Past Questions — 2010

ASCII stands for American Standard Code for Information Interchange. The standard ASCII character set was originally designed as a 7-bit code, which allows for 2^7 (128) unique characters (representing English letters, digits, punctuation marks, and control characters). While 'Extended ASCII' uses 8 bits to allow for 256 characters, the fundamental definition of the standard ASCII code is a 7-bit code. In most computer systems, these 7 bits are stored in an 8-bit byte, with the 8th bit (the most significant bit) often used as a parity bit for error checking or set to 0.

In network theory, electrical networks are classified into two main categories: Passive and Active. 1. **Passive Networks**: These consist of components that cannot generate energy or amplify a signal. They can only absorb, dissipate, or store energy. Examples include resistors (dissipate energy), inductors, and capacitors (store energy). Therefore, options (a), (b), and (c) describe passive components. 2. **Active Networks**: An active network is one that contains at least one active element. An active element is a component capable of delivering net energy to the circuit for an infinite period of time or providing power gain. The primary examples of active elements are independent and dependent voltage sources and current sources (e.g., batteries, generators, operational amplifiers). While an active network may contain resistors, inductors, and capacitors (as mentioned in option 'e'), the defining characteristic that makes a network 'active' is the presence of the voltage or current sources. Without the sources, the network remains passive regardless of the other components present.

The Maximum Power Transfer Theorem states that to obtain maximum power from a source with a finite internal impedance to a load, the load impedance must be the complex conjugate of the source impedance. 1. Mathematical Definition: If the source impedance is given by Zs = Rs + jXs, then for maximum power transfer, the load impedance ZL must be ZL = Zs* = Rs - jXs. 2. Physical Meaning: This means the resistive part of the load must equal the resistive part of the source (Rs = RL), and the reactive part of the load must be equal in magnitude but opposite in sign to the reactive part of the source (Xs = -XL). This effectively cancels out the reactances, leaving only the resistive components to determine the power transfer. 3. Analysis of Options: - 'Exponentially' and 'Indirectly' have no basis in the theorem of power transfer. - 'Directly' (ZL = Zs) would only be correct for purely resistive circuits. In AC circuits involving impedance (which includes reactance), a direct match (ZL = Zs) does not maximize power because the reactances would add up rather than cancel out. - 'Conjugately' is the precise technical requirement for AC network theory to ensure the load is matched to the source for maximum power delivery.

1. Let the delta-connected load be represented by an equivalent star-connected load. For a balanced load, the equivalent star impedance is Z_star = Z_delta / 3. Since the impedance is scaled by a real constant, the phase angle of the impedance remains the same. Therefore, the phase angle between the phase voltage and phase current in the equivalent star is also 'q'. 2. In a star connection, the line current (I_line) is identical to the phase current (I_phase). Thus, the angle between the line current and the star-phase voltage is 'q'. 3. To verify via phasor analysis: - Let the delta phase voltage be V_ph_delta. The delta phase current is I_ph_delta = V_ph_delta / Z_delta, lagging/leading by angle q. - The line current I_line is the phasor difference of two delta phase currents. In a balanced system, I_line lags I_ph_delta by 30° and its magnitude is √3 * I_ph_delta. - The equivalent star-phase voltage V_ph_star is the line-to-neutral voltage. V_ph_star lags the line-to-line voltage (V_ph_delta) by 30°. - Since both the line current and the star-phase voltage are shifted by 30° in the same direction relative to the delta phase current and delta phase voltage respectively, the relative angle between them remains q. Therefore, the angle between the line current and its equivalent star-phase voltage is the same as the angle between the delta phase current and delta phase voltage.

To solve the recurrence relation T(n) = 2T(n-1) + n with T(1) = 1, we can use the method of iteration or the method of undetermined coefficients. Method 1: Iteration/Expansion - T(1) = 1 - T(2) = 2(1) + 2 = 4 - T(3) = 2(4) + 3 = 11 - T(4) = 2(11) + 4 = 26 - T(5) = 2(26) + 5 = 57 Now, let's test the options with n = 3 (where T(3) = 11): - Option a: 2^(3+1) - 3 - 2 = 2^4 - 5 = 16 - 5 = 11. (Matches) - Option b: 2^3 - 3 = 8 - 3 = 5. (Incorrect) - Option c: 2^3 + 3 = 8 + 3 = 11. (Matches for n=3, let's check n=4) - Option d: 2^(3+1) - 2(3) - 2 = 16 - 6 - 2 = 8. (Incorrect) Testing Option c with n = 4: - Option c: 2^4 + 4 = 16 + 4 = 20. (Incorrect, T(4) should be 26) - Option a: 2^(4+1) - 4 - 2 = 2^5 - 6 = 32 - 6 = 26. (Matches) Method 2: General Solution The recurrence is a linear non-homogeneous recurrence relation. 1. Homogeneous part: T_h(n) = c * 2^n. 2. Particular part: Since the non-homogeneous part is 'n' (a polynomial of degree 1), we guess T_p(n) = An + B. Substitute into recurrence: An + B = 2(A(n-1) + B) + n An + B = 2An - 2A + 2B + n Rearranging: (A + 1)n + (B - 2A) = 0 Equating coefficients: A + 1 = 0 => A = -1; B - 2(-1) = 0 => B = -2. So, T(n) = c * 2^n - n - 2. Using T(1) = 1: 1 = c * 2^1 - 1 - 2 => 1 = 2c - 3 => 2c = 4 => c = 2. Therefore, T(n) = 2 * 2^n - n - 2 = 2^(n+1) - n - 2.

High-pass filter passes frequencies above cut-off, e.g., above 5 kHz region, low-pass is below 4 kHz.

To find the number of functions from a set A (domain) to a set B (codomain), we consider the choices available for each element in set A. 1. Let set A have 'm' elements: {a1, a2, ..., am}. 2. Let set B have 'n' elements: {b1, b2, ..., bn}. 3. For a mapping to be a function, every element in set A must be mapped to exactly one element in set B. 4. For the first element (a1), there are 'n' possible choices in set B. 5. For the second element (a2), there are also 'n' possible choices in set B (since functions do not require uniqueness of the output, i.e., they don't have to be injective). 6. This continues for all 'm' elements of set A. By the fundamental principle of counting (multiplication rule), the total number of functions is: n * n * n * ... * n (m times) = n^m. Comparing this to the options: - Option a (m + n) is incorrect as it represents a simple sum. - Option b (m^n) is the number of functions from an n-element set to an m-element set. - Option d (m * n) is the number of elements in the Cartesian product A x B, not the number of functions.

To determine the properties of the relation R on set A = {1, 2, 3, 4}, we test it against the definitions of reflexive, symmetric, and transitive relations: 1. **Reflexive**: A relation is reflexive if for every element 'a' in A, (a, a) is in R. Here, A = {1, 2, 3, 4}. While (1, 1) and (2, 2) are present, (3, 3) and (4, 4) are missing. Therefore, R is NOT reflexive. 2. **Symmetric**: A relation is symmetric if whenever (a, b) is in R, then (b, a) must also be in R. - (1, 1) is symmetric with itself. - (2, 2) is symmetric with itself. - (2, 3) is in R, and its reverse (3, 2) is also in R. - (3, 2) is in R, and its reverse (2, 3) is also in R. - (3, 4) is in R, and its reverse (4, 3) is also in R. - (4, 3) is in R, and its reverse (3, 4) is also in R. Since all pairs satisfy this condition, R is Symmetric. 3. **Transitive**: A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. - We have (2, 3) and (3, 2) in R. For transitivity, (2, 2) must be in R (which it is). - We have (2, 3) and (3, 4) in R. For transitivity, (2, 4) must be in R. However, (2, 4) is NOT in R. Therefore, R is NOT transitive. 4. **Equivalence**: A relation is an equivalence relation only if it is reflexive, symmetric, and transitive. Since R is neither reflexive nor transitive, it is NOT an equivalence relation. Conclusion: The only property that holds is symmetry.

An Encoder is a combinational logic circuit that converts an active input signal into a coded output signal, such as converting a decimal number into a binary number (e.g., a Decimal-to-BCD encoder). This process effectively converts one type of number system representation into another. Why other options are incorrect: - b. Logic gate: These are the basic building blocks of digital circuits (AND, OR, NOT, etc.) that perform basic logical operations, but they do not perform number system conversion on their own. - c. Half adder: This is a circuit used to perform the addition of two single-bit binary numbers; it is an arithmetic circuit, not a conversion circuit. - d. FA (Full Adder): Similar to the half adder, a full adder is used for binary addition (including a carry-in bit) and does not convert one number system to another.

To subtract (1010)2 from (1101)2 using 1's complement, follow these steps: 1. **Identify the numbers**: Minuend (A) = (1101)2 Subtrahend (B) = (1010)2 2. **Find the 1's complement of the subtrahend (B)**: The 1's complement is obtained by flipping all bits (0 to 1 and 1 to 0). 1's complement of (1010)2 = (0101)2 3. **Add the 1's complement of B to A**: (1101)2 + (0101)2 Calculation: 1101 + 0101 ------ 10010 4. **Handle the Carry**: The result is (10010)2. Since there is an end-around carry (the leftmost '1'), it indicates that the result is positive. In 1's complement subtraction, if a carry is generated, we remove the carry and add it to the Least Significant Bit (LSB) of the result. This is called the 'End-Around Carry'. 5. **Perform End-Around Carry addition**: Remaining bits: (0010)2 Add carry: (0010)2 + 1 = (0011)2 Verification: (1101)2 is 13 in decimal, and (1010)2 is 10 in decimal. 13 - 10 = 3. (0011)2 is 3 in decimal. Therefore, the answer is correct.

In digital electronics and mathematics, the 'base' (or radix) of a number system refers to the number of unique digits used to represent values. 1. **Binary System**: Uses only two digits, 0 and 1. Therefore, its base is 2. This is the fundamental system used by digital computers because it maps directly to the two states of a transistor (ON/OFF). 2. **Octal System**: Uses eight digits (0-7), so its base is 8 (Option a). 3. **Decimal System**: Uses ten digits (0-9), so its base is 10 (Option c). This is the standard system used by humans. 4. **Hexadecimal System**: Uses sixteen symbols (0-9 and A-F), so its base is 16 (Option d). Since the question asks for the base of the Binary system, the correct answer is 2.

To determine which statement is false, we must check if the given sets satisfy the axioms of an Abelian Group (Closure, Associativity, Identity, Inverse, and Commutativity). 1. **Option (a): The set of rational numbers (ℚ) under addition.** - Closure: Sum of two rationals is rational. (Yes) - Associativity: (a+b)+c = a+(b+c). (Yes) - Identity: 0 is a rational number. (Yes) - Inverse: For every a, -a is also rational. (Yes) - Commutativity: a+b = b+a. (Yes) - Result: This is an abelian group. Statement is TRUE. 2. **Option (b): The set of rational integers (ℤ) under addition.** - Closure: Sum of two integers is an integer. (Yes) - Associativity: (a+b)+c = a+(b+c). (Yes) - Identity: 0 is an integer. (Yes) - Inverse: For every a, -a is also an integer. (Yes) - Commutativity: a+b = b+a. (Yes) - Result: This is an abelian group. Statement is TRUE. 3. **Option (c): The set of rational numbers (ℚ) under multiplication.** - Closure: Product of two rationals is rational. (Yes) - Associativity: (a*b)*c = a*(b*c). (Yes) - Identity: 1 is a rational number. (Yes) - Inverse: For a group under multiplication, every element must have a multiplicative inverse (1/a). However, the set of rational numbers includes **0**. The multiplicative inverse of 0 (1/0) is undefined. Therefore, the element 0 does not have an inverse in ℚ. - Result: Because 0 lacks an inverse, the set of all rational numbers is NOT a group under multiplication. (Note: The set of *non-zero* rationals ℚ\{0} would be an abelian group, but the statement specifies 'the set of rational numbers'). Statement is FALSE. Therefore, option (c) is the false statement.

In a parallel RLC circuit, the damped natural frequency (ωd) is the frequency at which the system oscillates when damping is present. 1. The undamped resonant frequency (natural frequency) is given by: ω₀ = 1 / √(LC). 2. The damping factor (α) for a parallel RLC circuit is given by: α = 1 / (2RC). 3. The relationship between the damped frequency (ωd), the undamped frequency (ω₀), and the damping factor (α) is: ωd² = ω₀² - α². 4. Substituting the values: ωd² = (1 / LC) - (1 / 2RC)². Looking at the options provided in the question, there appears to be a slight typographical error in the signs or the arrangement of the formula in the options (as the standard formula is ω₀² - α²). However, Option (b) represents the components of the damped frequency calculation: the damping term (1/2RC)² and the resonant term (1/LC). In many competitive exam contexts for this specific question, Option (b) is identified as the intended answer, representing the subtraction of the damping factor squared from the natural frequency squared, despite the inverted order in the text provided: ((1/2RC)^2) - 1/LC (which would technically be -ωd²). Given the choices, (b) is the only one containing the correct mathematical terms for a parallel RLC damped frequency calculation.

By definition, a lattice is a partially ordered set (poset) in which every pair of elements has a unique least upper bound (LUB, also called the join) and a unique greatest lower bound (GLB, also called the meet). 1. **Inductive Property**: If every pair of elements in a set has a LUB and a GLB, then by mathematical induction, every finite non-empty subset of that set also possesses a unique LUB and a unique GLB. 2. **Uniqueness**: In any poset, if a least upper bound or greatest lower bound exists, it is unique. Therefore, it is impossible to have 'many' LUBs or GLBs for a single subset, which eliminates options b, c, and d. 3. **Conclusion**: Since the question specifies a 'lattice' and a 'finite subset', the fundamental property of lattices guarantees that both a unique LUB and a unique GLB exist for that subset.

Let R1 and R2 be two equivalence relations on a set A. 1. **Union of Equivalence Relations**: The union R = R1 ∪ R2 is always reflexive (since both R1 and R2 are reflexive) and symmetric (since both R1 and R2 are symmetric). However, the union of two equivalence relations is not necessarily transitive. 2. **Transitive Closure**: The transitive closure of a relation R, denoted as R+, is the smallest transitive relation that contains R. 3. **Properties of the Transitive Closure (R+)**: - **Reflexivity**: Since R = R1 ∪ R2 is reflexive, its transitive closure R+ is also reflexive. - **Symmetry**: If (a, b) is in R+, there exists a path a = x0, x1, ..., xn = b such that (xi, xi+1) is in R. Since R is symmetric (because R1 and R2 are), there exists a reverse path b = xn, xn-1, ..., x0 = a. Thus, (b, a) is also in R+, making R+ symmetric. - **Transitivity**: By definition, the transitive closure is transitive. 4. **Conclusion**: Since R+ is reflexive, symmetric, and transitive, it satisfies all the requirements of an equivalence relation. Comparing with options: - (a) Partial order: Requires antisymmetry, which R+ does not generally possess. - (b) Total order: Requires antisymmetry and comparability, which R+ does not generally possess. - (c) Equivalence relation: Correct, as proven above.

A High-Pass Filter (HPF) is an electronic filter that allows signals with a frequency higher than a certain cutoff frequency to pass through while attenuating (reducing the amplitude of) signals with frequencies lower than that cutoff frequency. - Option (a) is the most complete and technically accurate definition. It specifies the 'cut-off frequency' as the boundary and describes the behavior for both the stop-band (below cut-off) and the pass-band (above cut-off). - Option (b) is partially correct but vague, as it doesn't define the boundary (cut-off frequency). - Option (c) is partially correct but incomplete, as it doesn't mention the attenuation of low frequencies. - Option (d) is incorrect; filters do not pass frequencies 'intermittently' (which would imply a time-based switching or a comb filter), but rather based on the frequency spectrum.

In the Theory of Computation, it is a fundamental theorem that Deterministic Finite Automata (DFA) and Non-deterministic Finite Automata (NFA) are equivalent in power. This means that for every NFA, there exists an equivalent DFA that accepts the exact same language. This equivalence is proven through the 'Subset Construction Algorithm' (also known as the Powerset Construction), where each state in the resulting DFA represents a set of states in the original NFA. Since any language recognized by an NFA can also be recognized by a DFA, a DFA can simulate any NFA. While the DFA may have up to 2^n states (where n is the number of states in the NFA), the computational capability remains the same.

Circuit visiting each vertex exactly is Hamiltonian; visiting each edge is Eulerian.

Subset relation is reflexive, antisymmetric, transitive but not total => partial ordering.

The evolution of computers is categorized into generations based on the primary switching technology used. 1. First Generation (1940-1956): Used Vacuum Tubes. These were large, consumed high power, and generated significant heat. 2. Second Generation (1956-1963): Used Transistors. Transistors were smaller, faster, cheaper, and more energy-efficient than vacuum tubes, marking a significant leap in computing power. 3. Third Generation (1964-1971): Used Integrated Circuits (IC-Chips). Multiple transistors were placed on a single silicon chip. 4. Fourth Generation (1971-Present): Used Very Large Scale Integration (VLSI) and Microprocessor chips, allowing thousands of ICs to be built onto a single silicon chip. Therefore, the second generation specifically utilized transistors.

A Multiplexer (MUX) is a combinational logic circuit designed to select one of several input signals and forward the selected input into a single output line. It acts as a digital switch. 1. **Functionality**: It takes 'n' input lines and uses 'm' selection lines to determine which specific input is routed to the single output line (where n = 2^m). 2. **Analysis of Options**: - **Option a (one in to many)**: This describes a **Demultiplexer (DEMUX)**, which takes one input and distributes it to one of many outputs. - **Option b (many in to one)**: This accurately describes a **Multiplexer**, as it consolidates multiple data sources into one path. - **Option c (many in to many)**: This does not describe a standard basic logic gate or selector circuit like a MUX or DEMUX. Therefore, the correct definition of a multiplexer is 'many in to one'.

A Low-Pass Filter (LPF) is a circuit designed to allow signals with a frequency lower than a certain specified threshold, called the cut-off frequency (f_c), to pass through while attenuating (reducing the amplitude of) signals with frequencies higher than the cut-off frequency. Analysis of options: - Option (a) is partially correct but imprecise; it doesn't define the boundary (cut-off frequency) where the filtering begins. - Option (b) is also partially correct but describes only the stop-band behavior, ignoring the pass-band. - Option (c) is incorrect as this would describe a total block or an open circuit, not a filter. - Option (d) is the most comprehensive and technically accurate definition. It specifies the 'pass-band' (frequencies up to the cut-off) and the 'stop-band' (frequencies above the cut-off), which defines the fundamental operation of a low-pass filter.

Primary storage (also known as main memory) refers to the memory that the CPU can access directly and quickly. The most common examples are RAM (Random Access Memory) and Cache memory. Let's evaluate the options provided: 1. Magnetic tape: This is a sequential access secondary storage device used primarily for backups and archiving. It is not primary storage. 2. PROM (Programmable Read-Only Memory): While ROM is technically a form of non-volatile primary memory, PROM is a specific type of ROM used for firmware. However, in the context of standard computer fundamentals exams, 'Primary Storage' typically refers to the main working memory (RAM). Even if ROM is considered primary, the options provided are designed to test the distinction between volatile main memory and secondary storage. 3. Floppy disk: This is a removable magnetic secondary storage device. It is not primary storage. Since none of the primary options (RAM/Cache) are listed and the provided options are either secondary storage or specialized firmware memory, 'None of the above' is the most accurate choice.

Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) are based on the assumption that the electrical circuit is a 'lumped parameter' system. In a lumped system, it is assumed that the physical dimensions of the circuit are negligible compared to the wavelength of the signals passing through them, meaning the electromagnetic field does not store significant energy in the space between components. 1. **Lumped Parameters (Option a):** KVL and KCL are specifically designed for lumped circuits where resistance, inductance, and capacitance are concentrated in discrete components. Therefore, they do not fail here. 2. **Distributed Parameters (Option b):** In high-frequency circuits or long transmission lines, the parameters (R, L, C, G) are distributed uniformly along the length of the conductor. In such cases, the phase of the voltage and current varies with position, and the effects of electromagnetic radiation and propagation delay become significant. Here, the lumped-element approximation fails, and Maxwell's equations (specifically wave equations) must be used instead of Kirchhoff's laws. 3. **Nonlinear (Option c):** Kirchhoff's laws are based on the conservation of charge and energy. They apply regardless of whether the components are linear (like a resistor) or nonlinear (like a diode). 4. **Linear (Option d):** Linear circuits are the primary application of Kirchhoff's laws.

In graph theory, a tree is defined as a connected graph that contains no cycles (circuits). By definition, a circuit is a path that starts and ends at the same vertex without repeating any edges. One of the fundamental properties of a tree with n nodes is that it is 'acyclic,' meaning it contains zero circuits. If a graph with n nodes is connected and has exactly n-1 edges, it is a tree and cannot contain any circuits. If any edge were added to a tree, it would create exactly one circuit; however, in its base state, a tree has none. Therefore, the number of circuits in a tree is always zero, regardless of the number of nodes n.