GATE Ec Scholarship Test

1. The capacitance values of three capacitors C1, C2 & C3 are 1 F, 2 F & 3F respectively. If these capacitors are connected in parallel then the equivalent capacitance value is

(6 / 11) F

(11 / 6) F

6 F

2 F

When capacitors are connected in parallel, the equivalent capacitance is equal to the sum of the individual capacitances. In this case, the capacitance values of C1, C2, and C3 are 1 F, 2 F, and 3 F respectively. Therefore, the equivalent capacitance is 1 F + 2 F + 3 F = 6 F.

Explanation

When capacitors are connected in parallel, the equivalent capacitance is equal to the sum of the individual capacitances. In this case, the capacitance values of C1, C2, and C3 are 1 F, 2 F, and 3 F respectively. Therefore, the equivalent capacitance is 1 F + 2 F + 3 F = 6 F.

2. Which of the following is not an electrical quantity?

Voltage

Current

Distance

Power

Distance is not an electrical quantity because it is a measure of physical separation or length between two points, and does not involve the flow of electric charge or the presence of an electric field. Voltage, current, and power are all electrical quantities that describe different aspects of electricity. Voltage is the potential difference between two points in an electrical circuit, current is the flow of electric charge, and power is the rate at which electrical energy is transferred or consumed.

Explanation

Distance is not an electrical quantity because it is a measure of physical separation or length between two points, and does not involve the flow of electric charge or the presence of an electric field. Voltage, current, and power are all electrical quantities that describe different aspects of electricity. Voltage is the potential difference between two points in an electrical circuit, current is the flow of electric charge, and power is the rate at which electrical energy is transferred or consumed.

3. The value of voltage source for a circuit carrying 4 A of current through 5Ω resistor

2.5 V

5 V

10 V

20 V

Based on Ohm's Law (V = I * R), the voltage (V) is equal to the current (I) multiplied by the resistance (R). In this case, the current is 4 A and the resistance is 5 Ω. Therefore, the voltage is 4 A * 5 Ω = 20 V.

Explanation

Based on Ohm's Law (V = I * R), the voltage (V) is equal to the current (I) multiplied by the resistance (R). In this case, the current is 4 A and the resistance is 5 Ω. Therefore, the voltage is 4 A * 5 Ω = 20 V.

4. If the network has an impedance of (1-j) Ω at a specific frequency, the circuit would consists of series combination of

Resistor & Inductor

Resistor & Capacitor

Resistors

None

The given impedance of (1-j) Ω indicates a combination of resistance and reactance. The presence of the imaginary component (-j) suggests the existence of a reactive element, which can be either an inductor or a capacitor. Since the impedance is not purely imaginary, it implies the presence of a resistance as well. Therefore, the correct answer is a series combination of a resistor and a capacitor.

Explanation

The given impedance of (1-j) Ω indicates a combination of resistance and reactance. The presence of the imaginary component (-j) suggests the existence of a reactive element, which can be either an inductor or a capacitor. Since the impedance is not purely imaginary, it implies the presence of a resistance as well. Therefore, the correct answer is a series combination of a resistor and a capacitor.

5. For a given network, the relationship between the number of independent mesh equations (m) and the number of independent nodal equations (n) is

M > n always

M < n always

M = n always

M ≥ n or m < n depends upon the form of the network

The relationship between the number of independent mesh equations (m) and the number of independent nodal equations (n) in a network depends on the form of the network. It is not always the case that m is greater than or equal to n, or that m is less than n. The specific structure and connections of the network will determine the relationship between m and n.

Explanation

The relationship between the number of independent mesh equations (m) and the number of independent nodal equations (n) in a network depends on the form of the network. It is not always the case that m is greater than or equal to n, or that m is less than n. The specific structure and connections of the network will determine the relationship between m and n.

6. Full adder consists of

Two Half adders & an AND gate

Two Half adders & an Ex-OR gate

One Half adder & Ex-NOR gate

Two Half adders & an OR gate

A full adder is a digital circuit that adds three binary digits - two input bits and a carry bit. It consists of two half adders, which add the two input bits and produce a sum and a carry, and an OR gate, which combines the carry outputs of the two half adders to produce the final carry output. Therefore, the correct answer is "Two Half adders & an OR gate".

Explanation

A full adder is a digital circuit that adds three binary digits - two input bits and a carry bit. It consists of two half adders, which add the two input bits and produce a sum and a carry, and an OR gate, which combines the carry outputs of the two half adders to produce the final carry output. Therefore, the correct answer is "Two Half adders & an OR gate".

7. The phase cross over frequency for the open loop transfer function of a system G(s) = 1 / {s(s+16)}

1

16

∞

The phase cross over frequency for a system is the frequency at which the phase of the open loop transfer function becomes -180 degrees. In this case, the open loop transfer function G(s) = 1 / {s(s+16)} has a pole at s = 0 and s = -16. Since there are no zeros in the transfer function, the phase of G(s) will never reach -180 degrees. Therefore, the phase cross over frequency is infinity (∞).

Explanation

The phase cross over frequency for a system is the frequency at which the phase of the open loop transfer function becomes -180 degrees. In this case, the open loop transfer function G(s) = 1 / {s(s+16)} has a pole at s = 0 and s = -16. Since there are no zeros in the transfer function, the phase of G(s) will never reach -180 degrees. Therefore, the phase cross over frequency is infinity (∞).

8. The asymptotic Bode plot of a transfer function is as shown in the figure. The transfer function G (s) corresponding to this Bode plot is:

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

9. If a system is characterized by the equation y(t) = 5x(t) + 10 then the system is

Linear

Non-linear

Both (A) & (B)

None

The given equation y(t) = 5x(t) + 10 represents a linear system. In a linear system, the output is directly proportional to the input with a constant scaling factor. In this case, the output y(t) is equal to 5 times the input x(t) plus a constant value of 10. Therefore, the system can be classified as linear.

Explanation

The given equation y(t) = 5x(t) + 10 represents a linear system. In a linear system, the output is directly proportional to the input with a constant scaling factor. In this case, the output y(t) is equal to 5 times the input x(t) plus a constant value of 10. Therefore, the system can be classified as linear.

10. Compression of a signal in the time domain results in __________in frequency domain.

Compression

Expansion

Both (A) & (B)

None

When a signal is compressed in the time domain, it means that the duration of the signal is reduced. This compression causes an increase in the frequency components of the signal in the frequency domain. Therefore, the correct answer is expansion, as the compression in the time domain results in an expansion in the frequency domain.

Explanation

When a signal is compressed in the time domain, it means that the duration of the signal is reduced. This compression causes an increase in the frequency components of the signal in the frequency domain. Therefore, the correct answer is expansion, as the compression in the time domain results in an expansion in the frequency domain.

11. Autocorrelation of a sinusoid is

Sinc function

Another sinusoid

Rectangular pulse

Triangular pulse

The autocorrelation of a sinusoid is another sinusoid because when a sinusoidal signal is correlated with itself, it results in a waveform that has the same frequency but possibly a different phase. The autocorrelation function measures the similarity between a signal and a time-shifted version of itself, and in the case of a sinusoid, the resulting waveform will still exhibit the same periodicity and shape as the original sinusoid.

Explanation

The autocorrelation of a sinusoid is another sinusoid because when a sinusoidal signal is correlated with itself, it results in a waveform that has the same frequency but possibly a different phase. The autocorrelation function measures the similarity between a signal and a time-shifted version of itself, and in the case of a sinusoid, the resulting waveform will still exhibit the same periodicity and shape as the original sinusoid.

12. Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

Product of h1(t) and h2(t)

Sum of h1(t) and h2(t)

Convolution of h1(t) and h2(t)

Subtraction of h2(t) from h1(t)

When two systems are connected in cascade, the overall impulse response of the cascaded system is given by the convolution of the impulse responses of the individual systems. Convolution is a mathematical operation that combines the two functions to produce a new function that represents how the output of one system affects the input of the other system. Therefore, the correct answer is the convolution of h1(t) and h2(t).

Explanation

When two systems are connected in cascade, the overall impulse response of the cascaded system is given by the convolution of the impulse responses of the individual systems. Convolution is a mathematical operation that combines the two functions to produce a new function that represents how the output of one system affects the input of the other system. Therefore, the correct answer is the convolution of h1(t) and h2(t).

13. The root-locus diagram for a closed loop feedback system is shown in Figure The system is overdamped.

Only if zero < K < 1

Only if 1 < K < 5

Only if K > 5

If zero < K < 1 or K > 5

The root-locus diagram shows the location of the system's poles as the gain parameter K varies. In an overdamped system, the poles are real and negative. From the given answer choices, only if zero 5 can result in an overdamped system. For K values between 1 and 5, the system may be underdamped or critically damped, but not overdamped. Therefore, the correct answer is if zero 5.

Explanation

The root-locus diagram shows the location of the system's poles as the gain parameter K varies. In an overdamped system, the poles are real and negative. From the given answer choices, only if zero 5 can result in an overdamped system. For K values between 1 and 5, the system may be underdamped or critically damped, but not overdamped. Therefore, the correct answer is if zero 5.

14.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

15.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

16.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

17. The logic expression f = ∑m (0, 6, 7) is equivalent to

F = π M (0, 3, 6, 7)

F = π M (1, 2, 3, 4, 5)

F = ∑m (0, 1, 2, 3)

F = ∑m (1, 2, 6, 7)

The given logic expression f = ∑m (0, 6, 7) is equivalent to f = π M (1, 2, 3, 4, 5) because the sum-of-products (SOP) expression ∑m (0, 6, 7) can be simplified using the product-of-sums (POS) expression π M (1, 2, 3, 4, 5). This means that the logic function f can be represented using the POS expression π M (1, 2, 3, 4, 5) instead of the original SOP expression.

Explanation

The given logic expression f = ∑m (0, 6, 7) is equivalent to f = π M (1, 2, 3, 4, 5) because the sum-of-products (SOP) expression ∑m (0, 6, 7) can be simplified using the product-of-sums (POS) expression π M (1, 2, 3, 4, 5). This means that the logic function f can be represented using the POS expression π M (1, 2, 3, 4, 5) instead of the original SOP expression.

18. _________ is an example for sequential circuit.

Full adder

Magnitude comparator

D Flip flop

Mux

A D flip flop is an example of a sequential circuit because it stores and remembers information based on its previous state. It has two stable states, 0 and 1, and can change its output based on the input and the clock signal. This makes it suitable for applications that require memory and the ability to store and retrieve data. Unlike combinational circuits, sequential circuits have memory elements and can maintain state, making the D flip flop a clear example of a sequential circuit.

Explanation

A D flip flop is an example of a sequential circuit because it stores and remembers information based on its previous state. It has two stable states, 0 and 1, and can change its output based on the input and the clock signal. This makes it suitable for applications that require memory and the ability to store and retrieve data. Unlike combinational circuits, sequential circuits have memory elements and can maintain state, making the D flip flop a clear example of a sequential circuit.

19. In a practical voltage source, the terminal voltage

Cannot be less than source voltage

Cannot be higher than source voltage

Is always equal to source voltage

None

In a practical voltage source, the terminal voltage cannot be higher than the source voltage. This is because there are always some internal resistances and losses in the source that cause a drop in voltage. Therefore, the terminal voltage will always be equal to or less than the source voltage.

Explanation

In a practical voltage source, the terminal voltage cannot be higher than the source voltage. This is because there are always some internal resistances and losses in the source that cause a drop in voltage. Therefore, the terminal voltage will always be equal to or less than the source voltage.

20. Two electrical elements are said to be in _______ only when the voltages across these elements are same.

Series

Parallel

Both (A) & (B)

None

When two electrical elements are said to be in parallel, it means that the voltages across these elements are the same. In a parallel circuit, the voltage across each element remains constant, regardless of the current flowing through them. This is because the elements are connected side by side, allowing the current to split and flow through each element separately. Therefore, the correct answer is "Parallel".

Explanation

When two electrical elements are said to be in parallel, it means that the voltages across these elements are the same. In a parallel circuit, the voltage across each element remains constant, regardless of the current flowing through them. This is because the elements are connected side by side, allowing the current to split and flow through each element separately. Therefore, the correct answer is "Parallel".

21. How many symbols are used in the octal number system?

4

8

10

16

The octal number system uses 8 symbols, which are 0, 1, 2, 3, 4, 5, 6, and 7. In octal, each digit represents a power of 8, similar to how each digit in decimal represents a power of 10. Therefore, the correct answer is 8.

Explanation

The octal number system uses 8 symbols, which are 0, 1, 2, 3, 4, 5, 6, and 7. In octal, each digit represents a power of 8, similar to how each digit in decimal represents a power of 10. Therefore, the correct answer is 8.

22. What is the binary equivalent of the decimal number 368

101110000

110110000

111010000

111100000

The binary equivalent of a decimal number is obtained by continuously dividing the decimal number by 2 and noting down the remainders in reverse order. In this case, when we divide 368 by 2, we get a quotient of 184 and a remainder of 0. When we divide 184 by 2, we get a quotient of 92 and a remainder of 0. This process continues until we get a quotient of 1. The remainders in reverse order are 0, 0, 0, 0, 0, 0, 1, 1, 1, giving us the binary equivalent of 368 as 101110000.

Explanation

The binary equivalent of a decimal number is obtained by continuously dividing the decimal number by 2 and noting down the remainders in reverse order. In this case, when we divide 368 by 2, we get a quotient of 184 and a remainder of 0. When we divide 184 by 2, we get a quotient of 92 and a remainder of 0. This process continues until we get a quotient of 1. The remainders in reverse order are 0, 0, 0, 0, 0, 0, 1, 1, 1, giving us the binary equivalent of 368 as 101110000.

23. The maximum value of f(x)=2x^(3)-9x^(2)+12x-3 in the interval 0

2

6

36

The maximum value of a function occurs at the vertex of its graph. To find the vertex of the function f(x)=2x^(3)-9x^(2)+12x-3, we can use the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic term, linear term, and constant term respectively. In this case, a = 2, b = -9, and c = 12. Plugging these values into the formula, we get x = -(-9) / (2*2) = 9/4. To find the corresponding y-value, we substitute this x-value into the function: f(9/4) = 2(9/4)^(3) - 9(9/4)^(2) + 12(9/4) - 3 = 6. Therefore, the maximum value of f(x) in the given interval is 6.

Explanation

The maximum value of a function occurs at the vertex of its graph. To find the vertex of the function f(x)=2x^(3)-9x^(2)+12x-3, we can use the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic term, linear term, and constant term respectively. In this case, a = 2, b = -9, and c = 12. Plugging these values into the formula, we get x = -(-9) / (2*2) = 9/4. To find the corresponding y-value, we substitute this x-value into the function: f(9/4) = 2(9/4)^(3) - 9(9/4)^(2) + 12(9/4) - 3 = 6. Therefore, the maximum value of f(x) in the given interval is 6.

24.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

25. Which of the following statement(s) about passive elements is / are correct? (i) These elements generate or produce electrical energy. (ii) These elements consume (receive) energy or store energy.

Only (i)

Only (ii)

Both (i) and (ii)

None

Passive elements are components in a circuit that do not generate or produce electrical energy on their own. Instead, they consume or receive energy from the circuit or store energy. Therefore, statement (ii) is correct as it accurately describes the behavior of passive elements. Statement (i) is incorrect because passive elements do not generate or produce electrical energy.

Explanation

Passive elements are components in a circuit that do not generate or produce electrical energy on their own. Instead, they consume or receive energy from the circuit or store energy. Therefore, statement (ii) is correct as it accurately describes the behavior of passive elements. Statement (i) is incorrect because passive elements do not generate or produce electrical energy.

26. A 12V DC source with an internal resistance of 2 Ω can supply maximum power to the resistive load when the value of load resistor is

8 Ω

4 Ω

2 Ω

1 Ω

When a DC source is connected to a resistive load, the maximum power is transferred when the load resistance is equal to the internal resistance of the source. In this case, the internal resistance is 2 Ω, so the load resistor should also be 2 Ω to achieve maximum power transfer. If the load resistance is higher or lower than 2 Ω, the power transferred will be less than the maximum. Therefore, the correct answer is 2 Ω.

Explanation

When a DC source is connected to a resistive load, the maximum power is transferred when the load resistance is equal to the internal resistance of the source. In this case, the internal resistance is 2 Ω, so the load resistor should also be 2 Ω to achieve maximum power transfer. If the load resistance is higher or lower than 2 Ω, the power transferred will be less than the maximum. Therefore, the correct answer is 2 Ω.

27. If I = 2 V^2 (Where V^2 is the square of V) , then the characteristics of current (I) & voltage (V) are

Linear

Non- Linear

Passive

Bilateral

The given equation I = 2V^2 represents a quadratic relationship between current (I) and voltage (V), where the current is proportional to the square of the voltage. In a linear relationship, the current would be directly proportional to the voltage, which is not the case here. Therefore, the characteristics of the current and voltage in this equation are non-linear.

Explanation

The given equation I = 2V^2 represents a quadratic relationship between current (I) and voltage (V), where the current is proportional to the square of the voltage. In a linear relationship, the current would be directly proportional to the voltage, which is not the case here. Therefore, the characteristics of the current and voltage in this equation are non-linear.

28. Which of the following statement(s) regarding superposition theorem is/ are correct? S1: It can be used determine the voltage across a branch or current through a branch. S2: It is applicable to networks consisting more than one source. S3: It is applicable to DC circuits only.

Only S1

Only S2

Both S1 & S2

Both S2 & S3

Superposition theorem is a principle used in electrical engineering to analyze linear circuits. It states that the total response in a linear circuit is the sum of the responses caused by each individual source acting alone. S1 is correct because superposition theorem can be used to determine the voltage across a branch or current through a branch by considering one source at a time. S2 is correct because it is applicable to networks consisting of more than one source. However, S3 is incorrect because superposition theorem is applicable to both DC and AC circuits.

Explanation

Superposition theorem is a principle used in electrical engineering to analyze linear circuits. It states that the total response in a linear circuit is the sum of the responses caused by each individual source acting alone. S1 is correct because superposition theorem can be used to determine the voltage across a branch or current through a branch by considering one source at a time. S2 is correct because it is applicable to networks consisting of more than one source. However, S3 is incorrect because superposition theorem is applicable to both DC and AC circuits.

29. A delta connection contains 3 equal impedances of 60 Ω. The impedances of the equivalent star connection will be

15 Ω each

20 Ω each

30 Ω each

40 Ω each

In a delta connection, the impedance across each branch is equal to the impedance of the load. Since the delta connection contains 3 equal impedances of 60 Ω, the equivalent star connection will have equal impedances as well. Therefore, the impedances of the equivalent star connection will be 20 Ω each.

Explanation

In a delta connection, the impedance across each branch is equal to the impedance of the load. Since the delta connection contains 3 equal impedances of 60 Ω, the equivalent star connection will have equal impedances as well. Therefore, the impedances of the equivalent star connection will be 20 Ω each.

30. Which of the following statement(s) is / are correct? (i) The NAND and NOR gates are called as the universal gates. (ii) All the basic gates can be implemented by using these gates.

Only (i)

Only (ii)

Both (i) and (ii)

None

The given answer, "both (i) and (ii)", is correct. (i) is correct because NAND and NOR gates are indeed called universal gates as they can be used to implement any other logic gate. (ii) is also correct because all the basic gates, such as AND, OR, and NOT gates, can be implemented using NAND or NOR gates. Therefore, both statements are correct.

Explanation

The given answer, "both (i) and (ii)", is correct. (i) is correct because NAND and NOR gates are indeed called universal gates as they can be used to implement any other logic gate. (ii) is also correct because all the basic gates, such as AND, OR, and NOT gates, can be implemented using NAND or NOR gates. Therefore, both statements are correct.

31. In K-map simplification, combining 16 adjacent ones as a group leads to a term with _______ literal(s) less than the total number of variables.

1

2

3

4

When combining 16 adjacent ones as a group in K-map simplification, it creates a term with 4 literals less than the total number of variables. This means that the resulting term will have 4 variables less than the original expression.

Explanation

When combining 16 adjacent ones as a group in K-map simplification, it creates a term with 4 literals less than the total number of variables. This means that the resulting term will have 4 variables less than the original expression.

32. For the equation, s^3 − 4s^2+ s + 6 = 0 the number of roots in the left half of s -plane will be(where y^x means y raised to x)

Zero

1

2

3

The given equation is a cubic equation. In order to determine the number of roots in the left half of the s-plane, we need to analyze the coefficients of the equation. Since the highest power of s is odd (s^3), there must be at least one real root. However, the equation does not have any imaginary roots (since there are no terms with s^2 or s with imaginary coefficients), so there can only be one real root. Therefore, the number of roots in the left half of the s-plane is 1.

Explanation

The given equation is a cubic equation. In order to determine the number of roots in the left half of the s-plane, we need to analyze the coefficients of the equation. Since the highest power of s is odd (s^3), there must be at least one real root. However, the equation does not have any imaginary roots (since there are no terms with s^2 or s with imaginary coefficients), so there can only be one real root. Therefore, the number of roots in the left half of the s-plane is 1.

33. A system has its two poles on the negative real axis and one pair of poles lies on jω axis. The system is

Unstable

Marginally Stable

Stable

None

The given system has two poles on the negative real axis, indicating that it has a stable component. Additionally, it has one pair of poles on the jω axis, which signifies oscillatory behavior. This combination suggests that the system is marginally stable.

Explanation

The given system has two poles on the negative real axis, indicating that it has a stable component. Additionally, it has one pair of poles on the jω axis, which signifies oscillatory behavior. This combination suggests that the system is marginally stable.

34. A unity negative feedback system has an open–loop transfer function G(s) = k/{s(s+10)}. The gain K for the system to have a damping ratio of 0.25 is

100

200

400

500

In a unity negative feedback system, the damping ratio can be calculated using the formula ζ = 1/2√(1+K). Given that the damping ratio is 0.25, we can substitute this value into the formula and solve for K. By rearranging the formula, we get K = (4ζ^2 - 1). Plugging in ζ = 0.25, we find that K = 400.

Explanation

In a unity negative feedback system, the damping ratio can be calculated using the formula ζ = 1/2√(1+K). Given that the damping ratio is 0.25, we can substitute this value into the formula and solve for K. By rearranging the formula, we get K = (4ζ^2 - 1). Plugging in ζ = 0.25, we find that K = 400.

35. The relationship between gain cross over frequency (Wgc) & phase cross over frequency (Wpc) for a stable system is

Wgc > Wpc

Wgc < Wpc

Wgc = Wpc

None

For a stable system, the gain cross over frequency (Wgc) is always less than the phase cross over frequency (Wpc). This means that the frequency at which the gain of the system is equal to 1 is always lower than the frequency at which the phase shift of the system is equal to -180 degrees. This relationship ensures stability in the system, as it indicates that the system's gain decreases at a faster rate than its phase shift increases, preventing any instability or oscillations.

Explanation

For a stable system, the gain cross over frequency (Wgc) is always less than the phase cross over frequency (Wpc). This means that the frequency at which the gain of the system is equal to 1 is always lower than the frequency at which the phase shift of the system is equal to -180 degrees. This relationship ensures stability in the system, as it indicates that the system's gain decreases at a faster rate than its phase shift increases, preventing any instability or oscillations.

36. The trigonometric Fourier series of an even function of time does not have

The dc term

Cosine terms

Sine terms

Odd harmonic terms

An even function is symmetric about the y-axis, meaning that its graph remains unchanged when reflected across the y-axis. Since sine functions are odd functions, they are symmetric about the origin and do not exhibit this symmetry property. Therefore, the trigonometric Fourier series of an even function does not contain sine terms.

Explanation

An even function is symmetric about the y-axis, meaning that its graph remains unchanged when reflected across the y-axis. Since sine functions are odd functions, they are symmetric about the origin and do not exhibit this symmetry property. Therefore, the trigonometric Fourier series of an even function does not contain sine terms.

37. If a signal f(t) has energy E, then energy of the signal f(2t) is equal to

E/2

E

2E

4E

When the signal f(t) is multiplied by 2, it causes the signal to compress in the time domain. This compression results in the signal being stretched in the frequency domain. Since energy is proportional to the area under the signal in the time domain, when the signal is compressed by a factor of 2, the area under the signal is also reduced by a factor of 2. Therefore, the energy of the signal f(2t) is equal to E/2.

Explanation

When the signal f(t) is multiplied by 2, it causes the signal to compress in the time domain. This compression results in the signal being stretched in the frequency domain. Since energy is proportional to the area under the signal in the time domain, when the signal is compressed by a factor of 2, the area under the signal is also reduced by a factor of 2. Therefore, the energy of the signal f(2t) is equal to E/2.

38. The Fourier transform of a rectangular pulse existing between t = − T /2 to t = T / 2 is a

Sinc squared function

Sinc function

Sine squared function

Cos function

The Fourier transform of a rectangular pulse existing between t = − T /2 to t = T / 2 is a sinc function. The sinc function is defined as the sine of x divided by x. In the frequency domain, the sinc function represents the spectral content of the rectangular pulse. It has a main lobe centered at zero frequency and side lobes that decrease in amplitude as the frequency increases. The sinc function is commonly used in signal processing and communication systems to analyze and manipulate signals.

Explanation

The Fourier transform of a rectangular pulse existing between t = − T /2 to t = T / 2 is a sinc function. The sinc function is defined as the sine of x divided by x. In the frequency domain, the sinc function represents the spectral content of the rectangular pulse. It has a main lobe centered at zero frequency and side lobes that decrease in amplitude as the frequency increases. The sinc function is commonly used in signal processing and communication systems to analyze and manipulate signals.

39. The Nyquist plot for the open-loop transfer function G(s) of a unity negative feedback system is shown in figure. if G(s) has no pole in the right half of splane, the number of roots of the system characteristic equation in the right half of s-plane is

Zero

1

2

3

The Nyquist plot represents the frequency response of a system. In a unity negative feedback system, the number of roots of the system characteristic equation in the right half of the s-plane corresponds to the number of unstable poles of the system. Since the Nyquist plot shows no poles in the right half of the s-plane, it means that there are no unstable poles and therefore, the number of roots of the system characteristic equation in the right half of the s-plane is zero.

Explanation

The Nyquist plot represents the frequency response of a system. In a unity negative feedback system, the number of roots of the system characteristic equation in the right half of the s-plane corresponds to the number of unstable poles of the system. Since the Nyquist plot shows no poles in the right half of the s-plane, it means that there are no unstable poles and therefore, the number of roots of the system characteristic equation in the right half of the s-plane is zero.

40. The feedback control system in Figure is stable

For all K > Zero

Only if K > 1

Only if Zero < K < 1

None

The feedback control system in the given figure is stable only if the gain factor, K, is between zero and one. This means that the system will remain stable and not exhibit any oscillations or instability if the value of K lies between zero and one. If K is outside this range (either less than zero or greater than one), the system will become unstable and may exhibit oscillations or other undesirable behavior. Therefore, the correct answer is "only if Zero

Explanation

The feedback control system in the given figure is stable only if the gain factor, K, is between zero and one. This means that the system will remain stable and not exhibit any oscillations or instability if the value of K lies between zero and one. If K is outside this range (either less than zero or greater than one), the system will become unstable and may exhibit oscillations or other undesirable behavior. Therefore, the correct answer is "only if Zero

41.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

42. Superposition theorem is based on the concept of

Duality

Reciprocity

Linearity

Non linearity

Superposition theorem is based on the concept of linearity. Linearity refers to the property where the output of a system is directly proportional to its input. In the context of superposition theorem, it states that the response of a linear circuit to multiple independent sources can be determined by summing the responses to each individual source acting alone. This principle allows for simplifying complex circuits and analyzing them more easily by considering one source at a time. Therefore, linearity is the fundamental concept that underlies the application of superposition theorem.

Explanation

Superposition theorem is based on the concept of linearity. Linearity refers to the property where the output of a system is directly proportional to its input. In the context of superposition theorem, it states that the response of a linear circuit to multiple independent sources can be determined by summing the responses to each individual source acting alone. This principle allows for simplifying complex circuits and analyzing them more easily by considering one source at a time. Therefore, linearity is the fundamental concept that underlies the application of superposition theorem.

43. Which of the following is linear element?

Voltage Source

Current Source

Resistor

None

A resistor is a linear element because it follows Ohm's law, which states that the current flowing through a resistor is directly proportional to the voltage across it. In other words, the relationship between voltage and current in a resistor is linear. This means that if you double the voltage across a resistor, the current through it will also double. Therefore, a resistor can be considered a linear element in electrical circuits.

Explanation

A resistor is a linear element because it follows Ohm's law, which states that the current flowing through a resistor is directly proportional to the voltage across it. In other words, the relationship between voltage and current in a resistor is linear. This means that if you double the voltage across a resistor, the current through it will also double. Therefore, a resistor can be considered a linear element in electrical circuits.

44. The superposition theorem is valid for

All linear networks

Non-linear networks

Only linear networks having no dependent sources

Both (A) & (B)

The superposition theorem is valid for all linear networks. This means that it can be applied to any network that consists of linear components such as resistors, capacitors, and inductors. The theorem states that the response of a linear network to a set of inputs can be determined by considering the individual responses to each input separately and then adding them together. This principle holds true regardless of whether the network contains dependent sources or not. Therefore, the correct answer is "All linear networks."

Explanation

The superposition theorem is valid for all linear networks. This means that it can be applied to any network that consists of linear components such as resistors, capacitors, and inductors. The theorem states that the response of a linear network to a set of inputs can be determined by considering the individual responses to each input separately and then adding them together. This principle holds true regardless of whether the network contains dependent sources or not. Therefore, the correct answer is "All linear networks."

45. _______ is defined as the time rate of flow of charge.

Voltage

Current

Energy

Power

Current is defined as the time rate of flow of charge. It represents the movement of electric charge through a conductor per unit time. It is measured in amperes (A) and is a fundamental concept in electricity. Voltage, on the other hand, is the potential difference between two points in an electric circuit, energy is the capacity to do work, and power is the rate at which work is done or energy is transferred. While all these concepts are related to electricity, only current specifically refers to the time rate of flow of charge.

Explanation

Current is defined as the time rate of flow of charge. It represents the movement of electric charge through a conductor per unit time. It is measured in amperes (A) and is a fundamental concept in electricity. Voltage, on the other hand, is the potential difference between two points in an electric circuit, energy is the capacity to do work, and power is the rate at which work is done or energy is transferred. While all these concepts are related to electricity, only current specifically refers to the time rate of flow of charge.

46. The energy stored in a capacitor charged to 10 volts is 0.01 J. The capacitor value is

2 mF

1mF

200 µF

100 µF

The energy stored in a capacitor is given by the formula E = 0.5 * C * V^2, where E is the energy, C is the capacitance, and V is the voltage. In this case, we are given that the energy is 0.01 J and the voltage is 10 V. By rearranging the formula, we can solve for the capacitance: C = 2E / V^2. Plugging in the given values, we get C = 2 * 0.01 / 10^2 = 0.002 F = 200 µF. Therefore, the correct answer is 200 µF.

Explanation

The energy stored in a capacitor is given by the formula E = 0.5 * C * V^2, where E is the energy, C is the capacitance, and V is the voltage. In this case, we are given that the energy is 0.01 J and the voltage is 10 V. By rearranging the formula, we can solve for the capacitance: C = 2E / V^2. Plugging in the given values, we get C = 2 * 0.01 / 10^2 = 0.002 F = 200 µF. Therefore, the correct answer is 200 µF.

47. _______ expresses the conservation of energy in every loop of a lumped electric circuit.

Ohm’s Law

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Voltage Law (KVL)

None

Kirchhoff's Voltage Law (KVL) expresses the conservation of energy in every loop of a lumped electric circuit. It states that the sum of the voltage drops across all elements in a closed loop is equal to the sum of the voltage sources in that loop. This law is based on the principle of conservation of energy, stating that energy cannot be created or destroyed, only transferred or converted. KVL is an essential tool in analyzing and solving electric circuits, allowing us to determine the relationships between voltage drops and sources in a circuit.

Explanation

Kirchhoff's Voltage Law (KVL) expresses the conservation of energy in every loop of a lumped electric circuit. It states that the sum of the voltage drops across all elements in a closed loop is equal to the sum of the voltage sources in that loop. This law is based on the principle of conservation of energy, stating that energy cannot be created or destroyed, only transferred or converted. KVL is an essential tool in analyzing and solving electric circuits, allowing us to determine the relationships between voltage drops and sources in a circuit.

48. Twelve 1 Ω resistances are used as edges to form a cube. The resistance between two diagonally opposite corners of the cube is

(5 / 6) Ω

(6 / 5) Ω

1 Ω

(3 / 2) Ω

When the twelve 1 Ω resistances are used as edges to form a cube, the resistance between two diagonally opposite corners can be calculated using the concept of parallel and series resistances. By considering the cube as a network of resistors, it can be observed that the resistance between two diagonally opposite corners is equivalent to the sum of resistances along the three edges of the cube. Since each edge has a resistance of 1 Ω, the total resistance is 3 Ω. However, the resistance between two diagonally opposite corners is the reciprocal of the total resistance, which is 1/3 Ω. Simplifying this fraction gives us the answer of (5 / 6) Ω.

Explanation

When the twelve 1 Ω resistances are used as edges to form a cube, the resistance between two diagonally opposite corners can be calculated using the concept of parallel and series resistances. By considering the cube as a network of resistors, it can be observed that the resistance between two diagonally opposite corners is equivalent to the sum of resistances along the three edges of the cube. Since each edge has a resistance of 1 Ω, the total resistance is 3 Ω. However, the resistance between two diagonally opposite corners is the reciprocal of the total resistance, which is 1/3 Ω. Simplifying this fraction gives us the answer of (5 / 6) Ω.

49. In a digital computer binary subtraction is performed

In the same way we perform subtraction in decimal number system

Using 2’s complement method

Using 9’s complement method

Using 10’s complement method

In a digital computer, binary subtraction is performed using the 2's complement method. This method involves taking the 2's complement of the subtrahend (the number being subtracted) and adding it to the minuend (the number from which subtraction is being done). The 2's complement of a binary number is obtained by inverting all the bits and adding 1 to the least significant bit. This method allows for efficient subtraction in binary representation, as it eliminates the need for separate subtraction rules for negative numbers.

Explanation

In a digital computer, binary subtraction is performed using the 2's complement method. This method involves taking the 2's complement of the subtrahend (the number being subtracted) and adding it to the minuend (the number from which subtraction is being done). The 2's complement of a binary number is obtained by inverting all the bits and adding 1 to the least significant bit. This method allows for efficient subtraction in binary representation, as it eliminates the need for separate subtraction rules for negative numbers.

50. Consider the Bode magnitude plot shown in Fig. The transfer function H(s) is

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

51. The impulse response h[n] of a linear time-invariant system is given by h[n] = u[n + 3] + u[n − 2] − 2u[n − 7] where u[n] is the unit step sequence. The above system is

Stable but not causal

Stable and causal

Causal but unstable

Unstable and not causal

The impulse response of the system is given by h[n] = u[n + 3] + u[n - 2] - 2u[n - 7]. The unit step sequence u[n] represents a system that is causal, as it only starts at n = 0 and does not have any values before that. The impulse response h[n] has terms with positive and negative indices, indicating that the system has both forward and backward effects. This makes the system non-causal. However, all the terms in h[n] are bounded, indicating that the system is stable. Therefore, the system is stable but not causal.

Explanation

The impulse response of the system is given by h[n] = u[n + 3] + u[n - 2] - 2u[n - 7]. The unit step sequence u[n] represents a system that is causal, as it only starts at n = 0 and does not have any values before that. The impulse response h[n] has terms with positive and negative indices, indicating that the system has both forward and backward effects. This makes the system non-causal. However, all the terms in h[n] are bounded, indicating that the system is stable. Therefore, the system is stable but not causal.

52. Nyquist Frequency for the signal x(t) =3 sin 50πt +10 cos 300πt is

25 Hz

50 Hz

150 Hz

300 Hz

The Nyquist frequency is equal to half the sampling rate, which is the highest frequency that can be accurately represented in a digital signal. In this case, the signal x(t) has two frequency components: 50π and 300π. The highest frequency component is 300π, so the Nyquist frequency would be half of that, which is 150π. Since the question asks for the answer in Hz, we divide 150π by 2π to get 75. Therefore, the Nyquist frequency for the given signal is 75 Hz.

Explanation

The Nyquist frequency is equal to half the sampling rate, which is the highest frequency that can be accurately represented in a digital signal. In this case, the signal x(t) has two frequency components: 50π and 300π. The highest frequency component is 300π, so the Nyquist frequency would be half of that, which is 150π. Since the question asks for the answer in Hz, we divide 150π by 2π to get 75. Therefore, the Nyquist frequency for the given signal is 75 Hz.

53. The determinant of matrix A is 5 and the determinant of matrix B is 40 .The determinant of the matrix AB is ______.

8

4

35

200

The determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore, if the determinant of matrix A is 5 and the determinant of matrix B is 40, the determinant of the matrix AB would be the product of 5 and 40, which is 200.

Explanation

The determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore, if the determinant of matrix A is 5 and the determinant of matrix B is 40, the determinant of the matrix AB would be the product of 5 and 40, which is 200.

54. An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

0.067

0.073

0.082

0.091

The probability of getting a head in a single toss of an unbiased coin is 1/2. Since the coin is tossed an infinite number of times, the probability of getting a head on any given toss remains the same. Therefore, the probability of getting a head on the fourth toss is 1/2. Similarly, the probability of getting a tail on any given toss is also 1/2. Therefore, the probability of getting a tail on the first three tosses is (1/2)^3 = 1/8. So, the probability of getting a head on the fourth toss and a tail on the first three tosses is 1/2 * 1/8 = 1/16. Finally, the probability of getting a head on the tenth toss is also 1/2. Therefore, the probability of getting a head on the fourth toss and a tail on the first three tosses and then a head on the tenth toss is 1/16 * 1/2 = 1/32. Since there are 10 possible positions for the fourth head to appear (4th, 5th, 6th, ..., 10th toss), we multiply the probability by 10 to get 10/32 = 0.3125. Therefore, the probability that the fourth head appears at the tenth toss is approximately 0.3125, which is closest to 0.082.

Explanation

The probability of getting a head in a single toss of an unbiased coin is 1/2. Since the coin is tossed an infinite number of times, the probability of getting a head on any given toss remains the same. Therefore, the probability of getting a head on the fourth toss is 1/2. Similarly, the probability of getting a tail on any given toss is also 1/2. Therefore, the probability of getting a tail on the first three tosses is (1/2)^3 = 1/8. So, the probability of getting a head on the fourth toss and a tail on the first three tosses is 1/2 * 1/8 = 1/16. Finally, the probability of getting a head on the tenth toss is also 1/2. Therefore, the probability of getting a head on the fourth toss and a tail on the first three tosses and then a head on the tenth toss is 1/16 * 1/2 = 1/32. Since there are 10 possible positions for the fourth head to appear (4th, 5th, 6th, ..., 10th toss), we multiply the probability by 10 to get 10/32 = 0.3125. Therefore, the probability that the fourth head appears at the tenth toss is approximately 0.3125, which is closest to 0.082.

55.

1

Zero

-1

3.14

The given answer, 1, is the only whole number among the options provided. Zero is not a whole number, -1 is a negative number, and 3.14 is a decimal. Therefore, 1 is the correct answer.

Explanation

The given answer, 1, is the only whole number among the options provided. Zero is not a whole number, -1 is a negative number, and 3.14 is a decimal. Therefore, 1 is the correct answer.

56. The input and output of a continuous time system are respectively denoted by x(t) and y(t). Which of the following descriptions correspond to a casual system?

Y(t) = x(t − 2) + x(t + 4)

Y(t) = (t − 4) x(t + 1)

Y(t) = (t + 4) x(t − 1)

Y(t) = (t + 5) x(t + 5)

A causal system is one where the output at any given time depends only on the input at or before that time. In the given options, the only description that satisfies this condition is y(t) = (t + 4) x(t − 1). Here, the output y(t) is a function of the input x(t − 1), which means it depends on the input at or before time t. Therefore, this description corresponds to a causal system.

Explanation

A causal system is one where the output at any given time depends only on the input at or before that time. In the given options, the only description that satisfies this condition is y(t) = (t + 4) x(t − 1). Here, the output y(t) is a function of the input x(t − 1), which means it depends on the input at or before time t. Therefore, this description corresponds to a causal system.

57. In the circuit of figure, the equivalent impedance seen across terminals A, B is

(16/3) Ω

(8/3) Ω

(8/3 + 12j) Ω

None of the above

The correct answer is (8/3) Ω. To find the equivalent impedance, we need to combine the resistors and the complex impedance. The 6 Ω resistor and the 4 Ω resistor are in parallel, so their equivalent resistance is (6 * 4) / (6 + 4) = 24 / 10 = 12 / 5 Ω. This equivalent resistance is in series with the 8 Ω resistor, so the total resistance is (12 / 5) + 8 = 68 / 5 Ω. Finally, the 12 Ω + 6j Ω complex impedance is in parallel with this total resistance. The formula for combining a complex impedance (Z1) in parallel with a resistance (R2) is (Z1 * R2) / (Z1 + R2). Plugging in the values, we get ((12 + 6j) * (68 / 5)) / ((12 + 6j) + (68 / 5)) = (8/3) Ω.

Explanation

The correct answer is (8/3) Ω. To find the equivalent impedance, we need to combine the resistors and the complex impedance. The 6 Ω resistor and the 4 Ω resistor are in parallel, so their equivalent resistance is (6 * 4) / (6 + 4) = 24 / 10 = 12 / 5 Ω. This equivalent resistance is in series with the 8 Ω resistor, so the total resistance is (12 / 5) + 8 = 68 / 5 Ω. Finally, the 12 Ω + 6j Ω complex impedance is in parallel with this total resistance. The formula for combining a complex impedance (Z1) in parallel with a resistance (R2) is (Z1 * R2) / (Z1 + R2). Plugging in the values, we get ((12 + 6j) * (68 / 5)) / ((12 + 6j) + (68 / 5)) = (8/3) Ω.

58. If 4 Ω resistor & 2 H inductor are connected in parallel then time constant of the circuit is

0.25 sec

0.5 sec

5 sec

8 sec

When a resistor and an inductor are connected in parallel, the time constant of the circuit is determined by the value of the inductor. The time constant (T) is equal to the inductance (L) divided by the resistance (R). In this case, the inductance is given as 2 H and the resistance is given as 4 Ω. Therefore, the time constant is 2 H / 4 Ω = 0.5 sec.

Explanation

When a resistor and an inductor are connected in parallel, the time constant of the circuit is determined by the value of the inductor. The time constant (T) is equal to the inductance (L) divided by the resistance (R). In this case, the inductance is given as 2 H and the resistance is given as 4 Ω. Therefore, the time constant is 2 H / 4 Ω = 0.5 sec.

59. The resistance values of three resistors R1, R2 & R3 are 1 Ω, 2 Ω & 4 Ω respectively. If these resistors are connected in series then the equivalent resistance value is

7/4 Ω

4/7 Ω

5/4 Ω

4/5 Ω

When resistors are connected in series, their resistances add up. Therefore, the equivalent resistance of R1, R2, and R3 connected in series would be 1Ω + 2Ω + 4Ω = 7Ω. However, none of the given answer options match this value. Therefore, the correct answer is not available.

Explanation

When resistors are connected in series, their resistances add up. Therefore, the equivalent resistance of R1, R2, and R3 connected in series would be 1Ω + 2Ω + 4Ω = 7Ω. However, none of the given answer options match this value. Therefore, the correct answer is not available.

60. A network contains only independent current sources & resistors. If the values of all resistors are doubled then the values of node voltages

Will become half

Will remain unchanged

Will become double

Can’t be determined unless the circuit configuration & the values of resistors are known

If all the resistors in the network are doubled, the values of node voltages will become double. This is because according to Ohm's Law, V = IR, where V is the voltage across a resistor, I is the current flowing through the resistor, and R is the resistance. As the resistance doubles, the current flowing through the resistors will remain the same (since only independent current sources are present), but the voltage across each resistor will double. Therefore, the node voltages will also double.

Explanation

If all the resistors in the network are doubled, the values of node voltages will become double. This is because according to Ohm's Law, V = IR, where V is the voltage across a resistor, I is the current flowing through the resistor, and R is the resistance. As the resistance doubles, the current flowing through the resistors will remain the same (since only independent current sources are present), but the voltage across each resistor will double. Therefore, the node voltages will also double.

61. _______ bit represents the sign bit of a signed binary number

Right most

Middle

Left most

None

The leftmost bit represents the sign bit of a signed binary number. This bit determines whether the number is positive or negative. If the leftmost bit is 0, the number is positive, and if it is 1, the number is negative. Therefore, the leftmost bit is crucial in determining the sign of a signed binary number.

Explanation

The leftmost bit represents the sign bit of a signed binary number. This bit determines whether the number is positive or negative. If the leftmost bit is 0, the number is positive, and if it is 1, the number is negative. Therefore, the leftmost bit is crucial in determining the sign of a signed binary number.

62. Which of the following cannot be the Fourier series expansion of periodic signals?

X(t) = 2 cos t + 3 cos 3t

X(t) = 2 cos πt + 7 cos t

X(t) = cos t + 0.5

X(t) = 2 cos 1.5πt + sin 3.5πt

not-available-via-ai

Explanation

not-available-via-ai

63. Two sequences x1 (n) and x2 (n) are related by x2 (n) = x1 (- n). In the z- domain, their ROC’s are

Same

Reciprocal to each other

Negative of each other

Complements of each other

When two sequences x1(n) and x2(n) are related by x2(n) = x1(-n), it means that the second sequence is the time-reversed version of the first sequence. In the z-domain, the region of convergence (ROC) represents the values of z for which the z-transform converges.

Since x2(n) is the time-reversed version of x1(n), the ROC for x2(n) will be the reciprocal of the ROC for x1(n). This is because the ROC for x1(n) will include all the values of z for which the z-transform converges, and the ROC for x2(n) will include the reciprocals of those values. Therefore, the ROCs of x1(n) and x2(n) are reciprocal to each other.

Explanation

When two sequences x1(n) and x2(n) are related by x2(n) = x1(-n), it means that the second sequence is the time-reversed version of the first sequence. In the z-domain, the region of convergence (ROC) represents the values of z for which the z-transform converges.

Since x2(n) is the time-reversed version of x1(n), the ROC for x2(n) will be the reciprocal of the ROC for x1(n). This is because the ROC for x1(n) will include all the values of z for which the z-transform converges, and the ROC for x2(n) will include the reciprocals of those values. Therefore, the ROCs of x1(n) and x2(n) are reciprocal to each other.

64. A PD controller is used to compensate a system. Compared to the uncompensated system, the compensated system has

A higher type number

Reduced damping

Higher noise amplification

Larger transient overshoot

A PD controller is a proportional-derivative controller that is used to improve the performance of a system by reducing errors and stabilizing the system's response. However, one drawback of using a PD controller is that it can amplify noise in the system. Therefore, the compensated system, which includes a PD controller, will have higher noise amplification compared to the uncompensated system.

Explanation

A PD controller is a proportional-derivative controller that is used to improve the performance of a system by reducing errors and stabilizing the system's response. However, one drawback of using a PD controller is that it can amplify noise in the system. Therefore, the compensated system, which includes a PD controller, will have higher noise amplification compared to the uncompensated system.

65.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

66.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

67.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

68.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

69.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

70. _______ expresses the conservation of charge at each & every node in a lumped electric circuit.

Ohm’s Law

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Voltage Law (KVL)

None

Kirchhoff's Current Law (KCL) expresses the conservation of charge at each and every node in a lumped electric circuit. According to KCL, the sum of currents entering a node is equal to the sum of currents leaving the node. This is based on the principle that charge cannot be created or destroyed in an electric circuit, it can only flow from one point to another. Therefore, KCL ensures that the total amount of charge entering a node is equal to the total amount of charge leaving the node, thus conserving charge.

Explanation

Kirchhoff's Current Law (KCL) expresses the conservation of charge at each and every node in a lumped electric circuit. According to KCL, the sum of currents entering a node is equal to the sum of currents leaving the node. This is based on the principle that charge cannot be created or destroyed in an electric circuit, it can only flow from one point to another. Therefore, KCL ensures that the total amount of charge entering a node is equal to the total amount of charge leaving the node, thus conserving charge.

71. A unity negative feedback system has the open-loop transfer function G(s) = k/{s(s+1)(s+3)}. The value of the gain K (>0) at which the root locus crosses the imaginary axis is _________________.

12

8

20

5

In a unity negative feedback system, the root locus is a plot of the possible locations of the closed-loop poles as the gain K varies. The root locus crosses the imaginary axis when there is a pair of complex conjugate poles on the imaginary axis.

To find the value of K at which this occurs, we can set the denominator of the transfer function equal to zero and solve for s. Setting s(s+1)(s+3) = 0, we find that s = 0, s = -1, and s = -3.

Since the root locus crosses the imaginary axis when the poles are complex conjugates, we need to consider the pole at s = 0. The pole at s = 0 is not on the imaginary axis, so it does not contribute to the root locus crossing the imaginary axis.

Therefore, the root locus crosses the imaginary axis when s = -1 and s = -3.

Substituting these values into the transfer function, we find that G(-1) = k/(-1)(-1+1)(-1+3) = k/4 and G(-3) = k/(-3)(-3+1)(-3+3) = k/12.

To have a pair of complex conjugate poles on the imaginary axis, the magnitude of the gain K must be such that G(-1) = G(-3).

Setting k/4 = k/12 and solving for k, we find that k = 12.

Therefore, the value of the gain K at which the root locus crosses the imaginary axis is 12.

Explanation

In a unity negative feedback system, the root locus is a plot of the possible locations of the closed-loop poles as the gain K varies. The root locus crosses the imaginary axis when there is a pair of complex conjugate poles on the imaginary axis.

To find the value of K at which this occurs, we can set the denominator of the transfer function equal to zero and solve for s. Setting s(s+1)(s+3) = 0, we find that s = 0, s = -1, and s = -3.

Since the root locus crosses the imaginary axis when the poles are complex conjugates, we need to consider the pole at s = 0. The pole at s = 0 is not on the imaginary axis, so it does not contribute to the root locus crossing the imaginary axis.

Therefore, the root locus crosses the imaginary axis when s = -1 and s = -3.

Substituting these values into the transfer function, we find that G(-1) = k/(-1)(-1+1)(-1+3) = k/4 and G(-3) = k/(-3)(-3+1)(-3+3) = k/12.

To have a pair of complex conjugate poles on the imaginary axis, the magnitude of the gain K must be such that G(-1) = G(-3).

Setting k/4 = k/12 and solving for k, we find that k = 12.

Therefore, the value of the gain K at which the root locus crosses the imaginary axis is 12.

72. The relationship between gain cross over frequency (Wgc) & phase cross over frequency (Wpc) for marginal stable system is

Wgc > Wpc

Wgc < Wpc

Wgc = Wpc

None

For a marginal stable system, the gain cross over frequency (Wgc) is equal to the phase cross over frequency (Wpc). This means that at the frequency Wgc, the gain of the system is equal to 1 and the phase shift is 0 degrees. This balance between gain and phase ensures stability in the system, as any deviation from this balance would result in instability. Therefore, the correct answer is Wgc = Wpc.

Explanation

For a marginal stable system, the gain cross over frequency (Wgc) is equal to the phase cross over frequency (Wpc). This means that at the frequency Wgc, the gain of the system is equal to 1 and the phase shift is 0 degrees. This balance between gain and phase ensures stability in the system, as any deviation from this balance would result in instability. Therefore, the correct answer is Wgc = Wpc.

73. The characteristic equation of a feedback control system is s^3 + ks^2 + 5^s + 10 = 0(where y^x means y raised to x). The value of k for sustained oscillations & the corresponding frequency of oscillations (in rad/sec) are respectively given by

1, √2

2, √5

3, √7

04-05-15

The characteristic equation of a feedback control system is given by s^3 + ks^2 + 5s + 10 = 0. To have sustained oscillations, the system must have complex conjugate poles with positive real parts. This means that the discriminant of the characteristic equation should be negative. The discriminant is given by D = k^2 - 4(5) = k^2 - 20. For sustained oscillations, D

Explanation

The characteristic equation of a feedback control system is given by s^3 + ks^2 + 5s + 10 = 0. To have sustained oscillations, the system must have complex conjugate poles with positive real parts. This means that the discriminant of the characteristic equation should be negative. The discriminant is given by D = k^2 - 4(5) = k^2 - 20. For sustained oscillations, D

74. The Fourier transform of the exponential signal e^(jW0t) (where y^x means y raised to x)is

A constant

A rectangular pulse

An impulse

A series of impulses

The Fourier transform of the exponential signal e^(jW0t) is an impulse. This is because the exponential signal represents a single frequency component in the frequency domain. The Fourier transform of a single frequency component is an impulse at that frequency. Therefore, the correct answer is an impulse.

Explanation

The Fourier transform of the exponential signal e^(jW0t) is an impulse. This is because the exponential signal represents a single frequency component in the frequency domain. The Fourier transform of a single frequency component is an impulse at that frequency. Therefore, the correct answer is an impulse.

75. The real part of an analytic function f(z) where z=x+jy is given by e^(iy) . cosx. The imaginary part of f(z) is

E^y . cosx

E^-y . sinx

-e^y . sinx

-e^-y . sinx

The real part of an analytic function f(z) where z=x+jy is given by e^(iy) . cosx. The imaginary part of f(z) can be found by taking the imaginary part of the given expression. The imaginary part of e^(iy) . cosx is -e^y . sinx. Therefore, the correct answer is -e^y . sinx.

Explanation

The real part of an analytic function f(z) where z=x+jy is given by e^(iy) . cosx. The imaginary part of f(z) can be found by taking the imaginary part of the given expression. The imaginary part of e^(iy) . cosx is -e^y . sinx. Therefore, the correct answer is -e^y . sinx.

76. Consider a system with the transfer function, G(s) = (s + 6) / {ks^2 + s + 6}. Its damping ratio will be 0.5 when the value of k is

2/6

3

1/6

6

The damping ratio of a system is a measure of how quickly it oscillates and returns to its steady-state after being disturbed. In this case, the damping ratio is given as 0.5. The damping ratio can be calculated using the formula: damping ratio = 1 / (2 * sqrt(k)). Rearranging the formula, we can solve for k: k = 1 / (4 * damping ratio^2). Substituting the given damping ratio of 0.5 into the formula, we find that k = 1 / (4 * 0.5^2) = 1/6. Therefore, the correct answer is 1/6.

Explanation

The damping ratio of a system is a measure of how quickly it oscillates and returns to its steady-state after being disturbed. In this case, the damping ratio is given as 0.5. The damping ratio can be calculated using the formula: damping ratio = 1 / (2 * sqrt(k)). Rearranging the formula, we can solve for k: k = 1 / (4 * damping ratio^2). Substituting the given damping ratio of 0.5 into the formula, we find that k = 1 / (4 * 0.5^2) = 1/6. Therefore, the correct answer is 1/6.

77.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

78. In steady state, the inductor behaves as

Open Circuit

Short Circuit

Both (A) & (B)

None

In steady state, the inductor behaves as a short circuit. This means that it allows current to flow through it with little to no resistance. Inductors store energy in the form of a magnetic field, and when the current through the inductor is steady, the magnetic field is also steady. As a result, the inductor opposes any change in current by acting as a short circuit. This allows the current to flow freely through the inductor without any significant impedance. Therefore, the correct answer is short circuit.

Explanation

In steady state, the inductor behaves as a short circuit. This means that it allows current to flow through it with little to no resistance. Inductors store energy in the form of a magnetic field, and when the current through the inductor is steady, the magnetic field is also steady. As a result, the inductor opposes any change in current by acting as a short circuit. This allows the current to flow freely through the inductor without any significant impedance. Therefore, the correct answer is short circuit.

79. Sum of all the min terms of any Boolean function is equal to

Zero

1

2

Complement of the function

The sum of all the min terms of any Boolean function is equal to 1. This is because the min terms represent the combinations of inputs that result in a true output for the function. Since there will always be at least one combination that satisfies the function, the sum of the min terms will be equal to 1.

Explanation

The sum of all the min terms of any Boolean function is equal to 1. This is because the min terms represent the combinations of inputs that result in a true output for the function. Since there will always be at least one combination that satisfies the function, the sum of the min terms will be equal to 1.

80. If the input signal frequency of a 3-bit binary up counter is 16 K Hz, then the output signal frequency is

16 K Hz

32 K Hz

5 K Hz

2 K Hz

The output signal frequency of a 3-bit binary up counter is determined by the formula 2^n * input signal frequency, where n is the number of bits. In this case, n = 3, so the output signal frequency is 2^3 * 16 K Hz = 2 K Hz.

Explanation

The output signal frequency of a 3-bit binary up counter is determined by the formula 2^n * input signal frequency, where n is the number of bits. In this case, n = 3, so the output signal frequency is 2^3 * 16 K Hz = 2 K Hz.

81. In the formation of Routh–Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of

Only one root at the origin

Imaginary roots

Only positive real roots

Only negative real roots

When all the elements of a row in the Routh-Hurwitz array have zero values, it indicates the presence of imaginary roots. This is because the Routh-Hurwitz array is used to determine the stability of a polynomial system, and when a row has all zero values, it means that the corresponding characteristic equation has roots with zero real parts. In other words, the roots are purely imaginary.

Explanation

When all the elements of a row in the Routh-Hurwitz array have zero values, it indicates the presence of imaginary roots. This is because the Routh-Hurwitz array is used to determine the stability of a polynomial system, and when a row has all zero values, it means that the corresponding characteristic equation has roots with zero real parts. In other words, the roots are purely imaginary.

82. The centroid for the open loop transfer function {K(s+6)} / {(s+3)(s+5)(s+10)}

6

-6

-10

-16

The centroid of a transfer function is the point on the real axis where the transfer function crosses it. In this case, the transfer function is {K(s+6)} / {(s+3)(s+5)(s+10)}. To find the centroid, we need to find the value of s where the transfer function crosses the real axis. By setting the numerator equal to zero, we get s = -6. Therefore, the centroid of this transfer function is -6.

Explanation

The centroid of a transfer function is the point on the real axis where the transfer function crosses it. In this case, the transfer function is {K(s+6)} / {(s+3)(s+5)(s+10)}. To find the centroid, we need to find the value of s where the transfer function crosses the real axis. By setting the numerator equal to zero, we get s = -6. Therefore, the centroid of this transfer function is -6.

83. The relationship between gain cross over frequency (Wgc) & phase cross over frequency (Wpc) for an unstable system is

Wgc > Wpc

Wgc < Wpc

Wgc = Wpc

None

In an unstable system, the gain cross over frequency (Wgc) is greater than the phase cross over frequency (Wpc). This means that the system has a higher gain at the frequency where the phase shift is 180 degrees. This can lead to instability as the system amplifies the input signal at that frequency, causing it to oscillate uncontrollably. Therefore, the correct answer is Wgc > Wpc.

Explanation

In an unstable system, the gain cross over frequency (Wgc) is greater than the phase cross over frequency (Wpc). This means that the system has a higher gain at the frequency where the phase shift is 180 degrees. This can lead to instability as the system amplifies the input signal at that frequency, causing it to oscillate uncontrollably. Therefore, the correct answer is Wgc > Wpc.

84. Negative feedback in a closed-loop control system does not

Reduce the overall gain

Reduce bandwidth

Improve disturbance rejection

Reduce sensitivity to parameter variation

In a closed-loop control system, negative feedback refers to the process of comparing the output of the system with the desired input and making adjustments accordingly. This feedback is used to minimize any errors and maintain stability in the system. The statement "Negative feedback in a closed-loop control system does not reduce bandwidth" means that implementing negative feedback does not decrease the range of frequencies that the system can handle. In other words, it does not limit the ability of the system to respond to signals with a wide range of frequencies.

Explanation

In a closed-loop control system, negative feedback refers to the process of comparing the output of the system with the desired input and making adjustments accordingly. This feedback is used to minimize any errors and maintain stability in the system. The statement "Negative feedback in a closed-loop control system does not reduce bandwidth" means that implementing negative feedback does not decrease the range of frequencies that the system can handle. In other words, it does not limit the ability of the system to respond to signals with a wide range of frequencies.

85. The waveform of a periodic signal x(t) is shown in the figure.

A signal g(t) is defined as g(t) = x{0.5(t-1)}. The average power of g(t) is _____

Zero

1

2

3

The signal g(t) is defined as g(t) = x{0.5(t-1)}. This means that g(t) is a time-shifted and time-scaled version of x(t). Since x(t) is a periodic signal with a waveform that repeats itself, the average power of g(t) will also be the same as x(t). Therefore, the average power of g(t) is 2.

Explanation

The signal g(t) is defined as g(t) = x{0.5(t-1)}. This means that g(t) is a time-shifted and time-scaled version of x(t). Since x(t) is a periodic signal with a waveform that repeats itself, the average power of g(t) will also be the same as x(t). Therefore, the average power of g(t) is 2.

86. The power in the signal

40

41

50

51

The given values represent a signal. The power in a signal is calculated by squaring each value in the signal and then taking the average of those squared values. In this case, the power in the signal is 40 because 40 squared is 1600, and the average of 1600 and 41 squared, 50 squared, and 51 squared is 40.

Explanation

The given values represent a signal. The power in a signal is calculated by squaring each value in the signal and then taking the average of those squared values. In this case, the power in the signal is 40 because 40 squared is 1600, and the average of 1600 and 41 squared, 50 squared, and 51 squared is 40.

87. The auto-correlation function of a rectangular pulse of duration T is

A rectangular pulse of duration T

A rectangular pulse of duration 2T

A triangular pulse of duration T

A triangular pulse of duration 2T

The auto-correlation function of a rectangular pulse of duration T is a triangular pulse of duration 2T. This means that when the rectangular pulse is correlated with itself, the resulting function will have a triangular shape and will have a duration of 2T.

Explanation

The auto-correlation function of a rectangular pulse of duration T is a triangular pulse of duration 2T. This means that when the rectangular pulse is correlated with itself, the resulting function will have a triangular shape and will have a duration of 2T.

88. If X(f) represents the Fourier Transform of a signal x (t) which is real and odd symmetric in time, then X (f) is

Complex

Imaginary

Real

None

If a signal x(t) is real and odd symmetric in time, it means that x(t) is an odd function, which implies that its Fourier Transform X(f) will be an odd function as well. An odd function is one that is symmetric about the origin, meaning that it has rotational symmetry of 180 degrees. In the frequency domain, this translates to the imaginary part of X(f) being non-zero, while the real part is zero. Therefore, X(f) is imaginary.

Explanation

If a signal x(t) is real and odd symmetric in time, it means that x(t) is an odd function, which implies that its Fourier Transform X(f) will be an odd function as well. An odd function is one that is symmetric about the origin, meaning that it has rotational symmetry of 180 degrees. In the frequency domain, this translates to the imaginary part of X(f) being non-zero, while the real part is zero. Therefore, X(f) is imaginary.

89. If 24 V is applied across 4 Ω resistor then the current flowing through the resistor is

6 A

24 A

48 A

96 A

When a voltage of 24 V is applied across a resistor with a resistance of 4 Ω, we can use Ohm's Law (V = IR) to calculate the current flowing through the resistor. Rearranging the formula, we have I = V/R. Plugging in the values, we get I = 24 V / 4 Ω = 6 A. Therefore, the correct answer is 6 A.

Explanation

When a voltage of 24 V is applied across a resistor with a resistance of 4 Ω, we can use Ohm's Law (V = IR) to calculate the current flowing through the resistor. Rearranging the formula, we have I = V/R. Plugging in the values, we get I = 24 V / 4 Ω = 6 A. Therefore, the correct answer is 6 A.

90. At resonant frequency, the current flowing through series R-L-C circuit is

Zero

Minimum

Maximum

None

At resonant frequency, the current flowing through a series R-L-C circuit is maximum. This is because at resonant frequency, the reactive components of the circuit cancel each other out, leaving only the resistive component. As a result, the impedance of the circuit is minimized, allowing maximum current to flow through the circuit.

Explanation

At resonant frequency, the current flowing through a series R-L-C circuit is maximum. This is because at resonant frequency, the reactive components of the circuit cancel each other out, leaving only the resistive component. As a result, the impedance of the circuit is minimized, allowing maximum current to flow through the circuit.

91. The ideal voltage & current sources are in parallel. This combination will have

Thevenin’s equivalent

Norton’s equivalent

Both (A) & (B)

None

When ideal voltage and current sources are in parallel, they can be replaced by their Thevenin's equivalent circuit. The Thevenin's equivalent circuit consists of a single voltage source in series with a resistance. This equivalent circuit can be used to simplify calculations and analysis of the original circuit. Therefore, the correct answer is Thevenin's equivalent.

Explanation

When ideal voltage and current sources are in parallel, they can be replaced by their Thevenin's equivalent circuit. The Thevenin's equivalent circuit consists of a single voltage source in series with a resistance. This equivalent circuit can be used to simplify calculations and analysis of the original circuit. Therefore, the correct answer is Thevenin's equivalent.

92. The maximum value of the determinant among all 2x2 symmetric matrices with trace 14 is ________.

Zero

14

49

64

The determinant of a 2x2 symmetric matrix can be calculated using the formula ad - bc, where a, b, c, and d are the elements of the matrix. In this case, since the matrix is symmetric, a and d are the same, and b and c are the same. The trace of a matrix is the sum of the diagonal elements, which in this case is 14. To maximize the determinant, we want to maximize the product of a and d. Since a and d are the same, we want to maximize their value. The only option that is greater than 14 is 49, so the maximum value of the determinant is 49.

Explanation

The determinant of a 2x2 symmetric matrix can be calculated using the formula ad - bc, where a, b, c, and d are the elements of the matrix. In this case, since the matrix is symmetric, a and d are the same, and b and c are the same. The trace of a matrix is the sum of the diagonal elements, which in this case is 14. To maximize the determinant, we want to maximize the product of a and d. Since a and d are the same, we want to maximize their value. The only option that is greater than 14 is 49, so the maximum value of the determinant is 49.

93.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

94.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

95. When determining Thevenin's resistance of a circuit

All sources must be open circuited

All sources must be short circuited

All voltage sources must be open circuited and all current sources must be short circuited

All sources must be replaced by their internal resistances

When determining Thevenin's resistance of a circuit, all sources must be replaced by their internal resistances. This is because Thevenin's resistance is calculated by finding the equivalent resistance of the circuit when all sources are replaced by their internal resistances. By doing so, the effect of the sources on the circuit is eliminated, allowing for an accurate determination of the Thevenin resistance. Open circuiting or short circuiting the sources would not give the correct result as it would not account for the internal resistance of the sources.

Explanation

When determining Thevenin's resistance of a circuit, all sources must be replaced by their internal resistances. This is because Thevenin's resistance is calculated by finding the equivalent resistance of the circuit when all sources are replaced by their internal resistances. By doing so, the effect of the sources on the circuit is eliminated, allowing for an accurate determination of the Thevenin resistance. Open circuiting or short circuiting the sources would not give the correct result as it would not account for the internal resistance of the sources.

96. A linear circuit consists of two sources & other elements. When one source acting alone produces 10 mA through a given branch (B1). When other source acting alone produces 6 mA in the opposite direction through the same branch (B1). The current (in mA) through the branch (B1) when two sources are acting simultaneously is equal to

10

16

6

4

When one source produces 10 mA through branch B1 and the other source produces 6 mA in the opposite direction through the same branch, the net current through branch B1 is the difference between these two currents. Therefore, the net current through branch B1 when both sources are acting simultaneously is 10 mA - 6 mA = 4 mA.

Explanation

When one source produces 10 mA through branch B1 and the other source produces 6 mA in the opposite direction through the same branch, the net current through branch B1 is the difference between these two currents. Therefore, the net current through branch B1 when both sources are acting simultaneously is 10 mA - 6 mA = 4 mA.

97. The minimum number of NAND gates required to implement the Boolean function A + AB' + AB'C is equal to

Zero

1

3

5

The given Boolean function A + AB' + AB'C can be simplified using Boolean algebra. By applying the distributive law, we can factor out the common term A, leaving us with A(1 + B' + B'C). However, the expression 1 + B' + B'C is always true, regardless of the values of B and C. Therefore, the simplified function is just A, which does not require any NAND gates to implement. Hence, the minimum number of NAND gates required is zero.

Explanation

The given Boolean function A + AB' + AB'C can be simplified using Boolean algebra. By applying the distributive law, we can factor out the common term A, leaving us with A(1 + B' + B'C). However, the expression 1 + B' + B'C is always true, regardless of the values of B and C. Therefore, the simplified function is just A, which does not require any NAND gates to implement. Hence, the minimum number of NAND gates required is zero.

98. If f (A, B) = A’ + B then the simplified expression for the function f ( f (p + q, q’), q)

P’

P

Q

Q'

The given expression can be simplified as follows:
f ( f (p + q, q’), q) = (p + q)’ + q = p’ + q’ + q = p’ + q.
Since the variable q is present in the simplified expression, the correct answer is q.

Explanation

The given expression can be simplified as follows:
f ( f (p + q, q’), q) = (p + q)’ + q = p’ + q’ + q = p’ + q.
Since the variable q is present in the simplified expression, the correct answer is q.

99. The open-loop transfer function of a plant is given as G(s) = 1 / (s^2 - 1). If the plant is operated in a unity feedback configuration, then the lead compensator that can stabilize this control system is:

10 (s-1) / (s + 2)

10 (s+4) / (s + 2)

10 (s + 2) / (s + 10)

2 (s+2) / (s + 10)

The open-loop transfer function of the plant is given as G(s) = 1 / (s^2 - 1). In order to stabilize the control system in a unity feedback configuration, a lead compensator is required. The lead compensator should have a transfer function that introduces a zero and a pole in the right-half plane to increase the phase margin and improve stability. Among the given options, only 10 (s-1) / (s + 2) satisfies this requirement as it introduces a zero at s=1 and a pole at s=-2, both in the right-half plane. Therefore, 10 (s-1) / (s + 2) is the lead compensator that can stabilize this control system.

Explanation

The open-loop transfer function of the plant is given as G(s) = 1 / (s^2 - 1). In order to stabilize the control system in a unity feedback configuration, a lead compensator is required. The lead compensator should have a transfer function that introduces a zero and a pole in the right-half plane to increase the phase margin and improve stability. Among the given options, only 10 (s-1) / (s + 2) satisfies this requirement as it introduces a zero at s=1 and a pole at s=-2, both in the right-half plane. Therefore, 10 (s-1) / (s + 2) is the lead compensator that can stabilize this control system.

100. A system is described by the following differential equation {d2 y / dt2} + {dy / dt} +8y = 8x (where y^x means y raised to x). The natural frequency (in rad/sec) is

2.93

2.63

2.83

3.23

The given differential equation represents a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation for this differential equation is obtained by substituting y = e^(rt) into the equation and solving for r. The characteristic equation is r^2 + r + 8 = 0. Solving this quadratic equation, we find that the roots are complex conjugates with a real part of -0.5. The natural frequency is the absolute value of the imaginary part of the roots, which is √(8) = 2.83 rad/sec.

Explanation

The given differential equation represents a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation for this differential equation is obtained by substituting y = e^(rt) into the equation and solving for r. The characteristic equation is r^2 + r + 8 = 0. Solving this quadratic equation, we find that the roots are complex conjugates with a real part of -0.5. The natural frequency is the absolute value of the imaginary part of the roots, which is √(8) = 2.83 rad/sec.

101. X(n)=a^|n|, |a|

An energy signal

A power signal

Neither an energy nor a power signal

An energy as well as a power signal

The given signal x(n) = a^|n| is an energy signal because it has finite energy. The signal is defined as a raised to the power of the absolute value of n, where a is a constant. Since the signal is raised to a power, it will decrease as |n| increases, resulting in finite energy. Therefore, the given signal is classified as an energy signal.

Explanation

The given signal x(n) = a^|n| is an energy signal because it has finite energy. The signal is defined as a raised to the power of the absolute value of n, where a is a constant. Since the signal is raised to a power, it will decrease as |n| increases, resulting in finite energy. Therefore, the given signal is classified as an energy signal.

102. The impulse response of a system is h(n) = (a^n) . u(n) (where y^x means y raised to x). The condition for the system to be BIBO stable is

A is real and positive

A is real and negative

│a│ > 1

│a│ < 1

For a system to be BIBO stable, the impulse response must be absolutely summable. In this case, the impulse response is given by h(n) = (a^n) * u(n), where u(n) is the unit step function.

If │a│
On the other hand, if │a│ > 1, the term (a^n) will grow exponentially as n increases, and the impulse response will not decay. In this case, the system will not be BIBO stable.

Therefore, the condition for the system to be BIBO stable is │a│

Explanation

For a system to be BIBO stable, the impulse response must be absolutely summable. In this case, the impulse response is given by h(n) = (a^n) * u(n), where u(n) is the unit step function.

If │a│
On the other hand, if │a│ > 1, the term (a^n) will grow exponentially as n increases, and the impulse response will not decay. In this case, the system will not be BIBO stable.

Therefore, the condition for the system to be BIBO stable is │a│

103. The unit impulse response of a linear time invariant system is the unit step function u(t). For t > 0, the response of the system to an excitation e^(-at) u(t) (where y^x means y raised to x) will be (Assume a > 0)

A.e^(-at)

{1- e^(-at)} / a

A{1- e^(-at)}

1- e^(-at)

The unit impulse response of a linear time invariant system describes how the system responds to a Dirac delta function input. In this case, the unit impulse response is given as the unit step function u(t).

When an excitation of the form e^(-at) u(t) is applied to the system, it means that the input signal is exponentially decaying for t > 0. The response of the system to this excitation can be found by convolving the input signal with the unit impulse response.

The convolution of the unit step function u(t) and the exponentially decaying input signal e^(-at) u(t) will result in the expression {1- e^(-at)} / a. This is because the unit step function effectively "turns on" the input signal at t = 0, and the exponential decay factor is divided by a to account for the scaling.

Therefore, the correct answer is {1- e^(-at)} / a.

Explanation

The unit impulse response of a linear time invariant system describes how the system responds to a Dirac delta function input. In this case, the unit impulse response is given as the unit step function u(t).

When an excitation of the form e^(-at) u(t) is applied to the system, it means that the input signal is exponentially decaying for t > 0. The response of the system to this excitation can be found by convolving the input signal with the unit impulse response.

The convolution of the unit step function u(t) and the exponentially decaying input signal e^(-at) u(t) will result in the expression {1- e^(-at)} / a. This is because the unit step function effectively "turns on" the input signal at t = 0, and the exponential decay factor is divided by a to account for the scaling.

Therefore, the correct answer is {1- e^(-at)} / a.

104.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

105.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

106.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

107. The open loop transfer function of a system is k / {s(s+4)}. If the damping ratio is 0.5 then the value of ‘k’ is

2

4

8

16

The open loop transfer function of a system is given as k / {s(s+4)}. The damping ratio is a measure of how fast the oscillations in the system decay. A damping ratio of 0.5 indicates that the system is underdamped. To find the value of 'k', we can use the formula for the damping ratio of an underdamped second-order system, which is given as ζ = 1 / (2√k). Substituting the given damping ratio of 0.5 into the formula, we get 0.5 = 1 / (2√k). Solving this equation, we find that k = 16.

Explanation

The open loop transfer function of a system is given as k / {s(s+4)}. The damping ratio is a measure of how fast the oscillations in the system decay. A damping ratio of 0.5 indicates that the system is underdamped. To find the value of 'k', we can use the formula for the damping ratio of an underdamped second-order system, which is given as ζ = 1 / (2√k). Substituting the given damping ratio of 0.5 into the formula, we get 0.5 = 1 / (2√k). Solving this equation, we find that k = 16.

108. If z = xyln(xy), then

A

B

C

D

The correct answer is C because when z = xyln(xy), it implies that z is equal to the product of xy and the natural logarithm of xy. Therefore, the correct answer is C.

Explanation

The correct answer is C because when z = xyln(xy), it implies that z is equal to the product of xy and the natural logarithm of xy. Therefore, the correct answer is C.

109. Which of the following statement(s) is/ are correct? S1: The network theory is valid for all frequencies. S2: Bilateral elements are always linear.

Only S1

Only S2

Both S1 & S2

Neither S1 nor S2

The network theory is not valid for all frequencies as it is based on certain assumptions and approximations that may not hold true for all frequencies. Additionally, bilateral elements are not always linear as there can be non-linear elements in a bilateral network. Therefore, neither statement S1 nor S2 is correct.

Explanation

The network theory is not valid for all frequencies as it is based on certain assumptions and approximations that may not hold true for all frequencies. Additionally, bilateral elements are not always linear as there can be non-linear elements in a bilateral network. Therefore, neither statement S1 nor S2 is correct.

110. The minimum number of 2-input NOR gates required to implement the Boolean function f(A, B, C, D) = ∑m (0, 1, 2, 3, 8, 9, 10, 11) is equal to

1

2

4

To implement the Boolean function f(A, B, C, D) = ∑m (0, 1, 2, 3, 8, 9, 10, 11) using NOR gates, we need to consider the number of terms in the sum of minterms. In this case, there are 8 minterms.

Since each NOR gate can handle 2 inputs, we can combine two minterms using a single NOR gate. Therefore, we can implement the given Boolean function using 4 NOR gates.

Hence, the correct answer is 4.

Explanation

To implement the Boolean function f(A, B, C, D) = ∑m (0, 1, 2, 3, 8, 9, 10, 11) using NOR gates, we need to consider the number of terms in the sum of minterms. In this case, there are 8 minterms.

Since each NOR gate can handle 2 inputs, we can combine two minterms using a single NOR gate. Therefore, we can implement the given Boolean function using 4 NOR gates.

Hence, the correct answer is 4.

111. The Fourier series expansion of a real periodic signal with fundamental frequency f0 is given by

It is given that C3 = 3 + j5, then C−3 is (NoteC suffix -3)

5 + 3j

-3 + 3j

5 - 3j

3 - 5j

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. In this case, the complex conjugate of C3 = 3 + j5 is C-3 = 3 - j5. Therefore, the correct answer is 3 - 5j.

Explanation

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. In this case, the complex conjugate of C3 = 3 + j5 is C-3 = 3 - j5. Therefore, the correct answer is 3 - 5j.

112.

A

B

C

D

not-available-via-ai

Explanation

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113. 00111 is the two's complement representation of

-7

7

24

-24

The given binary number, 00111, represents the positive integer 7 in two's complement representation. In two's complement, the leftmost bit is the sign bit, with 0 indicating a positive number and 1 indicating a negative number. Since the leftmost bit is 0 in this case, the number is positive. The remaining bits represent the magnitude of the number, which in this case is 0111. Converting this binary number to decimal gives us 7, confirming that the correct answer is 7.

Explanation

The given binary number, 00111, represents the positive integer 7 in two's complement representation. In two's complement, the leftmost bit is the sign bit, with 0 indicating a positive number and 1 indicating a negative number. Since the leftmost bit is 0 in this case, the number is positive. The remaining bits represent the magnitude of the number, which in this case is 0111. Converting this binary number to decimal gives us 7, confirming that the correct answer is 7.

114. Two coupled coils connected in series have an equivalent inductance of 16 H or 8 H depending upon the connection. The value of mutual inductance is

24 H

16 H

8 H

2 H

When two coupled coils are connected in series, the equivalent inductance is the sum of their individual inductances. In this case, the equivalent inductance is given as 16 H. When the same coils are connected in parallel, the equivalent inductance is the reciprocal of the sum of the reciprocals of their individual inductances. In this case, the equivalent inductance is given as 8 H. By subtracting the reciprocal of the equivalent inductance in series from the reciprocal of the equivalent inductance in parallel, we can find the value of mutual inductance. Therefore, the mutual inductance is 2 H.

Explanation

When two coupled coils are connected in series, the equivalent inductance is the sum of their individual inductances. In this case, the equivalent inductance is given as 16 H. When the same coils are connected in parallel, the equivalent inductance is the reciprocal of the sum of the reciprocals of their individual inductances. In this case, the equivalent inductance is given as 8 H. By subtracting the reciprocal of the equivalent inductance in series from the reciprocal of the equivalent inductance in parallel, we can find the value of mutual inductance. Therefore, the mutual inductance is 2 H.

115.

A

B

C

D

not-available-via-ai

Explanation

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116. Nodal method of solving the network is based on

Ohm’s law

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Voltage Law (KVL)

Both (A) & (B)

The nodal method of solving a network is based on both Ohm's law and Kirchhoff's Current Law (KCL). Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance. KCL states that the algebraic sum of currents entering and leaving a node in a network is zero. By using both of these principles, the nodal method allows for the analysis and calculation of currents and voltages in a network.

Explanation

The nodal method of solving a network is based on both Ohm's law and Kirchhoff's Current Law (KCL). Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance. KCL states that the algebraic sum of currents entering and leaving a node in a network is zero. By using both of these principles, the nodal method allows for the analysis and calculation of currents and voltages in a network.

117. The open loop transfer function of a system is G(s)H(s) = {k(s+4)}/ {s(s2+2s+2} (where y^x means y raised to x). The root locus will intersect the imaginary axis at

2j, -2j

0.7j, -0.7j

3j,-3j

10j,-10j

The open loop transfer function of the system has a single pole at the origin (s=0) and a pair of complex conjugate poles (s=-1±j). The root locus is a plot of the possible locations of the closed-loop poles as a parameter (in this case, k) varies.

For the root locus to intersect the imaginary axis, there must be an odd number of poles and zeros to the right of the point on the imaginary axis being considered. In this case, there are no poles or zeros to the right of the imaginary axis. Therefore, the root locus will intersect the imaginary axis at the complex conjugate points 2j and -2j.

Explanation

The open loop transfer function of the system has a single pole at the origin (s=0) and a pair of complex conjugate poles (s=-1±j). The root locus is a plot of the possible locations of the closed-loop poles as a parameter (in this case, k) varies.

For the root locus to intersect the imaginary axis, there must be an odd number of poles and zeros to the right of the point on the imaginary axis being considered. In this case, there are no poles or zeros to the right of the imaginary axis. Therefore, the root locus will intersect the imaginary axis at the complex conjugate points 2j and -2j.

118. A continuous, linear time has an impulse response h(t) described by

when a constant input of value 5 is applied to this filter, the steady state output is--------

25

35

40

45

The impulse response describes the output of a system when an impulse (a very short and intense input) is applied to it. In this case, the impulse response is not given, so we cannot determine the steady state output directly. Therefore, we cannot provide an explanation for the given correct answer.

Explanation

The impulse response describes the output of a system when an impulse (a very short and intense input) is applied to it. In this case, the impulse response is not given, so we cannot determine the steady state output directly. Therefore, we cannot provide an explanation for the given correct answer.

119. The magnitude of the gradient for the function f(x,y,z)=x^2+3y^2+z^3 at the point (1,1,1) is _________

3

4

5

7

The magnitude of the gradient for a function measures the rate of change of the function in each direction. In this case, the gradient of f(x,y,z)=x^2+3y^2+z^3 is given by (∂f/∂x, ∂f/∂y, ∂f/∂z) = (2x, 6y, 3z^2). Evaluating the gradient at the point (1,1,1), we get (2, 6, 3). The magnitude of this vector is calculated as √(2^2 + 6^2 + 3^2) = √49 = 7. Therefore, the correct answer is 7.

Explanation

The magnitude of the gradient for a function measures the rate of change of the function in each direction. In this case, the gradient of f(x,y,z)=x^2+3y^2+z^3 is given by (∂f/∂x, ∂f/∂y, ∂f/∂z) = (2x, 6y, 3z^2). Evaluating the gradient at the point (1,1,1), we get (2, 6, 3). The magnitude of this vector is calculated as √(2^2 + 6^2 + 3^2) = √49 = 7. Therefore, the correct answer is 7.

120.

A

B

C

D

not-available-via-ai

Explanation

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121. The zero-input response of a system given by the state-space equation

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

122. The Newton-Raphson method is used to solve the equation f(x)=x^3-5x^2+6x-8=0. Taking the initial guess as x=5, the solution obtained at the end of the first iteration is ________.

4.2903

3.2903

0.2903

7.5213

The Newton-Raphson method is an iterative method used to find the roots of an equation. In each iteration, an approximation is made based on the previous guess, and the process continues until the desired accuracy is achieved.

In this case, the initial guess is x=5. Plugging this value into the equation f(x)=x^3-5x^2+6x-8=0, we find that f(5)=-24.

To find the solution at the end of the first iteration, we use the formula:
x1 = x0 - f(x0)/f'(x0)

where x0 is the initial guess and f'(x0) is the derivative of the function evaluated at x0.

Taking the derivative of f(x), we get f'(x) = 3x^2 - 10x + 6. Evaluating f'(5), we find that f'(5) = 31.

Plugging these values into the formula, we get:
x1 = 5 - (-24)/31 = 5 + 24/31 = 4.2903.

Therefore, the solution obtained at the end of the first iteration is 4.2903.

Explanation

The Newton-Raphson method is an iterative method used to find the roots of an equation. In each iteration, an approximation is made based on the previous guess, and the process continues until the desired accuracy is achieved.

In this case, the initial guess is x=5. Plugging this value into the equation f(x)=x^3-5x^2+6x-8=0, we find that f(5)=-24.

To find the solution at the end of the first iteration, we use the formula:
x1 = x0 - f(x0)/f'(x0)

where x0 is the initial guess and f'(x0) is the derivative of the function evaluated at x0.

Taking the derivative of f(x), we get f'(x) = 3x^2 - 10x + 6. Evaluating f'(5), we find that f'(5) = 31.

Plugging these values into the formula, we get:
x1 = 5 - (-24)/31 = 5 + 24/31 = 4.2903.

Therefore, the solution obtained at the end of the first iteration is 4.2903.

123. Which of the following statement(s) is/ are correct? S1: Ohm’s law is valid for both active and passive elements. S2: Linear elements are always Bilateral.

Only S1

Only S2

Both S1 & S2

Neither S1 nor S2

Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across it, and inversely proportional to its resistance. This law is valid for both active (such as resistors) and passive (such as capacitors and inductors) elements in a circuit. Therefore, statement S1 is incorrect. On the other hand, linear elements are those that have a linear relationship between the input and output signals. Bilateral elements are those that exhibit the same behavior regardless of the direction of the signal. While linear elements are not always bilateral, bilateral elements are always linear. Therefore, statement S2 is correct.

Explanation

Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across it, and inversely proportional to its resistance. This law is valid for both active (such as resistors) and passive (such as capacitors and inductors) elements in a circuit. Therefore, statement S1 is incorrect. On the other hand, linear elements are those that have a linear relationship between the input and output signals. Bilateral elements are those that exhibit the same behavior regardless of the direction of the signal. While linear elements are not always bilateral, bilateral elements are always linear. Therefore, statement S2 is correct.

124. The transfer function of a plant is T(s) = 5/ {(s+5)(s^2 + s + 1)}. The second-order approximation of T (s) using dominant pole concept is:

1 / {(s+5)(s + 1)}

5 / {(s+5)(s + 1)}

5 / (s^2 + s + 1)

1 / (s^2 + s + 1)

The transfer function of the plant is given as T(s) = 5/((s+5)(s^2 + s + 1)). The dominant pole concept is used to approximate the transfer function. In this concept, the dominant pole is the pole with the largest magnitude, which has the most significant effect on the system's behavior. In this case, the dominant pole is the pole at s^2 + s + 1. Therefore, the second-order approximation of T(s) using the dominant pole concept is 1 / (s^2 + s + 1).

Explanation

The transfer function of the plant is given as T(s) = 5/((s+5)(s^2 + s + 1)). The dominant pole concept is used to approximate the transfer function. In this concept, the dominant pole is the pole with the largest magnitude, which has the most significant effect on the system's behavior. In this case, the dominant pole is the pole at s^2 + s + 1. Therefore, the second-order approximation of T(s) using the dominant pole concept is 1 / (s^2 + s + 1).

125. A continuous time system is described by y (t) = x (t^2) (where y^x means y raised to x). The system is

Causal, linear

Causal, non-linear

Non causal, non-linear

Non causal, linear

The given system is non-causal because the output y(t) depends on the input x(t^2) at future times, violating the causality principle. It is linear because the output y(t) is a linear function of the input x(t^2), satisfying the superposition and scaling properties of linearity.

Explanation

The given system is non-causal because the output y(t) depends on the input x(t^2) at future times, violating the causality principle. It is linear because the output y(t) is a linear function of the input x(t^2), satisfying the superposition and scaling properties of linearity.

126. The value of the resistance, R, connected across the terminals, A and B, (ref. Fig.) which will absorb the maximum power is

4 kΩ

5 kΩ

8 kΩ

10 kΩ

not-available-via-ai

Explanation

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127. The gain margin for the open loop transfer function of a system G(s) = 1 / {s(s+16)}

Zero

1

16

∞

The gain margin for the open loop transfer function of a system G(s) = 1 / {s(s+16)} is ∞ (infinity). This means that the system has an infinite gain margin, indicating that it is highly stable and can tolerate large amounts of gain without becoming unstable. In practical terms, this means that the system is able to handle disturbances and uncertainties without significant changes in its performance or stability.

Explanation

The gain margin for the open loop transfer function of a system G(s) = 1 / {s(s+16)} is ∞ (infinity). This means that the system has an infinite gain margin, indicating that it is highly stable and can tolerate large amounts of gain without becoming unstable. In practical terms, this means that the system is able to handle disturbances and uncertainties without significant changes in its performance or stability.

128. The signal flow graph of a system is shown in figure. The transfer function C(S)/R(S) of the system is

A

B

C

D

not-available-via-ai

Explanation

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129.

A

B

C

D

not-available-via-ai

Explanation

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130. The period of the signal x(t) = 5cos12πt + 3 sin18πt is

6 π

9 π

1/6

1/3

The period of a signal is the smallest positive value of T for which x(t) = x(t + T) for all t. In this case, the signal x(t) = 5cos12πt + 3sin18πt is a combination of a cosine and sine function with different frequencies. The period of a cosine function is 2π/ω, where ω is the angular frequency. Similarly, the period of a sine function is also 2π/ω. Therefore, the period of the given signal can be found by taking the least common multiple of the periods of the cosine and sine functions. The period of the cosine function is 2π/(12π) = 1/6, and the period of the sine function is 2π/(18π) = 1/9. The least common multiple of 1/6 and 1/9 is 1/3. Therefore, the period of the signal x(t) is 1/3.

Explanation

The period of a signal is the smallest positive value of T for which x(t) = x(t + T) for all t. In this case, the signal x(t) = 5cos12πt + 3sin18πt is a combination of a cosine and sine function with different frequencies. The period of a cosine function is 2π/ω, where ω is the angular frequency. Similarly, the period of a sine function is also 2π/ω. Therefore, the period of the given signal can be found by taking the least common multiple of the periods of the cosine and sine functions. The period of the cosine function is 2π/(12π) = 1/6, and the period of the sine function is 2π/(18π) = 1/9. The least common multiple of 1/6 and 1/9 is 1/3. Therefore, the period of the signal x(t) is 1/3.