2016 | GATE 2023 - Important Concepts, Short cuts, Complete Solutions, Examples and Practice Problems

Previous IES Questions on Number Systems, Binary Codes and 1's and 2's Complements

1. Find the value of ‘r’, if √(41)_r = (7)₁₀. (1988)

2. Find the number of 1’s in an unsigned binary representation of given number. 16₃*9 + 16₂*7 + 16*5 + 3. (1991)

3. Find the number of 1’s in an unsigned binary representation of given number. 3 * 4096 + 15 * 256 + 5 * 16 + 3. (1995)

4. Find the value of ‘r’, if √(224)_r = (13)_r. (1997)
a. 10
b. 8
c. 5
d. 6

5. Convert the following: (1993)
a. 110.011₍₂₎ = X₍₁₀₎
b. 1118₍₁₀₎ = Y_(H)

6. Consider a 3 bit error detection and 1 bit error correction hamming code for 4 bit data. The extra parity bits required would be ________________, and to detect 3 bit error, the minimum distance of the code should be ____________. (1992)

7. In 2’s compliment arithmetic, explain the condition for overflow. (1992)
Answer:

8. Following 7 bit single error correcting Hamming coded message is received as 1000110. Determine if the message is correct, by assuming that at most 1-bit error could be corrected. If the message contains an error, find the bit which is erroneous and give the correct message. (1994)

9. What is the distance of the following code? If the code contains 000000, 010101, 000111, 011001 and 111111. (1995)
a. 2
b. 3
c. 4
d. 1

10. The octal representation of an integer is 342₍₈₎. If this were to be treated as an 8 bit integer in an 8085 based computer, its decimal representation is ________. (1998)
a. 226
b. – 98
c. 76
d. – 30

11. Zero has two representations in___________ (1999)
a. Sign magnitude
b. 1’s complement
c. 2’s complement
d. Both a and b

12. The number 43 in 2’s complement representation is (2000)
a. 01010101
b. 11010101
c. 00101011
d. 10101011

13. 2’s complement representation of (– 539)₁₀ in hexadecimal is (2001)
a. ABE
b. DBC
c. DE5
d. 9E7

14. What is equivalent value for the decimal value 0.25 (2002)
a. 0.1₍₂₎
b. 0.01₍₂₎
c. 0.00111…₍₂₎
d. Can’t be represented precisely

15. 2’s complement representation of decimal value (– 15) is (2002)
a. 1111
b. 11111
c. 111111
d. 10001

16. Assume all numbers are in 2’s complement form, which of the following numbers is divisible by 11111011? (2003)
a. 11100111
b. 11100100
c. 11010111
d. 11011011

17. If 73_(x) = 54_(y) then the possible values of x and y are (2004 & 2011)
a. 8, 16
b. 10,12
c. 9,13
d. 8,11

18. Let A = 11111010 and B = 00001010 be two 8-bit 2’s complement numbers. Their product in 2’complement is (2004)
a. 11000100
b. 10011100
c. 10100101
d. 11010101

19. An 8085 microprocessor executes the following instructions: Two numbers are represented in signed 2’s complement form as P = 11101101 and Q = 11100110. If Q is subtracted from P, the value obtained in signed 2’s complement form is (2012)
a. 100000111
b. 00000111
c. 11111001
d. 011111001

20. If (11X1Y)₈ = (12C9)₁₆ then the values of X and Y are (2012)
a. 3, 1
b. 5, 7
c. 7, 5
d. 1, 5

21. Binary data is being represented in size of byte and in 2’s complement form. The number of 0’s present in representation of (– 127)₁₀ is (2012)
a. 8
b. 7
c. 6
d. 5

22. The decimal equivalent of binary number 10110.11 is (2013)
a. 16.75
b. 20.75
c. 16.50
d. 22.75

23. Hexadecimal conversion of decimal number 227 will be (2013)
a. A3
b. E3
c. CC
d. C3

24. A seven bit Hamming code is received as 1111101. What is the correct code? (2013)
a. 1101111
b. 1011111
c. 1111111
d. 1111011

25. The number of 1’s present in the binary representation of 15*256 + 5*16 + 3 are (2014)
a. 8
b. 9
c. 10
d. 11

26. Given (135)_X + (144)_X = (323)_X. what is the value of x? (2005 & 2014)
a. 5
b. 3
c. 12
d. 6

27. Which one of the following is correct answer when 11011₍₂₎ is subtracted from 11101₍₂₎ by using 1’s complement method? (2015)
a. 01001
b. 10001
c. 00011
d. 00010

28. An Excess-3 code arithmetic operation is used to perform (2015)
a. Binary addition
b. Binary subtraction
c. BCD addition
d. BCD subtraction

29. Convert the decimal 41.6875 into octal (2015)
a. 51.54
b. 51.13
c. 54.13
d. 51.51

30. The decimal equivalent of binary 110.001 is (2015)
a. 6.25
b. 6.125
c. 62.5
d. 0.612

31. Given (125)_r = (203)₅. The value of r will be (2015)
a. 16
b. 10
c. 8
d. 6

32. The 9’s complement of (25.639)10 is (2015)
a. 74.360
b. 0.6732
c. 6.732
d. 7.436

33. If (2.3)₄ + (1.2)₄ = (Y)₄, then what is the value of Y? (2005)
a. 10.1
b. 10.01
c. 10.2
d. 1.02

34. The number of 1’s in 8-bit representation of – 127 in 2’s complement form is ‘m’ and that in 1’s complement form is ‘n’. What is the value of m:n? (2005)
a. 2:1
b. 1:2
c. 3:1
d. 1:3

35. Assertion (A): There is no overflow after an addition, if one number is positive and the other is negative.
Reason (R): Adding a positive number to a negative number always produces a result, which is smaller than the larger of the two. (2005)
a. Both A and R are individually true and R is the correct explanation of A
b. Both A and R are individually true and R is not the correct explanation of A
c. A is true but R is false
d. A is false but R is true

36. A Gray code is a/an___________ (2005)
a. Binary weight code
b. Arithmetic code
c. Code which exhibits a single bit change between two consecutive codes
d. Alphanumeric code

37. Which of the following subtraction operations do not result in F₁₆?
1. (BA)₁₆ – (AB)₁₆
2. (BC)₁₆ – (CB)₁₆
3. (CB)₁₆ – (BC)₁₆
Select the correct answer using the codes given below: (2006)
a. Only 1 and 2
b. Only 1 and 3
c. Only 2 and 3
d. All, 1 , 2 and 3

38. What is the Gray code word for the binary number 101011? (2006)
a. 101011
b. 110101
c. 011111
d. 111110

39. What is the addition of (– 64)₁₀ and (80)₁₆? (2007)
a. (– 16)₁₀
b. (16)₁₆
c. (1100000)₂
d. (01000000)₂

40. (24)₈ is expressed in Gray code. Which one of the following is correct? (2008)
a. 11000
b. 10100
c. 11110
d. 11111

41. Two 2’s complement numbers having sign bits ‘x’ and ‘y’ are added and the sign bit of the result is ‘z’. which Boolean function indicates overflow condition? (2008)
a. xyz
b. x’y’z’
c. x’y’z + xyz’
d. xy + yz + zx

42. Consider the following statements: (2009)
1. Taking 2’s complement is equivalent to sign change
2. In the 2’s complement representation, the MSB is zero for a positive number
3. In a 4 bit binary representation of a binary number A, A + 1’s complement of A = 2⁴.
Which of the above statements are correct?
a. 1 and 2 only
b. 1 and 3 only
c. 2 and 3 only
d. 1, 2 and 3

43. The hexadecimal representation of (657)₈ is (2010)
a. 1AF_H
b. D78_H
c. D71_H
d. 32F_H

44. Find radix ‘r’, if √(41)_r = 5₍₁₀₎.

45. Find radix, if 302/20 = 12.1

46. In a positional weight system, x and y are two successive digits and xy = 25₍₁₀₎ and yx = 31₍₁₀₎. Determine radix, x and y.

Network Synthesis & Driving Point Impedance - Topicwise GATE Questions on Network Theory (from 2003)

2005
1. The first and Last critical frequency of an RC driving point impedance function must respectively be
a) a zero and a pole
b) a zero and a zero
c) a pole and a pole
d) a pole and a zero

Solution : https://www.youtube.com/watch?v=noq2x3cDycY

2006
2. The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by
a) RL network only
b) RC network only
c) LC network only
d) RC as well as RL networks

Solution : https://www.youtube.com/watch?v=L4AUTybSCjk

3. A negative resistance R_neg is connected to a passive network N having driving point impedance Z₁(s) as shown below. For Z₂(s) to be positive real,

Solution : https://www.youtube.com/watch?v=ESM5mIZBEeU

2008
4. The driving point impedance of the following network is given as shown. The component values are

Solution : https://www.youtube.com/watch?v=CI2Na_stYRg

Steady State Analysis of AC and DC circuits - Topicwise GATE Questions on Network Theory (from 2003)

2005
1. For the circuit in figure, the instantaneous current i₁(t) is

Solution : https://www.youtube.com/watch?v=BcmlFVwT5C4

2007
2. In the AC network shown in the figure, the phasor voltage V_AB(in volts) is:

Solution : https://www.youtube.com/watch?v=CNz1YebFlDw

2008
Common Data Questions: 3 & 4
The following series RLC circuit with zero initial conditions is excited by a unit impulse function δ(t).

3. For t > 0, the output voltage V_c(t) is

Solution : https://www.youtube.com/watch?v=DB4jmdoaFIk

4. For t > 0, the voltage across the resistor is

Solution (3 & 4) : https://www.youtube.com/watch?v=nn6PggFjKgA

2009
5. The time domain behaviour of an RL circuit is represented as shown below. For an initial current of i(0) = V_o/R, the steady state value of the current is given by

Solution : https://www.youtube.com/watch?v=VBFOXuUEuKE

2010
6. The current I in the circuit shown is

a) - j1 A
b) j1 A
c) 0 A
d) 20 A

Solution : https://www.youtube.com/watch?v=dKZILrD0vOM

2011
7. In the circuit shown below, the Norton equivalent current in amperes with respect to the terminals P and Q is

a) 6.4 - j 4.8
b) 6.56 - j 7.87
c) 10 + j 0
d) 16 + j 0

Solution : https://www.youtube.com/watch?v=HsLyDiBmxx4

8. The circuit shown below is driven by a sinusoidal input V_i = V_pcos(t/RC). The steady state output V_o is

Solution : https://www.youtube.com/watch?v=51KGart3zsQ

9. In the circuit shown below, the current I is equal to

Solution : https://www.youtube.com/watch?v=t9esPoyU5tY

2012
10. In the circuit shown below, the current through the inductor is

Solution : https://www.youtube.com/watch?v=bD4nQW3U4SA

2014
11. In the circuit shown in the figure, the value of node voltage V₂ (in volts) is

a) 22 + j 2
b) 2 + j 22
c) 22 – j 2
d) 2 – j 22

Solution : https://www.youtube.com/watch?v=L9li2Lu26iU

12. The steady state output of the circuit shown in the figure is given by y(t) = A(ω) sin(ωt + φ(ω)).

If the amplitude |A(ω)| = 0.25, then the frequency ω is

Solution : https://www.youtube.com/watch?v=IIlI8e2TeDg

13. In the circuit shown in the figure,

the value of V_o(t) (in volts) for t → ∞ is ___________

Solution : https://www.youtube.com/watch?v=SHT_OgLFZmE

2015
14. In the circuit shown, assume that diodes D₁ and D₂ are ideal. In the steady state condition, the average voltage V_ab (in volts) across 0.5 µF capacitor is ____________

Solution : https://www.youtube.com/watch?v=2ZfO7mplk4g

Resonance (Series and Parallel) - Topicwise GATE Questions on Network Theory (from 2003)

2003
1. A series RLC circuit has a resonance frequency of 1 kHz and a quality factor Q = 100. If each R, L and C is doubled from its original value, the new Q-factor of the circuit is
a) 25
b) 50
c) 100
d) 200

Solution : https://www.youtube.com/watch?v=DCfbRcxcOEM

2004
2. Consider the following statements S1 and S2
S1: at the resonant frequency, the impedance of a series RLC circuit is zero.
S2: In a parallel G-L-C circuit, increasing the conductance G results in increase in its Q-factor.
Which one of the following is correct?
a) S1 is FALSE and S2 is TRUE
b) Both S1 and S2 are TRUE
c) S1 is TRUE and S2 is FALSE
d) Both S1 and S2 are FALSE

Solution : https://www.youtube.com/watch?v=s3hM-idhg2s

2005
3. The condition on R, L and C such that the step response y(t) in figure has no oscillations, is

Solution : https://www.youtube.com/watch?v=JfYzFU9Oc-0

4. In a series RLC circuit, R = 2 kΩ, L = 1 H, and C = 1/400 µF. The resonant frequency is
a) 2 x 10⁴ Hz
b) (1/π) x 10⁴ Hz
c) 10⁴ Hz
d) 2π x 10⁴ Hz

Solution : https://www.youtube.com/watch?v=B9Uk_uA0JbA

2007
5. Two series resonant filters are as shown in the figure. Let the 3-dB bandwidth of filter 1 is B₁ and that of Filter 2 is B₂. The value of B₁/B₂ is

a) 4
b) 1
c) 1/2
d) 1/4

Solution : https://www.youtube.com/watch?v=lMKem0ESd64

2010
6. For parallel RLC circuit, which one of the following statements is NOT correct?
a) The bandwidth of the circuit decreases if R is increased
b) The bandwidth of the circuit remains same if L is increased
c) At resonance, input impedance is a real quantity
d) At resonance, the magnitude of input impedance attains its minimum value.

Solution : https://www.youtube.com/watch?v=wPSqMuj0YhY

2013
7. Two magnetically uncoupled inductive coils have Q factors q₁ and q₂ at the chosen operating frequency. Their respective resistances are R₁ and R₂. When connected in series, their effective Q factor at the same operating frequency is
a) q₁ + q₂
b) (1/q₁) + (1/q₂)
c) (q₁R₁ + q₂R₂) / (R₁ + R₂)
d) (q₁R₂ + q₂R₁) / (R₁ + R₂)

Solution : https://www.youtube.com/watch?v=v7q3ssCyLqw

2014
8. In the circuit shown in the figure, the value of capacitor C (in mF) needed to have critically damped response i(t) is ____________________

Solution : https://www.youtube.com/watch?v=MTY9pmGILi4

9. A series LCR circuit is operated at a frequency different from its resonant frequency. The operating frequency is such that the current leads the supply voltage. The magnitude of current is half the value at resonance. If the values of L, C and R are 1H, 1F and 1Ω respectively, then the operating angular frequency (in rad/sec) is ____________________

Solution : https://www.youtube.com/watch?v=qJluHG5_mmk

10. In the circuit shown in the figure, the angular frequency ω (in rad/sec), at which the Norton equivalent impedance as seen from terminals b – b' is purely resistive, is _____________

Solution : https://www.youtube.com/watch?v=doL-VE7tziY

2015
11. In the circuit shown, at resonance, the amplitude of the sinusoidal voltage (in volts) across the capacitor is ____________________

Solution : https://www.youtube.com/watch?v=YZq-lDiqadg

12. The voltage (V_c) across the capacitor (in volts) in the network shown is ______________

Solution : https://www.youtube.com/watch?v=BiEeuX6a218

13. An LC tank circuit consists of an ideal capacitor C connected in parallel with a coil of inductance L having an internal resistance R. The resonant frequency of the tank circuit is

Solution : https://www.youtube.com/watch?v=Y331K9dgrPs

14. In the circuit shown, the current I flowing through the 50 Ω resistor will be zero, if the value of the capacitor C (in µF) is __________________

Solution : https://www.youtube.com/watch?v=ZSJIgQWJiKU

Network Topology and Mutual Inductance - Topicwise GATE Questions on Network Theory (from 2003)

2003
1. The current flowing through the resistance R in the circuit shown has the form Pcos4t, where p is

a) (0.18 + j0.72)
b) (0.46 + j1.90)
c) - (0.18 + j1.90)
d) - (0.192 + j0.144)

Answer: Data is insufficient as inductor values are not given in the figure.

2004
2. Consider the network graph shown in figure. Which one of the following is NOT a ‘tree’ of this group?

Solution : https://www.youtube.com/watch?v=xjHxZjJq8PI

3. The equivalent inductance measured between the terminals 1 and 2 for the circuit shown in figure, is

a) L₁ + L₂ + M
b) L₁ + L₂ - M
c) L₁ + L₂ + 2M
d) L₁ + L₂ - 2M

Solution : https://www.youtube.com/watch?v=LwuHSGj_pqI

2005
4. Impedance Z as shown in figure is

a) j29 Ω
b) j9 Ω
c) j19 Ω
d) j39 Ω

Solution : https://www.youtube.com/watch?v=tPOeNM7EELI

2008
5. In the following graph, the number of trees (P) and the number of cut-sets (Q) are

a) P = 2, Q = 2
b) P = 2, Q = 6
c) P = 4, Q = 6
d) P = 4, Q = 10

2013
6. The following arrangement consists of an ideal transformer and an attenuator which attenuates by a factor of 0.8. An ac voltage V_WX1 = 100 volts is applied across WX to get an open circuit voltage V_YZ1 across YZ. Next, an ac voltage V_YZ2 = 100 volts is applied across YZ to get an open circuit voltage V_WX2 across WX. Then V_YZ1/V_WX1, V_WX2/V_YZ2 are respectively

Solution : https://www.youtube.com/watch?v=IJX4fwiz-Fo

2014
7. In the magnetically coupled circuit shown in the figure, 56% of the total flux emanating from one coil links the other coil.

The value of the mutual inductance (in H) is _________________

Solution : https://www.youtube.com/watch?v=y2udU1LL1As

Two-Port Networks - Topicwise GATE Questions on Network Theory (from 2003)

2003
1. The impedance parameters Z₁₁ and Z₁₂ of the two port network in figure are

Solution : https://www.youtube.com/watch?v=v16OcXXl_8E

2004
2. For the lattice circuit shown in figure, Z_a = j2Ω and Z_b = 2Ω. The values of the open circuit impedance parameters (Z-parameters) are

Solution : https://www.youtube.com/watch?v=Vn-O5Ll3CwU

2005
3. The ABCD parameters of an ideal n:1 transformer shown in figure are given below. The value of 'x' will be

a) n
b) 1/n
c) n²
d) 1/n²

Solution : https://www.youtube.com/watch?v=wjvAb6BEi9o

4. The h-parameters of the circuit shown in figure are

Solution : https://www.youtube.com/watch?v=lI92Vfd0F1o

2006
5. A two port network is represented by ABCD parameters as shown below. If port-2 is terminated by R_L, the input impedance seen at port-1 is given by

Solution : https://www.youtube.com/watch?v=-emJQU9Qsx8

6. In the two port network shown in the figure below, Z₁₂ and Z₂₁ are respectively

Solution : https://www.youtube.com/watch?v=JFOPiiE7frs

2008
Statement for Linked Answer Questions: 7 & 8
A two port network shown below is excited by external DC sources. The voltages and the currents are measured with voltmeters V₁, V₂ and ammeters A₁, A₂ (all assumed to be ideal), as indicated. Under following switch conditions, the readings obtained are:

7. The Z-parameter matrix for this network is

8. The h-parameter network for this network is

Solution (7 & 8): https://www.youtube.com/watch?v=_-paorTBZZg

2010
9. For the two-port network shown below, the short-circuit admittance parameter matrix is

Solution : https://www.youtube.com/watch?v=IGYPN1wEH2I

2011
10. In the circuit shown below, the network N is described by the following Y matrix: The voltage gain V₂/V₁ is

a) 1/90
b) - 1/90
c) - 1/99
d) - 1/11

Solution : https://www.youtube.com/watch?v=nyPW45iQq2c

2012
Common Data for Questions 11 and 12:
With 10 volts DC connected at port A in the linear nonreciprocal two-port network shown below,

the following were observed:
(i) 1Ω connected at port B draws a current of 3 amps
(ii) 2.5Ω connected at port B draws a current of 2 amps
11. With 10 volts DC connected at port A, the current drawn by 7Ω connected at port B is
a) 3/7 A
b) 5/7 A
c) 1 A
d) 9/7 A

12. For the same network, with 6 volts DC connected at port A, 1Ω connected at port B draws 7/3 amp. If 8 volts DC is connected to port A, the open circuit voltage at port B is
a) 6 V
b) 7 V
c) 8 V
d) 9 V

Solution (11 & 12): https://www.youtube.com/watch?v=ghASFX3RYBA

2014
13. In the h – parameter model of 2 – port network given in the figure shown,

the value of h₂₂ (in siemens) is _____________

Solution : https://www.youtube.com/watch?v=3xxsvowoMWc

14. For the two – port network shown in the figure,

the impedance (Z) matrix (in Ω) is

Solution : https://www.youtube.com/watch?v=eGw_ZeJwOec

2015
15. The 2 – port Admittance matrix of the circuit shown is given by____________

Solution : https://www.youtube.com/watch?v=jQZIEyucrjU

16. The ABCD parameters of the following 2 – port network are

Solution : https://www.youtube.com/watch?v=iIjMzXmILXk

Transient Analysis of AC and DC circuits - Topicwise GATE Questions on Network Theory (from 2003)

2003
Common circuit for Questions 1 & 2:
Assume that the switch S is in position 1 for a long time and thrown to position 2 at t=0.

1. At t = 0⁺, the current i₁ is
a) – V/2R
b) – V/R
c) – V/4R
d) Zero

2. I₁(s) and I₂(s) are the Laplace Transforms of i₁(t) and i₂(t) respectively. The equations for the loop currents I₁(s) and I₂(s) for the same circuit shown, after the switch is brought from position 1 to position 2 at t = 0, are

Solution (1 & 2): https://www.youtube.com/watch?v=IiAPEkdgqQQ

2008
3. In the following cirucit, the switch S is closed at t = 0. The rate of change of current di(0⁺)/dt is given by

Solution : https://www.youtube.com/watch?v=WiYzbTZyhi0

2009
4. The switch in the circuit shown was on position 'a' for a long time, and is moved to position 'b' at time t=0. The current i(t) for t > 0 is given by

Solution : https://www.youtube.com/watch?v=ptI9Yzy17OQ

2010
5. In the cirucit shown, the switch S is open for a long time and is closed at t = 0. The current i(t) for t ≥ 0⁺ is

Solution : https://www.youtube.com/watch?v=1oztLmwhBL0

2011
6. In the circuit shown below, the initial charge on the capacitor is 2.5 mC, with the voltage polarity as indicated. The switch is closed at time t = 0. The current i(t) at a time t after the switch is closed is

Solution : https://www.youtube.com/watch?v=hQdNpuDt3zU

2014
7. In the figure shown, the ideal switch has been open for a long time.

If it is closed at t = 0, then the magnitude of the current (in mA) through the 4 kΩ resistor at t = 0⁺ is ____________

Solution : https://www.youtube.com/watch?v=lr0Teo6g4nA

2015
8. In the circuit shown, the switch SW is thrown from position A to position B at time t=0. The energy (in µJ) taken from the 3 volts source to charge the 0.1 µF capacitor from 0 volts to 3 volts is

a) 0.3
b) 0.45
c) 0.9
d) 3

Solution : https://www.youtube.com/watch?v=ZbFt7Lkclp8

9. In the circuit shown, switch SW is closed at t =0. Assuming zero initial conditions, the value of V_c(t) (in volts) at t = 1 sec is __________________

Solution : https://www.youtube.com/watch?v=_hcH81bC-cg