________.

A 200/400 V, 50 Hz, two-winding transformer is rated at 20 kVA. Its windings are connected as an auto-transformer of rating 200/600 V. A resistive load of 12 Ω is connected to the high voltage (600 V) side of the auto-transformer. The value of equivalent load resistance (in Ohm) as seen from low voltage side is _____.

A 3-phase 50 Hz square wave (6-step) VSI feeds a 3-phase, 4 pole induction motor. The VSI line voltage has a dominant 5^th harmonic component. If the operating slip of the motor with respect to fundamental component voltage is 0.04, the slip of the motor with respect to 5^th harmonic component of voltage is _________.

Consider a function $$\overrightarrow f=\frac1{r^2}\widehat r$$ where r is the distance from the origin and $$\widehat r$$ is the unit vector in the radial direction. The divergence of the function over a sphere of radius R, which includes the origin, is

A parallel plate capacitor is partially filled with glass of dielectric constant 4.0 as shown below. The dielectric strengths of air and glass are 30 kV/cm and 300 kV/cm, respectively. The maximum voltage (in kilovolts), which can be applied across the capacitor without any breakdown, is ______.

A steady current I is flowing in the −x direction through each of two infinitely long wires at $$y=\pm\frac L2$$ as shown in the figure. The permeability of the medium is $$\mu_0$$. The $$\overrightarrow B$$ field at (0,L,0) is

Consider a one-turn rectangular loop of wire placed in a uniform magnetic field as shown in the figure. The plane of the loop is perpendicular to the field lines. The resistance of the loop is 0.4 Ω, and its inductance is negligible. The magnetic flux density (in Tesla) is a function of time, and is given by B(t) = 0.25 sinωt, where ω = 2𝜋×50 radian/second. The power absorbed (in Watt) by the loop from the magnetic field is ________.

If a continuous function $$f(x)$$ does not have a root in the interval $$\left[ {a,b} \right],\,\,$$ then which one of the following statements is TRUE?

If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant of the matrix is ____________.

The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real eigenvectors is

Two players, $$A$$ and $$B,$$ alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player $$A$$ starts the game, the probability that $$A$$ wins the game is

A random variable $$X$$ has probability density function $$f(x)$$ as given below: $$$\,\,f\left( x \right) = \left\{ {\matrix{ {a + bx} & {for\,\,0 < x < 1} \cr 0 & {otherwise} \cr } } \right.\,\,$$$
If the expected value $$\,\,E\left[ X \right] = 2/3,\,\,$$ then $$\,\,\Pr \left[ {X < 0.5} \right]\,\,$$ is __________.

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What is the probability that the sum of the two numbers equal $$16?$$

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in at least one, a $$50$$% chance of passing in at least two and a $$40$$% chance of passing in exactly two. Following relations are drawn in $$m, p, c:$$
$${\rm I}.$$ $$\,\,\,\,\,\,$$ $$p+m+c=27/20$$
$${\rm I}{\rm I}.$$ $$\,\,\,\,\,\,$$ $$p+m+c=13/20$$
$${\rm I}{\rm I}{\rm I}.$$ $$\,\,\,\,\,\,$$ $$\left( p \right) \times \left( m \right) \times \left( c \right) = 1/10$$

A solution of the ordinary differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + 5{{dy} \over {dt}} + 6y = 0\,\,$$ is such that $$y(0)=2$$ and $$y(1)=$$ $$ - \left( {{{1 - 3e} \over {{e^3}}}} \right).$$ The value of $${{dy} \over {dt}}\left( 0 \right)$$ is

In the following chopper, the duty ratio of switch $$S$$ is $$0.4.$$ If the inductor and capacitor are sufficiently large to ensure continuous inductor current and ripple free capacitor voltage, the charging current (in Ampere) of the $$5$$ $$V$$ battery, under steady-state, is_____

A buck converter feeding a variable resistive load is shown in the figure. The switching frequency of the switch $$S$$ is $$100$$ $$kHz$$ and the duty ratio is $$0.6.$$ The output voltage $${V_0}$$ is $$36$$ $$V.$$ Assume that all the components are ideal, and that the output voltage is ripple-free. The value of $$R$$ (in $$Ohm$$) that will make the inductor current ($${{\rm I}_L}$$) just continuous is _______.

A self commutating switch $$SW,$$ operated at duty cycle $$\delta $$ is used to control the load voltage as shown in the figure

Under steady state operating conditions, the average voltage across the indicator and the capacitor respectively, are

The circuit shown is meant to supply a resistive load $${R_L}$$ from two separate $$DC$$ voltage sources. The switches $$S1$$ and $$S2$$ are controlled so that only one of them is ON at any instant. $$S1$$ is turned on for $$0.2$$ $$ms$$ and $$S2$$ is turned on for $$0.3$$ $$ms$$ in a $$0.5$$ $$ms$$ switching cycle time period. Assuming continuous conduction of the inductor current and negligible ripple on the capacitor voltage, the output voltage $${V_0}$$ (in Volt) across $${R_L}$$ is __________.

The single-phase full-bridge voltage source inverter $$(VSI),$$ shown in figure, has an output frequency of $$50$$ $$Hz$$. It uses unipolar pulse width modulation with switching frequency of $$50$$ $$kHz$$ and modulation index of $$0.7.$$ For
$${V_{in}} = 100\,V\,\,DC,\,\,L = 9.55\,\,mH,\,C = 63.66\,\,\mu F,$$ and $$R = 5\,\Omega ,$$ the amplitude of

A 50 Hz generating unit has H-constant of 2 MJ/MVA. The machine is initially operating in steady state at synchronous speed, and producing 1 pu of real power. The initial value of the rotor angle $$\delta $$ is $${5^ \circ }$$, when a bolted three phase to ground short circuit fault occurs at the terminal of the generator. Assuming the input mechanical power to remain at 1 pu, the value of $$\delta $$ in degrees, 0.02 second after the fault is _____________

A sustained three phase fault occurs in the power system shown in the figure. The current and voltage phasors during the fault (on a common reference), after the natural transients have died down, are also shown. Where is the fault located?

Consider the economic dispatch problem for a power plant having two generating units. The fuel costs in $$Rs/MWh$$ along with the generation limits for the two units are given below:
$$\eqalign{ & {C_1}\left( {{P_1}} \right) = 0.01\,P_1^2 + 30{P_1} + 10; \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,100\,MW \le {P_1} \le 150\,MW \cr} $$
$$\eqalign{ & {C_2}\left( {{P_2}} \right) = 0.05\,P_2^2 + 10{P_2} + 10; \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,100\,MW \le {P_2} \le 180\,MW \cr} $$

The incremental cost (in $$Rs/MWh$$) of the power plant when it supplies $$200$$ $$MW$$ is __________.

Base load power plants are
$$P.$$ wind farms
$$Q.$$ run-of-river plants
$$R.$$ nuclear power plants
$$S.$$ diesel power plants

Determine the correctness or otherwise of the following Assertion (a) and the Reason (R).
Assertion (A): Fast decoupled load flow method gives approximate load flow solution because it uses several assumptions.
Reason (R): Accuracy depends on the power mismatch vector tolerance.

Consider a $$HVDC$$ link which uses thyristor based line commutated converters as shown in the figure. For a power flow of $$750$$ $$MW$$ from System $$1$$ to System $$2,$$ the voltages at the two ends, and the current, are given by: $${V_1} = 500\,kV,\,\,{V_2} = 485\,kV\,\,$$ and $$\,{\rm I} = 1.5\,kA.\,\,$$ If the direction of power flow is to be reversed (that is, from System $$2$$ to System $$1$$) without changing the electrical connections, then which one of the following combinations is feasible?

A moving average function is given by $$y\left(t\right)=\frac1T\int_{t-T}^tu\left(\tau\right)d\tau$$. If the input u is a sinusoidal signal of frequency $$\frac1{2T}Hz$$, then in steady state, the output y will lag u (in degree) by ________.

The impulse response g(t) of a system G, is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure (b)?

The signum function is given by $$$\mathrm{sgn}\left(\mathrm x\right)=\left\{\begin{array}{l}\frac{\mathrm x}{\left|\mathrm x\right|};\;\mathrm x\neq0\\0\;;\;\;\mathrm x=0\end{array}\right.$$$ The Fourier series expansion of sgn(cos(t)) has

Consider a discrete time signal given by
x[n]=(-0.25)ⁿu[n]+(0.5)ⁿu[-n-1]
The region of convergence of its Z-transform would be