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521
GATE Electrical 2017 Set 1 | GA Question: 2
Research in the workplace reveals that people work for many reasons ________. Money beside Beside money Money besides Besides money
Research in the workplace reveals that people work for many reasons ________.Money besideBeside moneyMoney besidesBesides money
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Verbal Aptitude
gate2017-ee-1
verbal-ability
english-grammar
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522
GATE Electrical 2017 Set 1 | GA Question: 3
Rahul, Murali, Srinivas and Arul are seated around a square table. Rahul is sitting to the left of Murali. Srinivas is sitting to the right of Arul. Which of the following pairs are seated opposite each other? Rahul and Murali Srinivas and Arul Srinivas and Murali Srinivas and Rahul
Rahul, Murali, Srinivas and Arul are seated around a square table. Rahul is sitting to the left of Murali. Srinivas is sitting to the right of Arul. Which of the followin...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Analytical Aptitude
gate2017-ee-1
analytical-aptitude
logical-reasoning
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523
GATE Electrical 2017 Set 1 | GA Question: 4
Find the smallest number $y$ such that $y \times 162$ is a perfect cube. $24$ $27$ $32$ $36$
Find the smallest number $y$ such that $y \times 162$ is a perfect cube.$24$$27$$32$$36$
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
numerical-computation
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524
GATE Electrical 2017 Set 1 | GA Question: 5
The probability that a $k$-digit number does NOT contain the digits $0, 5$ or $9$ is $0.3^{k}$ $0.6^{k}$ $0.7^{k}$ $0.9^{k}$
The probability that a $k$-digit number does NOT contain the digits $0, 5$ or $9$ is$0.3^{k}$$0.6^{k}$$0.7^{k}$$0.9^{k}$
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
probability
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525
GATE Electrical 2017 Set 1 | GA Question: 6
"The hold of the nationalist imagination on our colonial past is such that anything inadequately or improperly nationalist is just not history." Which of the following statements best reflects the author's opinion? ... filter of nationalism. Our colonial past never happened. Nationalism has to be both adequately and properly imagined.
"The hold of the nationalist imagination on our colonial past is such that anything inadequately or improperly nationalist is just not history."Which of the following sta...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Verbal Aptitude
gate2017-ee-1
verbal-ability
verbal-reasoning
statements-follow
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526
GATE Electrical 2017 Set 1 | GA Question: 7
Six people are seated around a circular table. There are at least two men and two women. There are at least three right-handed persons. Every women has a left-handed person to her immediate right. None of the women are right-handed. The number of women at the table is $2$ $3$ $4$ Cannot be determined.
Six people are seated around a circular table. There are at least two men and two women. There are at least three right-handed persons. Every women has a left-handed pers...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
round-table-arrangement
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0
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527
GATE Electrical 2017 Set 1 | GA Question: 8
The expression $\frac{(x+y)-|x-y|}{2}$ is equal to The maximum of $x$ and $y$ The minimum of $x$ and $y$ $1$ None of the above.
The expression $\frac{(x+y)-|x-y|}{2}$ is equal toThe maximum of $x$ and $y$The minimum of $x$ and $y$$1$None of the above.
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
absolute-value
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528
GATE Electrical 2017 Set 1 | GA Question: 9
Arun, Gulab, Neel and Shweta must choose one shirt each from a pile of four shirts coloured Red, Pink, Blue and White respectively. Arun dislikes the colour red and Shweta dislikes the colour white. Gulab and Neel like all the colours. In how many different ... the shirts so that no one has a shirt with a colour he or she dislikes? $21$ $18$ $16$ $14$
Arun, Gulab, Neel and Shweta must choose one shirt each from a pile of four shirts coloured Red, Pink, Blue and White respectively. Arun dislikes the colour red and Shwet...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
permutation-and-combination
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529
GATE Electrical 2017 Set 1 | GA Question: 10
A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at $25$m intervals in this plot. If in a flood, the water level rises to $525$m, which of the villages $P, Q, R, S, T$ get submerged? $P, Q$ $P, Q, T$ $R, S, T$ $Q, R, S$
A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at $25$m...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Quantitative Aptitude
gate2017-ee-1
numerical-ability
data-interpretation
contour-plots
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530
GATE Electrical 2017 Set 1 | Question: 42
Only one of the real roots of $f(x)=x^{6}-x-1$ lies in the interval $1 \leq x \leq 2$ and bisection method is used to find its value. For achieving an accuracy of $0.001$, the required minimum number of iterations is _________.
Only one of the real roots of $f(x)=x^{6}-x-1$ lies in the interval $1 \leq x \leq 2$ and bisection method is used to find its value. For achieving an accuracy of $0.001$...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Numerical Methods
gate2017-ee-1
numerical-answers
numerical-methods
bisection-method
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531
GATE Electrical 2017 Set 1 | Question: 43
In the circuit shown below, the maximum power transferred to the resistor $R$ is ______ W.
In the circuit shown below, the maximum power transferred to the resistor $R$ is ______ W.
Arjun
15.9k
points
Arjun
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Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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0
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0
answers
532
GATE Electrical 2017 Set 1 | Question: 44
The magnitude of magnetic flux density (B) in micro Teslas $(\mu T)$, at the centre of a loop of wire wound as a regular hexagon of side length $1 m$ carrying a current $(I=1 A)$, and placed in vacuum as shown in the figure is _________. (Give the answer up to two decimal places.)
The magnitude of magnetic flux density (B) in micro Teslas $(\mu T)$, at the centre of a loop of wire wound as a regular hexagon of side length $1 m$ carrying a current $...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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533
GATE Electrical 2017 Set 1 | Question: 45
A $375$W, $230$V, $50$Hz, capacitor start single-phase induction motor has the following constants for the main and auxiliary windings (at starting): $Z_{m}=(12.50+j15.75) \Omega$ ... of the capacitance (in $\mu F$) to be added in series with the auxiliary winding to obtain maximum torque at starting is ________.
A $375$W, $230$V, $50$Hz, capacitor start single-phase induction motor has the following constants for the main and auxiliary windings (at starting): $Z_{m}=(12.50+j15.75...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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534
GATE Electrical 2017 Set 1 | Question: 46
Two parallel connected, three-phase, $50$Hz, $11$kV, star-connected synchronous machines $A$ and $B$, are operating as synchronous condensers. They together supply $50$MVAR to a $11$kV grid. Current supplied by both the machines are equal. ... excitation current of machine $A$ to that of machine $B$ is _______. (Give the answer up to two decimal places.)
Two parallel connected, three-phase, $50$Hz, $11$kV, star-connected synchronous machines $A$ and $B$, are operating as synchronous condensers. They together supply $50$MV...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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535
GATE Electrical 2017 Set 1 | Question: 47
A $220$V DC series motor runs drawing a current of $30$ A from the supply. Armature and field circuit resistances are $0.4\Omega$ and $0.1\Omega$, respectively. The load torque varies as the square of the speed. The flux in the ... the resistance in ohms that should be added in series with the armature is ______. (Give the answer up to two decimal places.)
A $220$V DC series motor runs drawing a current of $30$ A from the supply. Armature and field circuit resistances are $0.4\Omega$ and $0.1\Omega$, respectively. The load ...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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536
GATE Electrical 2017 Set 1 | Question: 48
A three-phase, three winding $\Delta / \Delta / Y (1.1kV/6.6kV/400V)$ transformer is energized from $AC$ mains at the $1.1kV$ side. It supplies $900kVA$ load at $0.8$ power factor lag from the $6.6kV$ winding and $300kVA$ load ... $1.1kV$ winding from the mains is ________. (Give the answer up to one decimal place.)
A three-phase, three winding $\Delta / \Delta / Y (1.1kV/6.6kV/400V)$ transformer is energized from $AC$ mains at the $1.1kV$ side. It supplies $900kVA$ load at $0.8$ pow...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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537
GATE Electrical 2017 Set 1 | Question: 49
A separately excited $DC$ generator supplies $150$A to a $145$V DC grid. The generator is running at $800$RPM. The armature resistance of the generator is $0.1\Omega$. If the speed of the generator is increased to $1000$ RPM, the current in amperes supplied by the generator to the $DC$ grid is ______. (Give the answer up to one decimal place.)
A separately excited $DC$ generator supplies $150$A to a $145$V DC grid. The generator is running at $800$RPM. The armature resistance of the generator is $0.1\Omega$. If...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
+
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0
votes
0
answers
538
GATE Electrical 2017 Set 1 | Question: 50
For a system having transfer function $G(s)=\frac{-s+1}{s+1}$, a unit step input is applied at time $t=0$. The value of the response of the system at $t=1.5 sec$ (rounded off to three decimal places) is ________.
For a system having transfer function $G(s)=\frac{-s+1}{s+1}$, a unit step input is applied at time $t=0$. The value of the response of the system at $t=1.5 sec$ (rounded...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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539
GATE Electrical 2017 Set 1 | Question: 51
Consider a causal and stable $LTI$ system with rational transfer function $H(z)$, whose corresponding impulse response begins at $n=0$. Furthermore, $H(1)=\frac{5}{4}$. The poles of $H(z)$ ... $g[n]=j^{n}h[n]$. The value of $g[8]$ equals __________. (Give the answer up to three decimal places.)
Consider a causal and stable $LTI$ system with rational transfer function $H(z)$, whose corresponding impulse response begins at $n=0$. Furthermore, $H(1)=\frac{5}{4}$. T...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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answers
540
GATE Electrical 2017 Set 1 | Question: 52
The circuit shown in the figure uses matched transistors with a thermal voltage $V_{T}=25$m V. The base currents of the transistors are negligible. The value of the resistance $R$ in $k\Omega$ that is required to provide $1 \mu A$ bias current for the differential amplifier block shown is ________.(Give the answer up to one decimal place.)
The circuit shown in the figure uses matched transistors with a thermal voltage $V_{T}=25$m V. The base currents of the transistors are negligible. The value of the resis...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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541
GATE Electrical 2017 Set 1 | Question: 53
The figure below shows an uncontrolled diode bridge rectifier supplied from a $220$V, $50$Hz, $I$-phase ac source. The load draws a constant current $I_{0}=14A$. The conduction angle of the diode $D_{1}$ in degrees (rounded off to two decimal places) is _______.
The figure below shows an uncontrolled diode bridge rectifier supplied from a $220$V, $50$Hz, $I$-phase ac source. The load draws a constant current $I_{0}=14A$. The cond...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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0
votes
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answers
542
GATE Electrical 2017 Set 1 | Question: 31
Let the signal $x(t)=\sum_{k=-\infty }^{+\infty } (-1)^{k} \delta (t-\frac{k}{2000})$ be passed through an LTI system with frequency response $H(\omega )$ ... $2000+2000\cos (2000 \pi t)+2000\cos (4000 \pi t)$ $4000\cos (2000 \pi t)$ $2000\cos (2000 \pi t)$
Let the signal$x(t)=\sum_{k=-\infty }^{+\infty } (-1)^{k} \delta (t-\frac{k}{2000})$be passed through an LTI system with frequency response $H(\omega )$, as given in the ...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
+
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0
votes
0
answers
543
GATE Electrical 2017 Set 1 | Question: 32
In the system whose signal flow graph is shown in the figure, $U_{1}(s)$ and $U_{2}(s)$ are inputs. The transfer function $\frac{Y(s)}{U_{1}(s)}$ is $\frac{k_{1}}{J Ls^{2}+J Rs+k_{1}k_{2}}$ $\frac{k_{1}}{J Ls^{2}-J Rs-k_{1}k_{2}}$ ... $\frac{k_{1}-U_{2}(sL-R)}{J Ls^{2}-(J R+U_{2}L)s-k_{1}k_{2}+U_{2}R}$
In the system whose signal flow graph is shown in the figure, $U_{1}(s)$ and $U_{2}(s)$ are inputs. The transfer function $\frac{Y(s)}{U_{1}(s)}$ is$\frac{k_{1}}{J Ls^{2}...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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544
GATE Electrical 2017 Set 1 | Question: 33
The transfer function of the system $Y(s)/U(s)$ ... $\frac{(s-4)}{(s^{2}+s-4)} \\ $ $\frac{(s+4)}{(s^{2}-s-4)} \\$
The transfer function of the system $Y(s)/U(s)$ whose state-space equations are given below is:$\begin{bmatrix} \dot{x}_{1}(t)\\ \dot{x}_{2}(t) \end{bmatrix}=\begin{bma...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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545
GATE Electrical 2017 Set 1 | Question: 34
The load shown in the figure is supplied by a $400$V (line-to-line), $3$-phase source (RYB sequence). The load is balanced and inductive, drawing $3464$ VA. When the switch $S$ is in position $N$, the three watt-meters $W_{1}, W_{2},$ and $W_{3}$ ... $W_1=0$; $W_2=1732$; and $W_3=0$ $W_1=866$; $W_2=0$; and $W_3=866$ $W_1=W_2=0$ and $W_3=1732$
The load shown in the figure is supplied by a $400$V (line-to-line), $3$-phase source (RYB sequence). The load is balanced and inductive, drawing $3464$ VA. When the swit...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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546
GATE Electrical 2017 Set 1 | Question: 35
The approximate transfer characteristic for the circuit shown below with an ideal operational amplifier and diode will be
The approximate transfer characteristic for the circuit shown below with an ideal operational amplifier and diode will be
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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547
GATE Electrical 2017 Set 1 | Question: 36
The output expression for the Karnaugh map shown below is $B \overline{D}+BCD$ $B \overline{D}+AB$ $\overline{B}D+ABC$ $B \overline{D}+ABC$
The output expression for the Karnaugh map shown below is$B \overline{D}+BCD$$B \overline{D}+AB$$\overline{B}D+ABC$$B \overline{D}+ABC$
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Analog and Digital Electronics
gate2017-ee-1
analog-and-digital-electronics
boolean-algebra
k-map
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548
GATE Electrical 2017 Set 1 | Question: 37
The logical gate implemented using the circuit shown below where $V_{1}$ and $V_{2}$ are inputs (with $0$V as digital $0$ and $5$V as digital $1$) and $V_{out}$ is the output, is NOT NOR NAND XOR
The logical gate implemented using the circuit shown below where $V_{1}$ and $V_{2}$ are inputs (with $0$V as digital $0$ and $5$V as digital $1$) and $V_{out}$ is the ou...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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549
GATE Electrical 2017 Set 1 | Question: 38
The input voltage $V_{DC}$ of the buck-boost converter shown below varies from $32$V to $72$V. Assume that all components are ideal, inductor current is continuous, and output voltage is ripple free. The range of duty ratio $D$ of the converter for which the magnitude of the ... $0 \leq D \leq 1$ $\frac{1}{3}\leq D \leq \frac{2}{3}$
The input voltage $V_{DC}$ of the buck-boost converter shown below varies from $32$V to $72$V. Assume that all components are ideal, inductor current is continuous, and o...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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550
GATE Electrical 2017 Set 1 | Question: 39
A load is supplied by a $230$ V, $50$ Hz source. The active power $P$ and the reactive power $Q$ consumed by the load are such that $1kW \leq P \leq 2kW$ and $1kVAR \leq Q \leq 2kVAR$. A capacitor connected across the load for power factor ... power factor after power factor correction is $0.447 \text{ lag}$ $0.707 \text{ lag}$ $0.894 \text{ lag}$ $1$
A load is supplied by a $230$ V, $50$ Hz source. The active power $P$ and the reactive power $Q$ consumed by the load are such that $1kW \leq P \leq 2kW$ and $1kVAR \leq ...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
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551
GATE Electrical 2017 Set 1 | Question: 40
The bus admittance matrix for a power system network is $\begin{bmatrix} -j39.9 & j20 & j20\\ j20 & -j39.9 & j20 \\ j20 & j20 & -j39.9 \end{bmatrix} pu.$ There is a transmission line, connected between buses $1$ and $3$, ...
The bus admittance matrix for a power system network is$\begin{bmatrix}-j39.9 & j20 & j20\\ j20 & -j39.9 & j20 \\ j20 & j20 & -j39.9\end{bmatrix} pu.$There is a transmi...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Others
gate2017-ee-1
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552
GATE Electrical 2017 Set 1 | Question: 41
The switch in the figure below was closed for a long time. It is opened at $t=0$. The current in the inductor of $2$H for $t \geq 0$, is $2.5e^{-4t}$ $5e^{-4t}$ $2.5e^{-0.25t}$ $5e^{-0.25t}$
The switch in the figure below was closed for a long time. It is opened at $t=0$. The current in the inductor of $2$H for $t \geq 0$, is $2.5e^{-4t}$$5e^{-4t}$$2.5e^{-0.2...
Arjun
15.9k
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Arjun
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Feb 26, 2017
new
gate2017-ee-1
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553
GATE Electrical 2017 Set 1 | Question: 20
A three-phase, $50$Hz, star-connected cylindrical rotor synchronous machine is running as a motor. The machine is operated from a $6.6$ kV grid and draws current at unity power factor (UPF). The synchronous reactance of the motor is $30 \Omega$ ... $30^{\circ}$. The power delivered to the motor in kW is __________. (Give the answer up to one decimal place).
A three-phase, $50$Hz, star-connected cylindrical rotor synchronous machine is running as a motor. The machine is operated from a $6.6$ kV grid and draws current at unity...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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554
GATE Electrical 2017 Set 1 | Question: 21
A $10$-bus power system consists of four generator buses indexed as $G1, G2, G3, G4$ and six load buses indexed as $L1, L2, L3, L4, L5, L6$. The generator-bus $G1$ is considered as slack bus, and the load buses ... limit. The number of non-linear equations required for solving the load flow problem using Newton - Raphson method in polar form is ___________.
A $10$-bus power system consists of four generator buses indexed as $G1, G2, G3, G4$ and six load buses indexed as $L1, L2, L3, L4, L5, L6$. The generator-bus $G1$ is con...
Arjun
15.9k
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Arjun
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Feb 26, 2017
new
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numerical-answers
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555
GATE Electrical 2017 Set 1 | Question: 22
Consider the unity feedback control system shown. The value of $K$ that results in a phase margin of the system to be $30^{\circ}$ is _________. (Give the answer up to two decimal places.)
Consider the unity feedback control system shown. The value of $K$ that results in a phase margin of the system to be $30^{\circ}$ is _________. (Give the answer up to tw...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
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numerical-answers
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556
GATE Electrical 2017 Set 1 | Question: 23
The following measurements are obtained on a single phase load $V=220 V \pm 1$%, $I=5.0 A \pm 1$% and $W=555 W \pm 2$%. If the power factor is calculated using these measurements, the worst case error in the calculated power factor in percent is _______. (Give answer up to one decimal place.)
The following measurements are obtained on a single phase load $V=220 V \pm 1$%, $I=5.0 A \pm 1$% and $W=555 W \pm 2$%. If the power factor is calculated using these meas...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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557
GATE Electrical 2017 Set 1 | Question: 24
In the converter circuit shown below, the switches are controlled such that the load voltage $v_{0}(t)$ is a $400$ Hz square wave. The RMS value of the fundamental component of $v_{0}(t)$ in volts is ________.
In the converter circuit shown below, the switches are controlled such that the load voltage $v_{0}(t)$ is a $400$ Hz square wave.The RMS value of the fundamental compone...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
gate2017-ee-1
numerical-answers
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558
GATE Electrical 2017 Set 1 | Question: 25
A $3$-phase voltage source inverter is supplied from a $600V$ DC source as shown in the figure below. For a star connected resistive load of $20\Omega$ per phase, the load power for $120^{\circ}$ device conduction, in kW, is ________.
A $3$-phase voltage source inverter is supplied from a $600V$ DC source as shown in the figure below. For a star connected resistive load of $20\Omega$ per phase, the loa...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
new
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numerical-answers
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559
GATE Electrical 2017 Set 1 | Question: 26
A function $f(x)$ is defined as $f(x)= \begin{cases} e^{x}, & x < 1 \\ \text{In } x+ax^{2}+bx, & x\geq 1 \end{cases}$, where $x \in \mathbb{R}$ Which one of the following statement is TRUE? $f(x)$ is NOT differentiable at $x=1$ ... for all values of $a$ and $b$ such that $a+b=e$. $f(x)$ is differentiable at $x=1$ for all values of $a$ and $b$.
A function $f(x)$ is defined as$f(x)= \begin{cases} e^{x}, & x < 1 \\ \text{In } x+ax^{2}+bx, & x\geq 1 \end{cases}$, where $x \in \mathbb{R}$Which one of the followin...
Arjun
15.9k
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Arjun
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Calculus
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calculus
continuity-and-differentiability
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GATE Electrical 2017 Set 1 | Question: 27
Consider the differential equation $(t^{2}-81)\frac{dy}{dt}+5t y=\sin(t)$ with $y(1)=2 \pi$. There exists a unique solution for this differential equation when $t$ belongs to the interval $(-2, 2)$ $(-10, 10)$ $(-10, 2)$ $(0, 10)$
Consider the differential equation $(t^{2}-81)\frac{dy}{dt}+5t y=\sin(t)$ with $y(1)=2 \pi$. There exists a unique solution for this differential equation when $t$ belong...
Arjun
15.9k
points
Arjun
asked
Feb 26, 2017
Differential Equations
gate2017-ee-1
differential-equations
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