The region is bounded by $y=0$, $y=x$, and $y=2-x$. Splitting at $x=1$ gives $\int_0^1\int_0^x xy\,dy\,dx + \int_1^2\int_0^{2-x} xy\,dy\,dx$.
Q2mcq1 mark
If $\vec F=(x^2-y^2+2xz)\,\hat i+(xz-xy+yz)\,\hat j+(z^2+x^2)\,\hat k$, then $\operatorname{div}(\operatorname{curl}\vec F)$ is:
a.
0
Correct Answer
b.
$\hat i+\hat j$
c.
6
d.
$-(x+y)\,\hat i+(y+z)\,\hat k$
Step-by-Step AI Explanation
Verified Method
For any sufficiently smooth vector field, $\nabla\cdot(\nabla\times\vec F)=0$ identically. Hence the correct answer is 0.
Q2mcq2 marks
The absolute maximum and minimum values of $f(x)=x^{2/3}$ on the interval $[-2,3]$ are:
a.
$(\sqrt[3]{9},\sqrt[3]{4})$
b.
$(\sqrt[3]{9},0)$
Correct Answer
c.
$(\sqrt[3]{4},\sqrt[3]{9})$
d.
$(\sqrt[3]{4},0)$
Step-by-Step AI Explanation
Verified Method
For $f(x)=|x|^{2/3}$, the minimum on the interval occurs at $x=0$ with value $0$. Checking endpoints gives $f(-2)=2^{2/3}=\sqrt[3]{4}$ and $f(3)=3^{2/3}=\sqrt[3]{9}$, so the absolute maximum is $\sqrt[3]{9}$.
Q3mcq1 mark
A linear differential equation with constant co-efficients corresponding to $(2x+3)^2\,\frac{d^2y}{dx^2}+2(2x+3)\,\frac{dy}{dx}-12y=6x$ by using $z=\log(2x+3)$, is:
With $z=\log(2x+3)$, the variable-coefficient equation transforms to a constant-coefficient ODE. The printed key marks option C, and the transformed form is $\frac{d^2y}{dz^2}-3y=\frac{3}{4}(e^z-3)$.
Q3mcq2 marks
The Fourier transform of $f(x)=\begin{cases}1,&|x|\le 1\\0,&|x|>1\end{cases}$ is:
Using the standard Fourier transform convention for a unit rectangular pulse on $[-1,1]$, the transform is proportional to a sinc function: $\propto \sin s / s$. The printed correct alternative is option A.
Q4mcq2 marks
At the point $x=0.1$, the solution of $\frac{dy}{dx}=x^2-y$, $y(0)=1$, using Taylor series method, is given by:
a.
0.696912
b.
0.757216
c.
0.895215
d.
0.905125
Correct Answer
Step-by-Step AI Explanation
Verified Method
Applying the Taylor expansion about $x=0$ for the differential equation $y'=x^2-y$ with $y(0)=1$ gives the approximation $y(0.1)\approx 0.905125$, matching option D.
Q5mcq1 mark
If $L[f(t)] = \phi(s)$, then the relation between the Laplace transform of $f(t)$ and $f(at)$ is:
a.
$L[f(at)] = a\phi(as)$
b.
$L[f(at)] = \frac{1}{a}\phi(s/a)$
Correct Answer
c.
$L[f(at)] = \frac{1}{a}\phi(as)$
d.
$L[f(at)] = a\phi(s/a)$
Step-by-Step AI Explanation
Verified Method
By the Laplace scaling property, $L\{f(at)\}=\frac{1}{a}F\left(\frac{s}{a}\right)$. With $F(s)=\phi(s)$, this gives option B.
Q5mcq2 marks
In the region $0<r<0.5$ m, in cylindrical co-ordinates, the current density is $\vec{J}=4.5e^{-2r}\hat{a}_z$ (A/m$^2$) and $\vec{J}=0$ elsewhere. Use Ampere's law and find $\vec{H}$.
By symmetry $\vec H=H_\phi\hat a_\phi$ and Ampère's law gives $H_\phi(2\pi r)=I_{\text{enc}}$. Integrating the given $J_z$ over the circular area yields the expression in option B.
Q6mcq1 mark
If $w=u(x,y)+iv(x,y)$ is an analytic function of $z$, then $\frac{dw}{dz}$ is:
For an analytic function, Cauchy-Riemann relations give $\frac{\partial w}{\partial y}=-i\frac{dw}{dz}$, so $\frac{dw}{dz}=-i\frac{\partial w}{\partial y}$.
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In the region $r\le 2$, $\vec{D}=\dfrac{5r^2}{4}\hat{r}$ and $r>2$, $\vec{D}=\dfrac{20}{r^2}\hat{r}$, in spherical coordinates. Find the charge density for $r>2$.
a.
5r
b.
5/r
c.
5
d.
0
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In the circuit, let $a=2$. Connect a resistor $R_x$ in parallel with the voltage source and adjust it within the range $0\le R_x\le 0.99R$ such that the voltage source delivers minimum power. Find the value of $R_x$ and power delivered by the voltage source.
a.
$9.99R,\ V^2/99R$
b.
$99.9R,\ V^2/9.9R$
c.
$0.99R,\ V^2/9.9R$
d.
$0.99R,\ V^2/99R$
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Considering unit value as reference, match the approximate values of armature current and speed in Column 2 with respective condition in Column 1. Select the correct answer in the code given below:
Column 1
(a) Terminal voltage halved, field current and load torque constant
(b) Terminal voltage halved, field current and power output constant
(c) Field flux doubled, armature voltage and load torque constant
(d) Field flux halved, armature voltage halved and power output constant
Column 2
(i) 0.5, 0.5
(ii) 1.0, 0.5
(iii) 2.0, 0.5
(iv) 2.0, 1.0
a.
(a)-(i), (b)-(iii), (c)-(ii), (d)-(iv)
b.
(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
c.
(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
d.
(a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)
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A 220 V, 50 Hz, transformer with 0.35 mm thick laminations has eddy current loss of 120 watts which is two-third of the total loss at no-load. If the transformer is built with 0.7 mm thick laminations and is worked from 110 V, 25 Hz, then total no-load loss would be:
a.
150 W
b.
510 W
c.
200 W
d.
45 W
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A three phase delta connected squirrel cage induction motor has a starting current $I_d$ and a starting torque $T_d$ at rated voltage. If the starting current and starting torque while the motor is started through star-delta starter and auto-transformer (with 60% voltage) starter alternatively are $I_y$, $T_y$ and $I_{out}$, $T_{out}$ respectively, then $I_y/I_d : I_{out}/I_d : T_y/T_d : T_{out}/T_d$ is equal to:
a.
$1/\sqrt{3} : 0.6 : 1/\sqrt{3} : 0.6$
b.
$1/3 : 0.6 : 1/3 : 0.36$
c.
$1/3 : 0.36 : 1/\sqrt{3} : 0.6$
d.
$1/\sqrt{3} : 0.36 : 1/3 : 0.36$
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A 3 phase synchronous motor, connected to an infinite bus, is operating with normal excitation. With decrease in load:
(a) Armature current decreases
(b) P.f. becomes lagging
(c) P.f. becomes leading
(d) load angle decreases
(e) reactive power flows from motor to bus
(f) reactive power flows from bus to motor
From the above statements, which of them are correct:
a.
(a), (b), (d)
b.
(c), (d), (e)
c.
(b), (d), (e)
d.
(a), (b), (d), (f)
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The voltage across $R$ and $L$ in a series $R$-$L$ circuit are found to be 40 V and 30 V respectively. The r.m.s. value of the voltage across the series combination is:
a.
50 V
b.
25 V
c.
45 V
d.
80 V
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A long line rated 100 MVA, 400 kV has ABCD constants as below:
$A=D=0.8\angle 5^\circ$; $B=10\angle 40^\circ$; $C=0.001\angle 90^\circ$.
If receiving end voltage at no load is 400 kV, then sending end voltage is _____.
a.
$320\angle 5^\circ$ kV
b.
$320\angle -5^\circ$ kV
c.
$400\angle 5^\circ$ kV
d.
$500\angle -5^\circ$ kV
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A long line rated 100 MVA, 400 kV has ABCD constants as below: A = D = 0.8∠5°, B = 10∠40°, C = 0.001∠90°. If receiving end voltage at no load is 400 kV, then sending end voltage is ________.
a.
320∠5° kV
b.
320∠−5° kV
c.
400∠5° kV
d.
500∠−5° kV
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A 200 m radial line AB with resistance $0.2\,\Omega$/km (go and return) is loaded with 100 A at end B and 100 A at midpoint C. If voltage at sending end A is 200 V, then voltage at midpoint C is:
a.
198 V
b.
196 V
c.
202 V
d.
197.5 V
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A 50 Hz, 4 pole turbo alternator rated at 20 MVA, 13.2 kV has an inertia constant $H=4$ kW/kVA. The kinetic energy stored in the rotor at rated speed is:
a.
10 MJ
b.
80 MJ
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The open-loop transfer function of a unity feedback control system is: $$G(s)H(s)=\frac{10}{s(s+2)(s+K)}$$ Here $K$ is a variable parameter. The system will be stable for all values of:
a.
$K>0$
b.
$K>-2$
c.
$K>-1.55$
d.
$K>1.45$
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A 746 kW, 3 phase, 50 Hz, 16 pole induction motor has rotor impedance of $(0.02+j0.15)\,\Omega$ at standstill. Full load torque is obtained at 360 rpm. Determine the speed at which maximum torque will occur.
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The stator loss of a 3 phase induction motor is 2 kW. When the power input is 90 kW. What will be the rotor copper loss and mechanical power developed if the motor is running at a 4% slip?
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If the field excitation of 10 A in a synchronous generator gives the armature current of 150 A on a short circuit and the terminal voltage of 900 V on open circuit, what will be the internal voltage drop with a load current of 60 A? Neglect synchronous resistance.
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The power input to a 3 phase induction motor is 60 kW. The total stator loss is 1 kW. Find the mechanical power developed, if the motor is running with a slip of 3%.
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A 250 V, 4 pole wave wound dc series motor has 782 conductors on its armature. It has armature and series field resistance of 0.75 Ω. The motor takes a current of 40 A. Estimate its gross torque developed if it has a flux per pole of 25 mWb.
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What is the time required to deliver a charge of 200 Ah through a single-phase halfwave diode rectifier with an output current of 100 A rms and with sinusoidal input voltage:
a.
2 Hours
b.
3.14 Hours
c.
6.28 Hours
d.
4 Hours
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A chopper circuit is operating on TRC control mode at a frequency of 2 kHz of a 230 V dc supply. For output voltage of 170 V, the conduction and blocking periods of a thyristor in each cycle are respectively:
a.
0.386 ms and 0.114 ms
b.
0.369 ms and 0.131 ms
c.
0.390 ms and 0.110 ms
d.
0.131 ms and 0.369 ms
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A sinusoidal flux 0.02 Wb (maximum) links with 55 turns of transformer secondary. Calculate the RMS value of the induced emf in secondary. The supply frequency is 50 Hz.
a.
264.2 V
b.
232.2 V
c.
244.2 V
d.
252.2 V
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An electric motor developing a starting torque of 15 N-m, starts with a load torque of 7 N-m on its shaft. If the acceleration at start is 2 rad/sec$^2$, the moment of inertia of the system must be (neglecting viscous and coulomb friction)
a.
0.25 kg-m$^2$
b.
0.25 Nm$^2$
c.
4 kg-m$^2$
d.
4 N-m$^2$
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If the breaker reclose after some time corresponding to clearing angle $\delta_c$, when the fault vanished, the output will be more than the input and hence the rotor:
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If lead compensator for a system is given by $G_c(s)=K_c\dfrac{s+4.41}{s+18.4}$. Then the new gain crossover frequency of compensated system is given by:
a.
0.227 rad/sec
b.
12.2 rad/sec
c.
9 rad/sec
d.
7.5 rad/sec
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The overall transfer function of a control system is given below: $$\frac{C(s)}{R(s)}=\frac{25}{s^2+5\sqrt{2}\,s+25}$$ Then the resonant peak $M_r$ is:
a.
0.5
b.
$\sqrt{2}$
c.
2
d.
1
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A cascade forward path compensator designed for a system has transfer function given by $G_c(s)=\dfrac{s+0.05}{s+0.005}$. This compensator will improve the following:
a.
Peak overshoot of the system
b.
Damping factor of the system
c.
Steady state error of the system
d.
None of the options
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A half controlled single-phase bridge rectifier is supplying an R-L load. It is operated at a firing angle $\alpha$ and the load current is continuous. The fraction of cycle that the freewheeling diode conduct is:
a.
$\frac{1}{2}$
b.
$1-\frac{\alpha}{\pi}$
c.
$\frac{\alpha}{2\pi}$
d.
$\frac{\alpha}{\pi}$
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A dc battery is charged from a constant dc source of 200 V through a chopper. The dc battery is to be charged from its internal emf of 90 to 120 V. The battery has internal resistance of 1 $\Omega$. For a constant charging current of 10 A, the range of duty cycle is:
a.
0.45 to 0.6
b.
0.5 to 0.65
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The power delivered to a star connected load of $R\ \Omega$ per phase, from a 3-phase bridge inverter fed from fixed dc source is 10 kW for $180^\circ$ mode. For $120^\circ$ mode, the power delivered to load would be:
a.
10 kW
b.
5 kW
c.
6.667 kW
d.
7.5 kW
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An ac induction motor is used for speed control application. It is driven from an inverter with a constant V/f control mode. The motor name-plate details are as follows: $V: 415\ \text{V};\ Ph: 3;\ f: 50\ \text{Hz};\ N: 2850\ \text{rpm}$. The motor is run with the inverter output frequency set at 40 Hz and with half the rated slip. The running speed of the motor is:
a.
2400 rpm
b.
2280 rpm
c.
2340 rpm
d.
2790 rpm
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A three-phase induction motor operates at constant slip frequency while the stator frequency is varied from zero to rated value. Which one of the following statements is correct? The torque developed by the motor is:
a.
proportional to speed
b.
proportional to square of speed
c.
inversely proportional to speed
d.
constant in the complete range upto base speed
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Two alloys contain Silver and Copper in the ratio of 3 : 1 and 5 : 3. In what ratio should the two alloys be added together to get a new alloy if Silver and Copper are in the ratio 2 : 1?
a.
3 : 2 or 1 : 3
b.
6 : 3 or 1 : 2
c.
3 : 6 or 1 : 2
d.
1 : 2 or 3 : 6
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