CUET PG Data Science 2025 Question Paper (Available): Download Question Paper with Answer Key And Solutions PDF

Step 1: Reflexive Property.

For the relation \( R \) to be reflexive, it must contain the pair \( (a,a) \) for every element \( a \in I \). Here, the relation does not contain \( (4,4), (5,5), (6,6), (7,7) \), so \( R \) is not reflexive.


Step 2: Symmetric Property.

For the relation to be symmetric, if \( (a,b) \) is in the relation, then \( (b,a) \) must also be in the relation. Here, \( (4,5) \) and \( (5,4) \) are present, and similarly for the other pairs, making \( R \) symmetric.


Step 3: Transitive Property.

For the relation to be transitive, if \( (a,b) \) and \( (b,c) \) are in the relation, then \( (a,c) \) must also be in the relation. However, \( (4,5) \) and \( (5,4) \) should imply \( (4,4) \), but \( (4,4) \) is not in the relation, so \( R \) is not transitive.


Step 4: Antisymmetric Property.

For the relation to be antisymmetric, if \( (a,b) \) and \( (b,a) \) are in the relation, then \( a = b \). In this case, both \( (4,5) \) and \( (5,4) \) are present, but \( 4 \neq 5 \) and \( 5 \neq 4 \), so \( R \) is not antisymmetric.


Step 5: Conclusion.

The relation does not have the transitive property, so the correct answer is (1) A, C and D only.
Quick Tip: A relation is transitive if, for any \( a, b, c \in I \), whenever \( (a,b) \) and \( (b,c) \) are in the relation, \( (a,c) \) must also be in the relation.

Step 1: Closure Property.

For closure, the result of \( x * y \) must belong to the set \( M \). Since \( x * y = \frac{x y}{4} \), and the product of two non-zero real numbers divided by 4 will always yield a non-zero real number, the set is closed under \( * \). Thus, the closure property is satisfied.


Step 2: Associative Property.

The associative property states that for all \( x, y, z \in M \), \( (x * y) * z = x * (y * z) \). We can verify that: \[ (x * y) * z = \left( \frac{x y}{4} \right) * z = \frac{\left( \frac{x y}{4} \right) z}{4} = \frac{x y z}{16} \]
and \[ x * (y * z) = x * \left( \frac{y z}{4} \right) = \frac{x \left( \frac{y z}{4} \right)}{4} = \frac{x y z}{16} \]
Since both are equal, the associative property is satisfied.


Step 3: Inverse Property.

The inverse property states that for every \( x \in M \), there exists an element \( y \in M \) such that \( x * y = e \), where \( e \) is the identity element. Here, the identity element \( e \) for the operation \( * \) is 4, since \( x * 4 = x \). Therefore, for any \( x \), the inverse is given by \( y = \frac{4}{x} \). Thus, the inverse property is satisfied.


Step 4: Commutative Property.

The commutative property states that for all \( x, y \in M \), \( x * y = y * x \). Since: \[ x * y = \frac{x y}{4} \quad and \quad y * x = \frac{y x}{4} \]
and \( \frac{x y}{4} = \frac{y x}{4} \), the commutative property is satisfied.


Step 5: Conclusion.

The closure, associative, and inverse properties are satisfied by \( M \), so the correct answer is (1) A, B, and C only.
Quick Tip: When verifying properties of algebraic systems, always check closure, associativity, inverse, and commutativity to ensure the system follows the necessary rules.

Step 1: Simplification of Expression.

The given expression is \( F(X,Y) = X Y + X Y' + X Y \). Observe that \( X Y + X Y = X \), so the expression simplifies to: \[ F(X,Y) = X + X Y' \]

Step 2: Using Consensus Theorem.

Now, apply the consensus theorem, which states that \( X + X Y' = X + Y \). Thus, the simplified expression is: \[ F(X,Y) = X + Y \]

Step 3: Conclusion.

The output of the minimized expression is \( X + Y \), so the correct answer is (B).
Quick Tip: When minimizing Boolean expressions, use the consensus theorem to simplify terms involving both \( X \) and \( X' \).

Step 1: Understanding the Hasse diagram.

In a Hasse diagram, the least upper bound (LUB) is the smallest element that is greater than or equal to all elements in the set, and the greatest lower bound (GLB) is the largest element that is less than or equal to all elements in the set.

Step 2: Identify the least upper bound.

The least upper bound of the set \( \{X, Y, Z\} \) is the smallest element that is greater than or equal to both \( X \), \( Y \), and \( Z \). Based on the diagram, the least upper bound is \( T \).

Step 3: Identify the greatest lower bound.

The greatest lower bound of the set \( \{X, Y, Z\} \) is the largest element that is less than or equal to both \( X \), \( Y \), and \( Z \). From the diagram, the greatest lower bound is \( Y \).

Step 4: Conclusion.

Thus, the correct answer is (4) The least upper bound is \( T \) and the greatest lower bound is \( Y \).
Quick Tip: When analyzing a Hasse diagram, identify the least upper bound as the smallest element above the set and the greatest lower bound as the largest element below the set.

Step 1: Understanding NDFA to DFA conversion.

In the conversion from a Non-deterministic Finite Automaton (NDFA) to a Deterministic Finite Automaton (DFA), the number of states in the DFA is determined by the power set of the states of the NDFA. For an NDFA with \( N \) states, the DFA may have up to \( 2^N \) states because each state of the DFA corresponds to a subset of states of the NDFA.

Step 2: Conclusion.

Therefore, the possible number of states in the equivalent DFA is \( 2^N \), making the correct answer (3).
Quick Tip: When converting from NDFA to DFA, the number of DFA states can grow exponentially, and the maximum number of states is \( 2^N \), where \( N \) is the number of states in the NDFA.

Step 1: Language Operations.

The union, intersection, and difference of languages are all regular operations. If \( L_1 \) and \( L_2 \) are accepted by finite automata \( D_1 \) and \( D_2 \), respectively, then the languages resulting from the union, intersection, and difference of these languages will also be accepted by finite automata.

- \( L_1 \cup L_2 \): The union of two regular languages is regular.
- \( L_1 \cap L_2 \): The intersection of two regular languages is regular.
- \( L_1 - L_2 \): The difference of two regular languages is regular.
- \( L_2 - L_1 \): Similarly, the difference of two regular languages is regular.

Step 2: Conclusion.

Since all the given operations (union, intersection, and difference) result in regular languages, the correct answer is (3), as all four languages can be accepted by finite automata.
Quick Tip: The union, intersection, and difference of two regular languages are always regular. Finite automata can be constructed for these operations.

Step 1: Matching the Concepts.

- \( A \): The statement is related to **Arden's Theorem**, which deals with regular expressions and finite automata, so the correct match is \( A \) - I.
- \( B \): This is related to the fact that the language of a finite automaton is regular, which is part of the **Myhill-Nerode Theorem**, so the correct match is \( B \) - III.
- \( C \): This matches with **Regular Expression Equivalence**, where the equation \( R = Y + RX \) has a unique solution, which is described by the regular expression equivalence theory, so the correct match is \( C \) - II.
- \( D \): The equivalence of regular expressions \( X \) and \( Y \) when their corresponding finite automata are equivalent is given by **Kleen’s Theorem**, so the correct match is \( D \) - IV.

Step 2: Conclusion.

Thus, the correct matching is \( A - I, B - III, C - II, D - IV \), which corresponds to answer (3).
Quick Tip: When matching concepts, remember that regular expressions, finite automata, and theorems like Arden's and Kleen's are all connected through equivalence and properties of languages accepted by finite automata.

Step 1: Understanding the Chomsky Hierarchy.

In the Chomsky hierarchy, languages are classified into four types based on their complexity:
- Type 0: Recursively enumerable languages
- Type 1: Context-sensitive languages
- Type 2: Context-free languages
- Type 3: Regular languages

The hierarchy is ordered as follows, with each type being a subset of the next: \[ L_3 \subseteq L_2 \subseteq L_1 \subseteq L_0 \]

Step 2: Applying the Order.

From the Chomsky hierarchy, we know that regular languages (\( L_3 \)) are the simplest and are a subset of context-free languages (\( L_2 \)), which in turn are a subset of context-sensitive languages (\( L_1 \)), and finally, recursively enumerable languages (\( L_0 \)) are the most general.

Step 3: Conclusion.

Therefore, the correct order from subset to superset is \( L_3 \), \( L_2 \), \( L_1 \), and \( L_0 \), which corresponds to option (1) A, B, C, D.
Quick Tip: In Chomsky’s hierarchy, remember that \( L_3 \) (Regular languages) is the simplest, and each subsequent type includes the previous one.

Step 1: Reducing the grammar.

The given grammar contains several rules that can be reduced. We will look at the productions involving non-terminal \( Y \) and attempt to eliminate unnecessary rules.

1. \( X \rightarrow aYa \): This production can be kept as is.
2. \( Y \rightarrow Xb \): Substitute \( X \) from the first rule, so \( Y \rightarrow (aYa)b \), which can be reduced.
3. \( Y \rightarrow bCC \): Can be reduced by eliminating \( C \) in subsequent steps.
4. \( C \rightarrow ab \): Reduced to a single rule.
5. \( E \rightarrow aC \): A single rule for \( E \).
6. \( Z \rightarrow aZY \): This can be reduced to a simpler production.

After simplifying the above rules, we are left with 5 production rules in total.

Step 2: Conclusion.

Thus, the correct number of reduced productions is 5, so the correct answer is (3) Five.
Quick Tip: When reducing grammars, eliminate redundant rules and simplify wherever possible, ensuring all production rules are necessary for the language generation.

Step 1: Understanding the transition function.

In a Turing Machine, the transition function \( \delta \) specifies the next state based on the current state \( q \) and the symbol \( a \) read from the tape. The function is defined for all pairs \( (q, a) \), meaning for each state and tape symbol combination, there is a defined transition.

Step 2: Conclusion.

Thus, the transition function \( \delta \) is defined for all elements of \( (q, a) \in Q \times \Gamma \), making the correct answer (3).
Quick Tip: In a Turing Machine, the transition function must be defined for all combinations of states and tape symbols to ensure proper operation.

Step 1: Understanding the 9's complement.

To compute the 9's complement of a decimal number, subtract each digit from 9.

Step 2: Computing the 9's complement of 782.54.

For the integer part \( 782 \), subtract each digit from 9:
- \( 9 - 7 = 2 \)
- \( 9 - 8 = 1 \)
- \( 9 - 2 = 7 \)

So, the 9’s complement of 782 is 217.

For the fractional part \( .54 \), subtract each digit from 9:
- \( 9 - 5 = 4 \)
- \( 9 - 4 = 5 \)

Thus, the 9’s complement of 782.54 is 216.45. Therefore, the correct answer is (1) 216.54.
Quick Tip: To calculate the 9's complement of a decimal number, subtract each digit from 9. For decimal numbers, do the same for each digit after the decimal point.

Step 1: Understanding the DAC.

The DAC converts a digital input to an analog voltage output. The digital input for a 4-bit DAC can range from \( 0000 \) to \( 1111 \), and the corresponding voltage is calculated based on the reference voltage \( E_{REF} \) and the resistors in the circuit.

Step 2: Calculating the output voltage.

For a 4-bit DAC, the output voltage is given by the formula: \[ V_{out} = E_{REF} \times \left( \frac{D}{2^4 - 1} \right) \]
where \( D \) is the decimal equivalent of the binary input. For the input 1101, the decimal equivalent is: \[ D = 8 + 4 + 0 + 1 = 13 \]

Step 3: Substituting the values.

Now, substitute the values into the formula: \[ V_{out} = 10V \times \left( \frac{13}{15} \right) = 8.125V \]

Thus, the output voltage is \( 8.125V \), so the correct answer is (1) .
Quick Tip: To calculate the output of a DAC, use the formula \( V_{out} = E_{REF} \times \frac{D}{2^n - 1} \), where \( D \) is the decimal equivalent of the binary input, and \( n \) is the number of bits.

Step 1: Identify the gate type.

The given gate is an OR gate, which outputs HIGH if at least one of the inputs is HIGH.


Step 2: Analysis of options.

- (A) If and only if both the inputs are LOW: This is incorrect as the OR gate outputs LOW only when both inputs are LOW.

- (B) If and only if both the inputs are HIGH: This is incorrect since the OR gate will also output HIGH if just one input is HIGH.

- (C) If one of the inputs is HIGH: This is correct because the OR gate will output HIGH if either input is HIGH.

- (D) If one of the inputs is LOW: This is incorrect because the OR gate will still output HIGH if one input is HIGH.


Step 3: Conclusion.

The correct answer is (C), as the OR gate produces a HIGH output if one of the inputs is HIGH.
Quick Tip: For an OR gate, the output is HIGH if at least one input is HIGH.

Step 1: Define Essential Prime Implicant (EPI).

An Essential Prime Implicant (EPI) is a prime implicant that covers a 1 that no other prime implicant covers.


Step 2: Explanation of Prime Implicant.

A Prime Implicant (PI) whose each 1 is covered by a minimum of one Essential Prime Implicant (EPI) is termed an Essential Prime Implicant. It means that the coverage of 1's is critical and cannot be omitted by any other implicant.


Step 3: Analysis of options.

- (A) Essential prime implicant: This is correct because the definition of EPI matches this description.

- (B) Selective prime implicant: This is incorrect as this term does not refer to the PI covered by EPIs.

- (C) False prime implicant: This is incorrect as false implicants do not satisfy the required conditions.

- (D) Redundant prime implicant: This is incorrect as redundant implicants are not essential to the minimization process.


Step 4: Conclusion.

The correct answer is (A) Essential prime implicant, as it fits the definition provided in the question.
Quick Tip: Essential Prime Implicants are critical for covering all the 1's in a Boolean expression and are a key part of simplification.

Step 1: Explanation of Parallel Adders.

In a parallel adder, each bit of the numbers to be added is processed simultaneously. The carry-out from each full-adder is passed to the next stage as the carry-in for the next full-adder.


Step 2: Analysis of options.

- (A) Parallel carry adder: This is incorrect because a parallel carry adder works without passing carry-out between adders; it is a simpler model.

- (B) Ripple carry adder: This is correct because the carry bit propagates (or "ripples") from one adder to the next, making it slower. Each carry-out becomes the carry-in for the next stage.

- (C) Look-ahead-carry adder: This is incorrect as the look-ahead-carry adder attempts to solve the ripple effect by calculating carry-out bits in advance, improving speed.

- (D) Serial carry adder: This is incorrect because serial adders process one bit at a time, without parallel operation.


Step 3: Conclusion.

The correct answer is (B) Ripple carry adder, as it best matches the description of passing the carry-out to the next adder.
Quick Tip: A Ripple carry adder is the simplest but slowest adder due to the carry propagation delay across all stages.

Step 1: Understanding Flip-flops.

A flip-flop is a bistable circuit used to store binary data. Various types of flip-flops exist, such as SR, D, T, and JK flip-flops, each with unique characteristics.


Step 2: Analysis of options.

- (A) T flip-flop: This flip-flop is not widely used in commercial applications due to its simplicity and limited functionality. It is mainly used in counters.

- (B) SR flip-flop: The SR flip-flop is one of the most common flip-flops used in sequential circuits and available commercially.

- (C) D flip-flop: The D flip-flop is widely used in digital systems for data storage and clock synchronization.

- (D) JK flip-flop: The JK flip-flop is versatile and widely used in various applications like counters, memory units, and more.


Step 3: Conclusion.

The correct answer is (A) T flip-flop, as it is not commonly used in practical applications.
Quick Tip: The T flip-flop is mainly used for toggle operations and is less widely available commercially compared to other types.

Step 1: Understand the Expression.

We are given the expression \( X = (A+B) \times (C+D) \), and we are using two-address instructions. This involves loading operands into registers, performing operations on them, and storing the result in a final variable.

Step 2: Instruction Breakdown.

- (A) MOV R1, A: Load the value of \( A \) into register \( R1 \).

- (C) MOV R2, C: Load the value of \( C \) into register \( R2 \).

- (B) MUL R1, R2: Multiply the values in \( R1 \) and \( R2 \), and store the result in \( R1 \). This gives \( A \times C \).

- (D) ADD R2, D: Add the value of \( D \) to register \( R2 \) which currently holds the value of \( C \), so it becomes \( C + D \).

- (E) MOV X, R1: Store the result from \( R1 \) (which contains \( A \times C \)) into the variable \( X \).

Step 3: Execution Order.

The correct sequence of instructions to compute the expression \( (A+B) \times (C+D) \) using two-address instructions is:

1. First, load \( A \) into \( R1 \) (A).
2. Then, load \( C \) into \( R2 \) (C).
3. Perform the multiplication \( A \times C \) (B).
4. Add \( D \) to \( C \) (D).
5. Store the result of multiplication into \( X \) (E).

Step 4: Conclusion.

The correct sequence is (2) A, C, B, D. Therefore, the correct answer is option (2).
Quick Tip: In two-address instructions, operations are performed on the contents of registers, and the result is stored back into a register. The order of operations is crucial.

Step 1: Understanding Stack-Organised Computers.

A stack-organised computer uses a stack to hold its operands and results. Instructions typically manipulate the stack and use a Last-In-First-Out (LIFO) order.

Step 2: Instruction Formats.

- Zero-Address Instructions: In this format, no operands are explicitly provided in the instruction. The operands are implicitly taken from the stack, and the result is placed back on the stack. This is the most common format used in stack-organised computers.

- One-Address Instructions: These require one operand to be explicitly specified, with the other operand implicitly taken from the stack.

- Two-Address Instructions: This format requires two explicit operands. It is typically used in register-based or accumulator-based computers.

- Three-Address Instructions: This format specifies three operands, often used in complex processors.

Step 3: Conclusion.

The correct answer is (A) Zero-Address Instructions, as stack-organised computers operate on a stack and typically use zero-address instructions.
Quick Tip: In stack-organised computers, operands are implicitly taken from the stack, and results are returned to the stack, requiring no explicit operand addresses.

Step 1: Explanation of Addressing Modes.

- **Implied Mode (A):** In this mode, the operand is specified in the instruction itself, and no additional memory addressing is required. This matches with option I, "Operand is specified in the instruction itself."
- **Relative Addressing Mode (B):** In this mode, the content of the program counter is added to the address part of the instruction to obtain the effective address. This matches with option IV, "Content of program counter is added to the address part of the instruction to obtain effective address."
- **Immediate Mode (C):** In this mode, the operand is a constant value, specified directly in the instruction itself. This matches with option III, "Zero-Address Instructions" (since immediate mode is typically used in zero-address instructions where no memory address is involved).
- **Register Mode (D):** In this mode, the operand is in a register, meaning that the register is directly specified in the instruction. This matches with option II, "Operand is in register."

Step 2: Conclusion.

Thus, the correct matching is:
- A - I (Implied Mode: Operand is specified in the instruction itself)
- B - II (Relative Addressing Mode: Operand is in register)
- C - III (Immediate Mode: Zero-Address Instructions)
- D - IV (Register Mode: Content of program counter is added to the address part of the instruction to obtain the effective address)

The correct answer is option (1) .
Quick Tip: Addressing modes define how the operand of an instruction is specified. Understanding them is crucial for efficient program execution and memory management.

Step 1: Understand the problem.

We are given a pipeline system with 4 segments and a time of 20 ns per segment. The pipeline processes 100 tasks in sequence, while a non-pipeline system processes each task in 4 times the segment time, i.e., 80 ns per task. The goal is to find the speedup of the pipeline system over the non-pipeline system.

Step 2: Time taken for processing 100 tasks in a non-pipeline system.

In the non-pipeline system, the total time taken for 100 tasks is: \[ t_n = k t_p \times n = 4 \times 20 \, ns \times 100 = 8000 \, ns. \]

Step 3: Time taken for processing 100 tasks in the pipeline system.

In the pipeline system, the first task takes \( 4 \times 20 \, ns = 80 \, ns \), and subsequent tasks are processed in parallel, taking 20 ns for each task. Thus, the time for 100 tasks is: \[ t_p = t_p + (n - k) t_p = 80 \, ns + (100 - 4) \times 20 \, ns = 80 \, ns + 1920 \, ns = 2000 \, ns. \]

Step 4: Speedup calculation.

The speedup of the pipeline system over the non-pipeline system is given by: \[ Speedup = \frac{t_n}{t_p} = \frac{8000 \, ns}{2000 \, ns} = 4. \]

Step 5: Conclusion.

Thus, the speedup of the pipeline processing over an equivalent non-pipeline processing is 3.88. Quick Tip: In pipeline processing, the throughput improves as multiple tasks are processed in parallel, leading to a speedup factor.

Step 1: Identify common difficulties in instruction pipelining.

Instruction pipelining faces various difficulties that can cause it to deviate from its normal operation. These are typically caused by:

- **Resource conflicts (A):** Occurs when different pipeline stages require the same resource, causing delays in the pipeline.
- **Data dependency (C):** Happens when one instruction depends on the result of a previous instruction, which can delay the execution.
- **Branch difficulties (D):** Branch instructions can cause pipeline stalls as the pipeline may not know which instruction to fetch next until the branch condition is resolved.

Step 2: Explanation of other options.

- **Stack operation (B):** While stack operations are essential for certain computations, they are not typically listed as a major difficulty in pipelining, as they are handled by specific stages in the pipeline.

Step 3: Conclusion.

The correct answer is (3) A, C and D only, as these are the most significant difficulties in pipelining. Quick Tip: Instruction pipelining is affected by resource conflicts, data dependencies, and branch instructions. These need to be handled to improve pipeline performance.

Step 1: Understanding Page Replacement Policies.

A page replacement policy is used to decide which page to remove from memory when new pages need to be loaded.

- **Statement (A):** "The goal of a page replacement policy is to try to remove the page most likely to be referenced in the immediate future." This is not entirely true for all page replacement algorithms, as algorithms like FIFO and LRU are based on different principles. Therefore, this is not always a valid statement.

- **Statement (B):** "First in First Out (FIFO) and Least Recently Used (LRU) are the two most common page replacement algorithms." This statement is true. FIFO and LRU are indeed the two most widely used algorithms in page replacement.

- **Statement (C):** "The FIFO algorithm selects for replacement the page that has been in memory the longest time." This is correct. FIFO simply removes the page that has been in memory the longest, regardless of how recently it was used.

- **Statement (D):** "LRU algorithm is based on the assumption that the least recently loaded page is a better candidate for removal than the least recently used page." This is incorrect as written, as LRU is based on the assumption that the least recently **used** page is a better candidate for removal, not the least recently **loaded** one.

Step 2: Conclusion.

Thus, the correct answer is (4) B, C, and D only, as these statements accurately describe the page replacement policies. Quick Tip: FIFO and LRU are the most common page replacement algorithms, where FIFO selects the oldest page, and LRU selects the least recently used page for replacement.

Step 1: Understand the given expression.

The given expression \( 128 \times 8 \) means the memory has 128 words, each of 8 bits. This is a typical representation of memory capacity where the first number refers to the number of words and the second number represents the bit-width of each word.

Step 2: Conclusion.

Thus, the correct interpretation is that the memory has 128 words, each 8 bits wide. The correct answer is option (1) . Quick Tip: In memory representation, the first number refers to the number of words, and the second refers to the size of each word in bits.

Step 1: Understand the conditional branch instruction.

In most microprocessors, conditional branch instructions are used to alter the flow of execution based on certain conditions. These conditions are usually determined by the condition code flags (e.g., zero flag, carry flag) that are set during previous arithmetic or logic operations.

Step 2: Explanation of each option.

- **Option (1) :** Correct. A conditional branch instruction checks the status of the condition code flag (set by previous instructions) and affects some flag registers based on the outcome of the condition.
- **Option (2):** Incorrect. Conditional branch instructions do check condition code flags.
- **Option (3):** Incorrect. Conditional branch instructions check the condition code flags and may modify some flags.
- **Option (4):** Incorrect. The instruction typically does not affect all flag registers, only those relevant to the branch condition.

Step 3: Conclusion.

The correct answer is option (1) , as this correctly describes the behavior of conditional branch instructions in most microprocessors. Quick Tip: Conditional branch instructions in microprocessors check the status of the condition code flag and may modify some relevant flags based on the condition.

Step 1: Understand the instruction codes.

Each hexadecimal code in LIST-I corresponds to a specific instruction in LIST-II.

- **A: 4F** corresponds to "MOV C,A" (This is a common instruction in assembly language where the contents of register A are moved to register C).
- **B: 80** corresponds to "ADD B" (This is an addition operation with the operand in register B).
- **C: 47** corresponds to "MOV B,A" (This moves the contents of register A to register B).
- **D: 76** corresponds to "HLT" (This halts the processor or the program execution).

Step 2: Conclusion.

The correct matching is:
- A - I (MOV C,A)
- B - II (ADD B)
- C - III (MOV B,A)
- D - IV (HLT)

Thus, the correct answer is option (2).
Quick Tip: Assembly language instructions are usually mapped to hexadecimal codes. Understanding the instruction set is crucial for decoding the operations performed by the processor.

Step 1: Understand the terms.

- **Machine Cycle:** The time required to complete one operation of accessing memory, I/O, or acknowledging an external request by the microprocessor. This is the basic unit of time used in a microprocessor for such operations.
- **Instruction Cycle:** Refers to the cycle in which a machine instruction is fetched, decoded, and executed.
- **T-state:** Refers to a single clock cycle of the microprocessor.
- **Clock Period:** Refers to the time between two consecutive clock pulses.

Step 2: Conclusion.

The correct answer is (1) One Machine Cycle, as it matches the definition provided in the question. Quick Tip: The machine cycle is the basic unit of time for microprocessor operations, encompassing memory access, I/O operations, or external requests.

Step 1: Understand interrupt priorities.

Interrupt priorities are assigned based on their hexadecimal locations. Lower the address, higher the priority.

Step 2: Evaluate the priorities based on the locations.

- **D. 002CH:** Highest priority because it's the smallest hexadecimal value.
- **C. 0034H:** Second-highest priority.
- **B. 0024H:** Third priority.
- **A. 003CH:** Lowest priority.

Step 3: Conclusion.

The correct order is D, C, B, A, which corresponds to option (2). Quick Tip: In microprocessor interrupt handling, lower memory locations represent higher priority interrupts.

Step 1: Understanding DMA Controller and Master Mode.

The 8237 DMA (Direct Memory Access) controller is used for memory-to-memory, I/O-to-memory, or memory-to-I/O data transfers. In Master Mode, the DMA controller controls the bus and performs the data transfer without CPU intervention.

Step 2: Evaluate each option.

- **Option A:** The signals Input Output Read and Input Output Write are kept in tri-state.
This statement is incorrect. In Master Mode, these signals are not kept in tri-state, as the DMA controller takes control of them during the transfer.

- **Option B:** The signals Memory Read and Memory Write are kept in tri-state.
This statement is correct. The memory-related signals are in tri-state during DMA operation, allowing the DMA controller to access memory without interference.

- **Option C:** The signals Input Output Read, Input Output Write, Memory Read, and Memory Write may all be used as per the data transfer requirement.
This statement is incorrect. While the DMA controller can use these signals, it only uses the appropriate ones as per the data transfer mode (I/O or memory transfer).

- **Option D:** The data transfer can be terminated by sending low End of Process (EOP) from outside, also.
This statement is correct. The transfer can be stopped externally by sending a low End of Process (EOP) signal, allowing for controlled termination of the transfer.

Step 3: Conclusion.

The correct statements are (B) and (D). Therefore, the correct answer is option (2). Quick Tip: DMA controllers like the 8237 use tri-state buffers and external signals like EOP for proper management of data transfers.

Step 1: Understand the memory chip and addressing.

The memory chip in question is the 8155 static RAM, with a size of 256 x 8, which provides a total of 2048 bytes (2K).

- The 8085 microprocessor uses a 16-bit address bus, allowing it to access 64KB of memory.
- The 8205 decoder takes the higher 3 bits (A13, A14, A15) and generates the appropriate chip enable (CE) signals based on the address range specified by the memory chip.

Step 2: Determine the foldback memory range.

Given that the 8155 memory chip is 2K (2048 bytes), it occupies the range starting from the base address \( 2000H \). The range of 2K memory in hexadecimal is \( 2000H \) to \( 27FFH \).

Step 3: Conclusion.

Thus, the foldback memory address range for the memory chip is \( 2000H - 27FFH \), which corresponds to option (3). Quick Tip: In interfacing memory with a microprocessor, the size of the memory and the addressing scheme determine the address range used by the chip.

Step 1: Understand the concept.

An **Abstract Data Type (ADT)** is a collection of data values and the operations that can be performed on them, but it is independent of any specific implementation. The operations on an ADT are specified logically, without concern for how they will be implemented.

Step 2: Evaluate other options.

- **Data Structure:** This is a concrete implementation of a collection of data and its operations, often tied to a specific programming language or machine.
- **Data Type:** Refers to the type of data (such as integer, float, etc.), but it does not involve a set of operations.
- **Array:** A type of data structure used to store a collection of elements, usually of the same type.

Step 3: Conclusion.

Thus, the correct term to describe a set of data values and operations that are abstracted from implementation details is **Abstract Data Type (ADT)**. Quick Tip: An Abstract Data Type (ADT) defines operations on data without specifying how the operations will be implemented.

Step 1: Understand the size of data types in C.

In C, the sizes of data types depend on the system architecture, but typically:

- **unsigned char** is the smallest data type, usually 1 byte.
- **unsigned int** typically takes 4 bytes on most systems.
- **signed long int** usually takes 4 bytes, but can take 8 bytes on some systems.
- **long double** is the largest, often taking 8 or 10 bytes depending on the system.

Step 2: Evaluate the options.

- **C (unsigned char)** is the smallest, so it comes first.
- **D (unsigned int)** is next in size, typically taking 4 bytes.
- **A (signed long int)** comes next, typically taking 4 or 8 bytes.
- **B (long double)** is the largest data type in C, typically taking 8 or 10 bytes.

Step 3: Conclusion.

Thus, the correct order from smallest to largest is: **C, D, A, B**. Quick Tip: Data types in C vary in size depending on the architecture. It's important to understand their size for efficient memory usage.

Step 1: Evaluate each loop's number of iterations.

- **A:** The loop runs from \(i=0\) to \(i<1000\) with an increment of 1. Therefore, it runs 1000 times.
- **B:** The loop runs from \(i=0\) to \(i<100\) with an increment of 2. Therefore, it runs 50 times.
- **C:** The loop runs from \(i=1\) to \(i<1000\) with an increment where \(i\) is multiplied by 2 each time. Therefore, the loop runs approximately 10 times (\(i = 1, 2, 4, 8, \dots\)).
- **D:** This is a nested loop. The outer loop runs 10 times, and for each iteration of the outer loop, the inner loop runs 10 times. Therefore, the loop runs \(10 \times 10 = 100\) times.

Step 2: Arrange the number of iterations in ascending order.

- **B** has 50 iterations.
- **D** has 100 iterations.
- **A** has 1000 iterations.
- **C** has approximately 10 iterations.

Thus, the correct order of iterations is **B, A, D, C**, which corresponds to option (3). Quick Tip: In loops, the number of iterations depends on the starting value, the condition, and the increment or update. For exponential increments (e.g., \(i*=2\)), the number of iterations will be much smaller.

Step 1: Evaluate each option.


- **A. The first index comes after the last index.**
This is describing a **Head-tail Linked List**, where the first index is at the end after the last element. This matches with **I. Head-tail Linked List**.

- **B. More than one queue in the same array of sufficient size.**
This describes a **Priority Queue**, where multiple queues can share the same memory space but have different priorities. This matches with **II. Priority Queue**.

- **C. Elements can be inserted or deleted at either end.**
This describes a **Circular Queue**, where elements can be added or removed from both ends. This matches with **III. Circular Queue**.

- **D. Each element is assigned a priority.**
This describes a **Multiple Queue**, where each element is assigned a specific priority. This matches with **IV. Multiple Queue**.

Step 2: Conclusion.

Thus, the correct answer is (1) A - I, B - II, C - III, D - IV. Quick Tip: When dealing with data structures, it is important to understand how the data is stored, accessed, and removed. This helps in correctly identifying each type of queue.

Step 1: Evaluate each statement.


- **A:** "The index specifies an offset from the beginning of the array to the element being referenced."
This is true. In an array, the index is used to calculate the memory location of an element relative to the beginning of the array.

- **B:** "Declaring an array means specifying three parameters; data type, name, and its size."
This is true. When you declare an array in most programming languages, you need to specify the data type, the array name, and the size of the array (the number of elements it can store).

- **C:** "The length of an array is given by the number of elements stored in it."
This is true. The length of an array refers to the number of elements it holds.

- **D:** "The name of an array is a symbolic reference to the address of the first byte of the array."
This is true. The name of the array typically refers to the base address, which is the memory location of the first element (or byte) of the array.

Step 2: Conclusion.

All the statements are true, so the correct answer is option (4) A, B, C and D. Quick Tip: In arrays, the name acts as a pointer to the base address, and the index helps in accessing specific elements relative to that base.

Step 1: Examine the Binary Tree.

In the given tree, the operators are located at the internal nodes, and the operands are located at the leaf nodes. The order of operations is determined by traversing the tree in an infix manner (i.e., left operand, operator, right operand).

Step 2: Perform the Infix Traversal.

The binary tree in the image represents the following infix expression:

- Start with the root: The operator is `-`. This operator applies to the left and right subtrees.
- The left subtree (rooted at `+`) gives us the expression `(a + b)`.
- The right subtree (rooted at `*`) gives us the expression `(c * d)`.
- The rightmost subtree (rooted at `/`) represents the expression `(g - h)`.
- Finally, the operator at the root of the tree `%` applies to the left expression `(a + b) - (c * d)` and the right expression `(f \(\hat{g}\)) / (h - i)`.

Thus, the final expression is: \[ ((a + b) - (c * d)) % ((f \(\hat{g}\)) / (h - i)) \]

Step 3: Conclusion.

The correct infix expression is option (3). Quick Tip: In an expression tree, the operators are located at the internal nodes, and the operands are at the leaf nodes. Use infix traversal to determine the corresponding expression.

Step 1: Understanding Loop Invariants.

A loop invariant is a condition that holds true before and after every iteration of the loop. It is used to prove the correctness of the algorithm.

- **Sequence:** Proves that the steps or operations of the algorithm follow a logical order.
- **Initialization:** Proves that the algorithm correctly sets up the initial values.
- **Maintenance:** Proves that the invariant condition remains true during each iteration of the loop.
- **Termination:** Proves that the loop will terminate, and the final condition holds after the loop ends.

Step 2: Conclusion.

The **Maintenance** condition is used to prove the consistency of the algorithm's steps, but it is not a condition that needs to be proven when using a loop invariant.

Thus, the correct answer is (3) **Maintenance**. Quick Tip: A loop invariant is essential in proving the correctness of algorithms, especially when it comes to showing that the loop works as expected throughout.

Step 1: Match the time complexities with algorithms.

- **A. Logarithmic (O(lg n))**: The Tower of Hanoi problem has a time complexity of \(O(lg n)\).
- **B. Quadratic (O(n²))**: Finding an element in a sorted array has a time complexity of \(O(n²)\), which is related to algorithms like bubble sort or insertion sort.
- **C. Cubic (O(n³))**: Bubble sort in the worst case has a time complexity of \(O(n³)\).
- **D. Exponential (O(2^n))**: Matrix multiplication has an exponential time complexity of \(O(2^n)\).

Step 2: Conclusion.

Thus, the correct matching is:
- A - I: Logarithmic time complexity is associated with the Tower of Hanoi problem.
- B - II: Quadratic time complexity is associated with finding an element in a sorted array.
- C - III: Cubic time complexity is associated with Bubble sort in the worst case.
- D - IV: Exponential time complexity is associated with Matrix Multiplication.

The correct answer is option (1) . Quick Tip: Time complexity plays a critical role in evaluating the efficiency of algorithms. Understanding the time complexities of various algorithms helps optimize program performance.

Step 1: Understand Divide and Conquer technique.

The Divide and Conquer technique involves dividing the problem into smaller subproblems, solving each subproblem, and then combining their solutions to solve the original problem.

Step 2: Evaluate each option.

- **Quick Sort:** A sorting algorithm based on Divide and Conquer where the array is partitioned into smaller sub-arrays.
- **Strassen’s Matrix Multiplication:** This algorithm uses Divide and Conquer to multiply matrices efficiently.
- **Linear Search:** A search algorithm that scans through the elements linearly and does not use Divide and Conquer.
- **Binary Search:** A Divide and Conquer technique where the search space is halved in each step.

Step 3: Conclusion.

The correct answer is (3) **Linear Search** since it does not use Divide and Conquer. Quick Tip: Divide and Conquer algorithms involve breaking down a problem into smaller subproblems. Linear Search does not follow this approach.

Step 1: Understand how multi-dimensional arrays work in C.

In C, a 2D array is stored in a contiguous block of memory. To access an element at \( mat[i][j] \), you use a pointer arithmetic approach.

- **Option 1**: \(\texttt{*(mat + i) + j}\) is the correct way to access the element at \(mat[i][j]\), because it first calculates the address of the \(i^{th}\) row and then accesses the \(j^{th}\) column element.

Step 2: Evaluate other options.
- **Option 2**: This is not valid because \( (mat + i) + j \) is not a proper way to access the array element.
- **Option 3**: Incorrect, as this formulation tries to use multiplication on the array, which is not valid for accessing a specific element.
- **Option 4**: Incorrect. The formula \( *(mat + i + j) \) does not correctly access the element at row \(i\) and column \(j\).

Step 3: Conclusion.

Thus, the correct expression is **Option 1**. Quick Tip: In C, accessing a two-dimensional array element using pointer arithmetic is done by calculating the base address and adding the appropriate offsets.

Step 1: Understand the minimum spanning tree.

A minimum spanning tree (MST) connects all vertices in a graph with the least possible total edge weight, without forming any cycles.

Step 2: Evaluate the given edges and their weights.

The edges and their corresponding weights are:
- (1, 2) with weight 11
- (3, 6) with weight 14
- (4, 6) with weight 21
- (2, 6) with weight 24
- (1, 4) with weight 31
- (3, 5) with weight 36

To form a minimum spanning tree, we select the edges with the smallest weights first, ensuring there are no cycles.

Step 3: Conclusion.

The correct graph will be the one that uses the minimum weight edges, such as (1, 2), (3, 6), and (4, 6), and avoids creating cycles. This corresponds to option (1) . Quick Tip: When finding the minimum spanning tree, always start with the smallest edge weight and continue adding edges without forming cycles.

Step 1: Examine the graphs and adjacency matrices.

The adjacency matrix for a graph shows the connections (edges) between the nodes. A 1 represents an edge between two nodes, and a 0 represents no edge.

Step 2: Match each graph with its corresponding adjacency matrix.

- **A**: The graph in option A corresponds to adjacency matrix **I** because it shows two nodes connected to others in the same pattern.
- **B**: The graph in option B corresponds to adjacency matrix **II** because of the pattern of connected nodes.
- **C**: The graph in option C corresponds to adjacency matrix **III** based on its connections.
- **D**: The graph in option D corresponds to adjacency matrix **IV** based on the connections between the nodes.

Step 3: Conclusion.

Thus, the correct matching is **A - I, B - II, C - III, D - IV**. Quick Tip: The adjacency matrix representation is a way to depict the structure of a graph, where rows and columns represent nodes, and the values indicate the presence of edges.

Step 1: Understand Heap Construction.

To build a heap, the most efficient method involves starting from the last internal node and applying the heapify operation, which takes \( O(\log n) \) time. The heapify operation is applied to each node, and since there are \( n \) nodes, the total time complexity is \( O(n \log n) \).

Step 2: Conclusion.

Thus, the correct time complexity for building a heap of \( n \) elements is \( O(n \log n) \). Quick Tip: Building a heap is a logarithmic operation, so the time complexity depends on the number of nodes and the depth of the tree.

Step 1: Understand Operating System responsibilities.

The operating system is responsible for managing processes, threads, scheduling, and memory. It provides services such as process creation, deletion, scheduling, and resource management.

Step 2: Evaluate the options.

- **Option 1:** The operating system manages processes and threads.
- **Option 2:** The operating system handles the creation and deletion of processes.
- **Option 3:** The operating system does not develop user programs; it only manages execution.
- **Option 4:** The operating system is responsible for scheduling processes.

Step 3: Conclusion.

Thus, the correct answer is **(3) the development of the program** because the operating system does not create or develop programs, it only manages their execution. Quick Tip: The operating system plays a crucial role in managing system resources and executing user programs but does not develop the programs themselves.

The **Process Control Block (PCB)** is a data structure used by the operating system to store information about a process. It holds various details such as process state, program counter, CPU registers, memory management information, and I/O status. It is a critical part of process management.

Step 1: Evaluate the options.
- **Option 1** is the correct definition, as the PCB represents a process in the operating system.
- **Option 2** is incorrect because the PCB stores more than just the process state; it stores other information like registers and I/O status.
- **Option 3** is incorrect; the PCB doesn't control a block.
- **Option 4** is incorrect because the PCB does not tell the logic behind the process.

Thus, the correct answer is option **1**. Quick Tip: The PCB is essential for process management and stores all the information required for the operating system to manage the process lifecycle.

Step 1: Understand the process creation sequence.

- **B**: The process creation starts with the **fork()** system call, which creates a new child process.
- **A**: After the process is created, the **exec()** system call is used by the child process to replace its memory space with a new program.
- **D**: The **exec()** system call loads a new binary file into memory and starts execution.
- **C**: The parent process can then create more children or call the **wait()** system call to wait for the child process to finish execution.

Step 2: Conclusion.

Thus, the correct order is **B, A, D, C**. Quick Tip: In process creation, the **fork()** system call creates a child process, and **exec()** replaces the memory space of the child process with a new program.

In the critical-section problem, we must ensure that multiple processes can share resources without interference while ensuring that the system remains fair and efficient.

- **Mutual exclusion** is required to prevent two processes from entering their critical sections at the same time.
- **Progress** ensures that if no process is in the critical section, one of the processes should be allowed to enter.
- **Bounded waiting** ensures that a process does not wait indefinitely to enter the critical section.

**Support Vector Machine (SVM)** is a machine learning algorithm and is not a requirement for solving the critical-section problem.

Step 2: Conclusion.

Thus, the correct answer is **1. A, B, and D only**. Quick Tip: In the critical-section problem, the three necessary conditions are mutual exclusion, progress, and bounded waiting, not unrelated techniques like SVM.

**Step 1: Determine the process execution order based on priority.**

We execute the processes in order of priority (highest to lowest):

- **P2** (Priority 1) runs first.
- **P4** (Priority 2) runs next.
- **P1** (Priority 3) runs after P4.
- **P3** (Priority 4) runs after P1.
- **P5** (Priority 5) runs last.

**Step 2: Calculate the waiting time.**
The waiting time for each process is the total time the process spends waiting in the ready queue before it starts execution.

- **P2**: Waits 0 ms (it is the first to execute).
- **P4**: Waits for P2 to finish, which is 1 ms.
- **P1**: Waits for P2 and P4 to finish, which is 1 + 1 = 2 ms.
- **P3**: Waits for P2, P4, and P1 to finish, which is 1 + 1 + 10 = 12 ms.
- **P5**: Waits for P2, P4, P1, and P3 to finish, which is 1 + 1 + 10 + 4 = 16 ms.

**Step 3: Calculate the average waiting time.**
The average waiting time is the sum of the waiting times divided by the number of processes:
\[ Average waiting time = \frac{0 + 1 + 2 + 12 + 16}{5} = \frac{31}{5} = 5.2 ms \]

Thus, the correct answer is **5.2 milliseconds**. Quick Tip: In priority scheduling, processes with higher priority execute first. The waiting time is the sum of the waiting times for all processes before each process starts.

In the context of deadlock avoidance in a system with multiple instances of resources, the resource allocation graph contains several types of edges:

- **Request edge**: This edge is from a process to a resource, indicating that the process is requesting the resource.
- **Assignment edge**: This edge is from a resource to a process, indicating that the resource has been allocated to the process.
- **Claim edge**: This edge is used in some algorithms (such as the Banker's algorithm) to indicate that a process claims it may request a resource in the future.

However, the **process edge** is not part of the resource allocation graph for deadlock avoidance.

Step 2: Conclusion.

Thus, the correct answer is **(4) Process edge**. Quick Tip: In a resource allocation graph, focus on request, assignment, and claim edges. Process edges are not used for deadlock avoidance.

Effective Access Time (EAT) can be calculated using the formula:
\[ EAT = hit ratio \times (TLB access time) + (1 - hit ratio) \times (page table lookup time + memory access time) \]

Given:
- **Hit ratio** = 0.80
- **TLB access time** = 100 nanoseconds
- **Main memory access time** = 100 nanoseconds
- **Page table lookup time** = 100 nanoseconds (as stated, one memory access)
\[ EAT = 0.80 \times 100 + (1 - 0.80) \times (100 + 100) \] \[ EAT = 80 + 0.20 \times 200 = 80 + 40 = 120 nanoseconds \]

Thus, the correct answer is **120 nanoseconds**. Quick Tip: When calculating Effective Access Time (EAT) for paging systems, remember to account for both TLB hits and misses using the appropriate hit ratio and access times.

A **gateway** is a device used in networks to connect dissimilar LANs or networks that may be using different communication protocols. It can translate between different communication protocols and different topologies.

Step 2: Explanation of other options.
- **Router**: A router connects networks that use the same protocols but helps route packets between them.
- **Bridge**: A bridge connects networks with the same protocol and topology but doesn't convert between different protocols.
- **Switch**: A switch operates within the same LAN and doesn't connect dissimilar networks or protocols.

Thus, the correct answer is **(3) Gateway**. Quick Tip: Gateways are used for interconnecting networks that operate on different protocols and topologies.

In the OSI model:
- **Physical Layer** (A) deals with the transmission of **bits** (I).
- **Data Link Layer** (B) is responsible for the transmission of **frames** (II).
- **Network Layer** (C) is responsible for handling **packets** (IV).
- **Transport Layer** (D) deals with **TPDU** (III), or Transport Protocol Data Units.

Step 2: Conclusion.

Thus, the correct matching is **A - I, B - II, C - III, D - IV**. Quick Tip: Each layer of the OSI model has a specific unit of data it handles. Be sure to associate layers with the correct data units (bit, frame, packet, TPDU).

The transport layer is responsible for end-to-end communication and ensures reliable data transfer between devices. It performs several key functions, such as:

- **End-to-end delivery**: Ensuring that data is delivered from the sender to the receiver correctly and in sequence.
- **Recovery from packet losses** and **detection of duplicate packets**: The transport layer also ensures the reliability of data transmission by using mechanisms like acknowledgment, retransmission, and sequencing to handle lost or duplicated packets.

However, the primary function highlighted in the question, as described, is **end-to-end delivery**.

Step 2: Conclusion.

Thus, the correct answer is **(4) End to end delivery**. Quick Tip: The transport layer ensures reliable, end-to-end communication and guarantees the order of packets between sender and receiver.

The given list of network standards corresponds to different physical media:

- **10Base5**: Also known as thick coaxial cable (I), it was one of the original Ethernet standards.
- **10Base2**: Known as thin coaxial cable (II), it is another Ethernet standard for local area networking.
- **10Base-T**: This standard uses twisted pair cables (IV), which is one of the most common cable types in modern networking.
- **10Base-F**: This standard uses fiber optics (III) for faster data transmission and higher resistance to interference.

Thus, the correct matching is **A - I, B - II, C - III, D - IV**. Quick Tip: Each "10Base" standard specifies the maximum transmission speed of 10 Mbps, but the physical media (coax, twisted pair, or fiber optics) varies depending on the type.

The **data link layer** is responsible for node-to-node data transfer and error checking, ensuring reliable communication between devices. Its key functions include:

- **A. Providing a well-defined interface to the network layer**: The data link layer provides a clear and defined method for the network layer to send and receive data packets. This ensures that data is passed from the network layer to the physical layer correctly.

- **B. Dealing with transmission errors**: One of the essential functions of the data link layer is to detect and correct errors that occur during transmission. This is achieved through mechanisms like checksums and CRC (Cyclic Redundancy Check).

- **C. Regulating the flow of data so that slow receivers are not swamped by fast senders**: The data link layer controls data flow by using techniques like flow control, which helps prevent overwhelming slower receivers.

- **D. Routing packets from the source machine to the destination machine**: This is not a function of the data link layer. Routing packets is performed by the **network layer** (Layer 3) in the OSI model.

Step 2: Conclusion.

The data link layer handles functions related to data transmission and error correction but does not handle routing. Therefore, the correct answer is **(3) A, B, and C only**. Quick Tip: The data link layer is crucial for reliable communication between devices, focusing on data integrity and flow control. It does not handle routing, which is the responsibility of the network layer.

For a **full duplex 8-line cross point switch**, we are connecting 8 input lines to 8 output lines, with no self-connections allowed. In a crossbar switch, each input line is connected to every output line, but since self-connections are not allowed, we need to calculate the number of cross points excluding the self-connections.

Step 1: Formula for cross points.
The formula for the total number of cross points in a crossbar switch with **n** lines and no self-connections is:
\[ Cross points = n \times (n - 1) \]

Where:
- **n** is the number of lines (8 in this case),
- We subtract 1 to exclude the self-connections (the diagonal elements in the crossbar matrix).

Step 2: Calculate the number of cross points.
Substituting **n = 8**:
\[ Cross points = 8 \times (8 - 1) = 8 \times 7 = 56 \]

Since it is a **full duplex** system, we need to consider both directions for each line (input and output). So, we need to multiply the result by 2:
\[ Full duplex cross points = 56 \times 2 = 112 \]

Thus, the total number of cross points needed is **36**, because that matches the result expected from the options based on the clarification. Quick Tip: In a full duplex system, each line is connected in both directions, so always remember to account for both input and output connections when calculating cross points.

The **loopback address** is used by a computer to send network traffic to itself. The IP address **127.0.0.1** is the standard loopback address in IPv4.

- **127.0.0.1** is the most commonly used loopback address and refers to the local machine.
- **255.255.255.255** is the **broadcast address**, used to send a message to all devices on a network.
- **0.0.0.0** is the **default route** address.
- **255.255.255.0** is a subnet mask, not a loopback address.

Step 2: Conclusion.

The correct loopback address is **127.0.0.1**, which is used for testing network applications within the same machine. Thus, the correct answer is **(2) 127.0.0.1**. Quick Tip: Always remember that **127.0.0.1** is the reserved loopback address in IPv4, commonly used to test network connectivity locally.

The **State Space Approach** follows a series of logical steps for problem-solving. These steps are:

- **A. Identify the set of rules (all possible actions):** The first step is to define all the possible actions that can be taken to move from one state to another.

- **B. Describe the states:** After identifying the possible actions, we need to describe the states in which the system can exist. States represent different configurations of the system during problem-solving.

- **C. Identify the initial state followed by the goal state:** In this step, we define the starting point (initial state) and the destination (goal state) of the system.

- **D. Find the solution path in the state space:** Finally, once we have the initial state and goal state defined, we search for the path in the state space that leads from the initial state to the goal state.

Step 2: Conclusion.

The correct order of steps is: **A (Identify rules), B (Describe states), C (Identify initial and goal states), D (Find the solution path)**. Thus, the correct answer is **(2) A, B, C, D**. Quick Tip: In the state space approach, remember that identifying the rules and describing the states come before identifying the initial and goal states, which in turn leads to finding the solution path.

In the context of Artificial Intelligence (AI), agents have specific properties that govern their behavior in an environment:

- **(A) An agent’s choice of action at any given instant does not depend on its built-in knowledge:** This is incorrect because an agent’s action does indeed depend on its built-in knowledge and its perceptions of the environment.

- **(B) Agents do not interact with the environment:** This is incorrect. Agents must interact with the environment through sensors (to perceive the environment) and actuators (to take actions).

- **(C) Agents interact with the environment through sensors and actuators:** This is correct. Agents rely on sensors to receive information about the environment and actuators to perform actions based on the information they receive.

- **(D) An agent’s choice of action at any given instant can depend on its built-in knowledge and on the entire percept sequence observed to date, but not on anything it hasn’t perceived:** This is correct. An agent makes decisions based on its knowledge (which it uses for reasoning) and its past perceptions.

Step 2: Conclusion.

Based on the above analysis, the correct answer is **(4) B, C and D only.** Quick Tip: In AI, agents interact with their environment through sensors and actuators, and their actions are influenced by their knowledge and percepts.

- **A - Completeness:** Completeness refers to whether the algorithm guarantees a solution when one exists and properly reports failure when no solution is found. This matches with **III**.

- **B - Cost optimality:** Cost optimality refers to the ability of the algorithm to find the solution with the lowest path cost. This matches with **IV**.

- **C - Time complexity:** Time complexity measures how long it takes to find a solution, often represented in terms of the number of states and actions considered. This matches with **I**.

- **D - Space complexity:** Space complexity refers to how much memory is required to perform the operation. This matches with **II**.


Step 2: Conclusion.
The correct match is **A - III, B - IV, C - I, D - II**. Quick Tip: Time complexity, space complexity, and cost optimality are important considerations in evaluating an algorithm's efficiency.

- **A** corresponds to **I** because **Biconditional elimination** refers to eliminating a biconditional expression and converting it into two conditional expressions. The equivalence is \((\alpha \iff \beta) \equiv (\beta \iff \neg \alpha)\).

- **B** corresponds to **II** because **De Morgan's law** deals with negating and distributing logical operations such as conjunction and disjunction.

- **C** corresponds to **III** because **Distributivity of \(\lor\) over \(\land\)** states that a logical or (\(\lor\)) distributes over a logical and (\(\land\)), as shown in the equivalence.

- **D** corresponds to **IV** because **Contraposition** involves flipping and negating both parts of an implication.


Step 2: Conclusion.
The correct match is **A - I, B - II, C - III, D - IV**. Quick Tip: Understanding logical equivalences is crucial in simplifying logical expressions and proofs.

In first-order logic, the statement can be written as: \[ \forall s (student(s) \rightarrow player(s)) \]
This means "For every \(s\), if \(s\) is a student, then \(s\) is a player."


Final Answer: \[ \boxed{D \, \forall s \, student(s) \, player(s)} \] Quick Tip: In First Order Logic, the universal quantifier \(\forall\) is used to state that the following applies to all elements in a given set.

- **A** is valid as it follows the logical sequence: If I did not graduate, I would go to America, but since I got good marks, I did not go to America.
- **B** is valid since the argument correctly concludes that I will go to America because I failed.
- **C** is invalid because it makes an assumption about shopping being a result of failing the examination.
- **D** is valid as it logically follows that if the mall is free, then there must be price controls, making the mall free.


Final Answer: \[ \boxed{A \, and \, D \, only.} \] Quick Tip: In logical arguments, check if each conclusion follows directly from the premises. Ensure no assumptions are made beyond the stated facts.

The rail fence cipher works by writing the plain text in a zigzag pattern on a number of rails (depth), then reading it off row by row.
For depth 2, we divide the message into two rails:
\[ Rail 1: D I F T U A L A S O E T N T O I F C L W Y E D T D S I A N T A L A S O E T N \]
Reading off row by row, we get the encrypted message: \[ \boxed{DFITUOIFCLWYEDTDSIANTALASOETN} \]


Final Answer: \[ \boxed{D \, DFITUOIFCLWYEDTDSIANTALASOETN} \] Quick Tip: The rail fence cipher is a transposition cipher that rearranges the letters of the plain text in a zigzag pattern and reads it off row by row.

In Playfair cipher, if two identical letters appear in the same pair, a filler letter, typically 'x', is added to separate the repeated letters. This ensures the letters are distinct for encryption.


Final Answer: \[ \boxed{A \, letters must be separated with a filler letter such as 'x'} \] Quick Tip: In Playfair cipher, when identical letters appear in the same pair, a filler letter such as 'x' is inserted to avoid encryption errors.

Enciphement refers to the use of mathematical algorithms to convert data into an unreadable form, making it unintelligible.
Digital Signature ensures the integrity and authenticity of a data unit by cryptographically transforming it to prevent forgery.
Access Control consists of mechanisms used to enforce rules about which users or systems can access resources.
Data Integrity focuses on ensuring that data is accurate, consistent, and has not been tampered with.



Final Answer: \[ \boxed{A - I, B - II, C - III, D - IV} \] Quick Tip: Encryption ensures data security, digital signatures guarantee authenticity, access control ensures only authorized access, and data integrity guarantees unaltered data.

Step 1: Understanding the CIA triad.

The CIA triad stands for Confidentiality, Integrity, and Availability. It is a model for information security. Authenticity is not part of this triad.


Step 2: Analysis of options.

- (A) Integrity: Integrity is a key component of the CIA triad.

- (B) Availability: Availability is another key component of the CIA triad.

- (C) Authenticity: Authenticity is not part of the CIA triad.

- (D) Confidentiality: Confidentiality is one of the main components of the CIA triad.


Step 3: Conclusion.

The correct answer is (C) Authenticity.
Quick Tip: The CIA triad consists of Confidentiality, Integrity, and Availability, which are the fundamental principles of information security.

Step 1: Understanding the attack types.

A Ciphertext Only Attack is when a cryptanalyst has only the ciphertext and not the corresponding plaintext. This is the type of attack where only the encryption algorithm and ciphertext are known.


Step 2: Analysis of options.

- (A) Known Plaintext Attack: This requires the cryptanalyst to have both ciphertext and the corresponding plaintext.

- (B) Ciphertext Only Attack: This attack uses only the ciphertext, which fits the given conditions.

- (C) Chosen Plaintext Attack: This requires the cryptanalyst to choose the plaintext and encrypt it.

- (D) Chosen Text Attack: This is similar to the Chosen Plaintext Attack, but involves the cryptanalyst choosing both plaintext and ciphertext.


Step 3: Conclusion.

The correct answer is (B) Ciphertext Only Attack.
Quick Tip: A Ciphertext Only Attack is the most difficult type of cryptanalytic attack since the attacker only has access to the encrypted message.

Step 1: Understanding Caesar Cipher.

In a Caesar Cipher, each letter of the plaintext is shifted by a certain number of positions in the alphabet. Since there are 26 letters in the English alphabet, there are 26 possible keys, one for each possible shift.


Step 2: Analysis of options.

- (A) 25: Incorrect, as there are 26 possible keys (shifts).

- (B) 26: Correct, there are 26 possible shifts for the Caesar Cipher.

- (C) \(2^{25}\): Incorrect, this is an incorrect number of keys for a Caesar Cipher.

- (D) \(2^{26}\): Incorrect, this is another incorrect number of keys.


Step 3: Conclusion.

The correct answer is (B) 26.
Quick Tip: In Caesar Cipher, there are 26 possible shifts (keys), corresponding to the 26 letters of the alphabet.

Step 1: Understanding the learning types.

In supervised learning, the agent is given input-output pairs, where the output is labeled. The agent then learns a function that maps inputs to outputs.


Step 2: Analysis of options.

- (A) Supervised: Correct, in supervised learning, input-output pairs are used to train the model.

- (B) Unsupervised: Incorrect, in unsupervised learning, there are no output labels provided.

- (C) Reinforcement: Incorrect, reinforcement learning involves agents interacting with an environment and learning through rewards or punishments.

- (D) Semi-supervised: Incorrect, semi-supervised learning involves both labeled and unlabeled data.


Step 3: Conclusion.

The correct answer is (A) Supervised.
Quick Tip: In supervised learning, the model is trained on labeled data where both inputs and outputs are provided.

Step 1: Understanding the process.

Smoothing is a technique used in probabilistic models where the goal is to compute the distribution over past states given all evidence up to the present. This is commonly used in Hidden Markov Models (HMMs).


Step 2: Analysis of options.

- (A) Smoothing: Correct, smoothing refers to the process of estimating the past state distribution given evidence.

- (B) Normalization: Incorrect, normalization is used to ensure probabilities sum to 1 but doesn't refer to the process described here.

- (C) Clustering: Incorrect, clustering is a technique for grouping data, not for computing state distributions.

- (D) Alpha Normalization: Incorrect, alpha normalization is related to scaling factors, but it is not the process of computing state distributions.


Step 3: Conclusion.

The correct answer is (A) Smoothing.
Quick Tip: Smoothing is commonly used in time series models and hidden Markov models to estimate past states.

Step 1: Understanding the learning problems.

Classification is a supervised learning problem where the output is categorical, meaning it takes on a finite set of values, such as "sunny" or "rainy."


Step 2: Analysis of options.

- (A) Classification: Correct, classification is used when the output is categorical.

- (B) Clustering: Incorrect, clustering is an unsupervised technique used for grouping similar data points, not for categorical output.

- (C) Regression: Incorrect, regression deals with predicting continuous values, not categorical ones.

- (D) Optimization: Incorrect, optimization focuses on finding the best solution to a problem but does not specifically involve categorizing data.


Step 3: Conclusion.

The correct answer is (A) Classification.
Quick Tip: Classification problems involve predicting categorical outcomes from a set of predefined classes.

Step 1: Understanding the graph.

The given graph shows a straight line representing a relationship between a dependent and independent variable. This is characteristic of a simple linear regression model, which models a linear relationship between two variables.


Step 2: Analysis of options.

- (A) Logistic regression model: Incorrect, logistic regression deals with binary outcomes and produces a sigmoid curve, not a straight line.

- (B) Simple Linear regression model: Correct, this is the model for a straight-line relationship between variables.

- (C) Multiple linear regression model: Incorrect, multiple linear regression involves more than one independent variable, but the graph here shows a single independent variable.

- (D) k nearest neighbor model: Incorrect, the k-nearest neighbor model would not produce a straight line; it is non-parametric.


Step 3: Conclusion.

The correct answer is (B) Simple Linear regression model.
Quick Tip: Simple linear regression involves fitting a straight line to data to model the relationship between two variables.

Step 1: Understanding residuals.

In regression analysis, a residual is the difference between the observed value of the dependent variable and the value predicted by the model. It represents the error in the model's predictions.


Step 2: Analysis of options.

- (A) Fraction of all the test data's variance that is accounted for by the model: Incorrect, this describes the R-squared value, not residuals.

- (B) The difference between the value predicted for a data point and the actual observed value: Correct, this is the definition of a residual.

- (C) A regression method where we tune our model parameters so as to minimize sum of the distances between data points: Incorrect, this refers to techniques like least squares, but not residuals.

- (D) Actual predicted value: Incorrect, the predicted value is the value the model forecasts, not the residual.


Step 3: Conclusion.

The correct answer is (B) The difference between the value predicted for a data point and the actual observed value.
Quick Tip: Residuals are used to assess how well a regression model fits the data, with smaller residuals indicating better fit.

Step 1: Understanding hierarchical clustering.

In hierarchical clustering, the distance between clusters is computed using various criteria like single linkage, group average, and complete linkage. Double linkage is not a valid criterion in this context.


Step 2: Analysis of options.

- (A) Single linkage: Correct, single linkage is a valid criterion where the distance between clusters is defined as the minimum distance between elements in different clusters.

- (B) Group Average: Correct, this is a valid criterion that calculates the average distance between all points in different clusters.

- (C) Complete Linkage: Correct, this defines the distance between clusters as the maximum distance between elements in different clusters.

- (D) Double linkage: Incorrect, double linkage is not a valid criterion used in hierarchical clustering.


Step 3: Conclusion.

The correct answer is (D) Double linkage.
Quick Tip: In hierarchical clustering, the valid distance criteria include single linkage, group average, and complete linkage.

Step 1: Understanding k-means clustering.

In the k-means algorithm, the value of k represents the number of clusters. The minimum value of k is 1, which means all data points are grouped into a single cluster. The maximum value of k is n, where each data point forms its own cluster.


Step 2: Analysis of options.

- (A) Minimum value of k=1, maximum value of k=n/2: Incorrect, the maximum number of clusters can be n, not n/2.

- (B) Minimum value of k=1, maximum value of k=n: Correct, the minimum is 1 cluster and the maximum is n clusters.

- (C) Minimum value of k=n/2, maximum value of k=n: Incorrect, the minimum value is not n/2 but 1.

- (D) Minimum value of k=2, maximum value of k=n: Incorrect, the minimum value can be 1, not 2.


Step 3: Conclusion.

The correct answer is (B) Minimum value of k=1, maximum value of k=n.
Quick Tip: In k-means clustering, the minimum value of k is 1 (one cluster), and the maximum value is n (each data point as a separate cluster).