Q: Add the following BCD-encoded numbers (grouped in 4-bit nibbles): 0110 0111 1001 0101 1000 1000. Provide the correctly BCD-adjusted sum as 6 nibbles.

Introduction / Context: BCD addition requires normal binary addition per digit plus a correction step (adding 0110) whenever a 4-bit sum exceeds 1001 or a carry out of the nibble occurs. Here you are asked to add three BCD numbers given as consecutive nibbles and return the BCD-correct result. Given Data / Assumptions: Nibbles: 0110 0111 1001 0101 1000 1000. These correspond to BCD digits: 6 7 9 5 8 8 → decimal 67, 95, and 88. We perform 67 + 95 + 88 in BCD with proper digit-wise correction. Concept / Approach: Add the numbers as ordinary decimal: 67 + 95 + 88 = 250. Then encode the result 250 in BCD as digits 2, 5, 0 → 0010 0101 0000. Alternatively, you can carry out full BCD nibble-by-nibble addition with correction (add 0110 when a nibble sum exceeds 1001 or generates a carry) to reach the same result. Step-by-Step Solution: 1) Convert BCD to decimal: 67, 95, 88.2) Add in decimal: 67 + 95 = 162; 162 + 88 = 250.3) Convert 250 to BCD: 2 → 0010, 5 → 0101, 0 → 0000.4) Final BCD result: 0010 0101 0000. Verification / Alternative check: Perform digit-wise BCD addition: units digit 7 + 5 + 8 = 20 → write 0 (0000) and carry 2; tens digit 6 + 9 + 8 + carry 2 = 25 → write 5 (0101) and carry 2 to hundreds; hundreds digit carry only → 2 (0010). Matches 0010 0101 0000. Why Other Options Are Wrong: Long strings with 1011 (B) or 1111 (F) are not valid BCD digits (greater than 9).Appended zeros beyond 6 nibbles change the encoding length unnecessarily.Other patterns do not encode 250 correctly in BCD. Common Pitfalls: Forgetting to apply the BCD correction (add 0110) when a nibble exceeds 1001 or when carries propagate. Also, mixing up the digit boundaries when reading consecutive nibbles. Final Answer: 0010 0101 0000

Q: Digital logic design – parity generation using adders Could the SUM output (S) of a full-adder be used to generate the parity bit for two data inputs?

Introduction / Context: Parity generators are common in digital systems for simple error detection. A full-adder’s SUM output implements exclusive-OR behavior among its three inputs. This question explores whether the SUM output can directly act as a parity generator for two bits. Given Data / Assumptions: Full-adder inputs: A, B, and Cin. Full-adder SUM output S = A XOR B XOR Cin. Two-bit parity requires S = A XOR B when Cin = 0 for odd parity. Concept / Approach: The parity bit for two data inputs is the XOR of those inputs. Because a full-adder SUM equals XOR of all three inputs, tying Cin to 0 makes S = A XOR B. That is exactly an odd-parity result for two bits. If even parity is required, Cin can be tied HIGH to invert the parity. Step-by-Step Solution: Define parity: For two bits, odd parity = A XOR B.Use full-adder: S = A XOR B XOR Cin.Tie Cin = 0: S = A XOR B → odd parity bit for two inputs.Tie Cin = 1: S = NOT(A XOR B) → even parity bit for two inputs. Verification / Alternative check: Truth-table check for A,B in {00,01,10,11} confirms S toggles exactly as a 2-bit parity output when Cin is fixed. Why Other Options Are Wrong: No / conditional answers: They overlook that selecting Cin = 0 or Cin = 1 sets odd or even parity respectively. Common Pitfalls: Confusing SUM with addition; remember that SUM is bitwise XOR with Cin, not a majority function. Final Answer: Yes

Q: Hex arithmetic practice Compute ACE₁₆ − 999₁₆ and report the result in hexadecimal.

Introduction / Context: Hexadecimal subtraction is routine in digital design for addresses and register values. This exercise reinforces place-value borrowing in base-16 arithmetic and rapid mental conversion skills. Given Data / Assumptions: Minuend: ACE₁₆. Subtrahend: 999₁₆. We need the hexadecimal result. Concept / Approach: Subtract directly in hex columns or convert to decimal for a quick check. Hex digits A–F represent 10–15. Standard borrowing rules apply in base 16. Step-by-Step Solution: Convert to decimal for a check: ACE₁₆ = 1016^2 + 1216 + 14 = 2560 + 192 + 14 = 2766.999₁₆ = 916^2 + 916 + 9 = 2304 + 144 + 9 = 2457.Subtract: 2766 − 2457 = 309.Convert 309 to hex: 309 / 16 = 19 remainder 5; 19 / 16 = 1 remainder 3; 1 / 16 = 0 remainder 1 → 0x135. Verification / Alternative check: Column subtraction in hex: E−9 → borrow → 5, C−9 with borrow → 3, A−9 with borrow → 1, yielding 0x135. Matches the decimal cross-check. Why Other Options Are Wrong: 23516, 03516, 33516, 12516: Each differs by one nibble or digit; none equal 0x135 (decimal 309). Common Pitfalls: Forgetting that borrowing in hex subtracts 16 (not 10) and misreading A=10, C=12, E=14. Final Answer: 13516

Q: Multiword binary addition Add the 24-bit value 0010 0110 0011 1011 0011 1100 to the 24-bit value formed by concatenating 0101 0101 0001 1110 0001 1111. Report the 24-bit sum.

Introduction / Context: Adding wide binary words is common for addresses, checksums, and data processing. Grouping bits into nibbles (4-bit chunks) lets us use hexadecimal to streamline arithmetic and avoid carry mistakes. Given Data / Assumptions: Operand 1 (24-bit): 0010 0110 0011 1011 0011 1100. Operand 2 (24-bit): built by concatenation 0101 0101 0001 1110 0001 1111. Report the full 24-bit sum. Concept / Approach: Convert each 24-bit operand to hexadecimal (1 hex digit per nibble), perform hex addition with proper carries, then convert the result back to binary nibble-by-nibble. Step-by-Step Solution: Operand 1: 0010 0110 0011 1011 0011 1100 → 0x26 3B 3C → 0x263B3C.Operand 2: 0101 0101 0001 1110 0001 1111 → 0x55 1E 1F → 0x551E1F.Hex addition: 0x263B3C + 0x551E1F = 0x7B595B.Convert back: 0x7B 59 5B → 0111 1011 0101 1001 0101 1011. Verification / Alternative check: Decimal cross-check: 0x263B3C = 2,501,820; 0x551E1F = 5,578,527; sum = 8,080,347 = 0x7B595B. The binary matches this hex. Why Other Options Are Wrong: Each incorrect option differs in one or more nibbles (for example 0x7B415B instead of 0x7B595B), indicating carry or conversion errors. Common Pitfalls: Mixing independent 8-bit additions without propagating carry across bytes and misreading the concatenation as separate adds rather than a single 24-bit operand. Final Answer: 0111 1011 0101 1001 0101 1011

Q: Hex-by-hex subtraction by bytes Compute the three independent byte differences: 47 − 25, 34 − 1C, FA − 2F. Present the result as three hex bytes separated by spaces.

Introduction / Context: Byte-wise subtraction is frequent in low-level programming and protocol parsing. Treating each pair as an independent eight-bit subtraction with base-16 arithmetic sharpens comfort with hex digits and borrows. Given Data / Assumptions: Operate on bytes: (47 − 25), (34 − 1C), (FA − 2F). All values are hexadecimal (base 16). Each subtraction is performed independently. Concept / Approach: Subtract nibble-by-nibble with proper borrowing in base 16, where A=10 through F=15. No cross-byte borrowing occurs because each pair is separate. Step-by-Step Solution: 47 − 25: 0x47 − 0x25 = (71 − 37)₁₀ = 34₁₀ = 0x22.34 − 1C: 0x34 − 0x1C = (52 − 28)₁₀ = 24₁₀ = 0x18.FA − 2F: 0xFA − 0x2F = (250 − 47)₁₀ = 203₁₀ = 0xCB. Verification / Alternative check: Perform column-wise hex subtraction with borrows: A−F requires borrow; confirms 0xCB for the last byte, while the first two differences are straightforward without long borrow chains. Why Other Options Are Wrong: Any variation such as 22 17 CB, 22 19 CB, 22 18 CC, or 21 18 CB results from nibble borrow mistakes or decimal-hex confusion. Common Pitfalls: Treating a borrow as 10 instead of 16 in hex, or confusing C=12 and F=15 when subtracting. Final Answer: 22 18 CB

Q: 4-bit parallel adder behavior If the carry-in (Cin) of a 4-bit parallel adder is tied to logic HIGH, what is the effect on the final sum output?

Introduction / Context: Carry-in is used to cascade adders or to implement two’s-complement subtraction. Understanding its effect on the sum is crucial in ALU and arithmetic block designs. Given Data / Assumptions: A standard 4-bit parallel adder with inputs A[3..0], B[3..0], and Cin. The carry-in Cin is tied HIGH (logic 1). Concept / Approach: The adder computes S = A + B + Cin. Thus, Cin = 1 simply adds an extra 1 to the least significant stage, which propagates as needed, making the overall sum equal to A + B + 1. Step-by-Step Solution: General formula: Sum = A + B + Cin.With Cin = 1: Sum = A + B + 1.This is sometimes used to add the +1 term in two’s-complement subtraction (A + B' + 1). Verification / Alternative check: Example: A=0101 (5), B=0011 (3): normal sum 1000 (8). With Cin=1 the sum is 1001 (9), confirming the +1 effect. Why Other Options Are Wrong: Cin is not ignored; it directly enters the LSB adder cell.Carry-out depends on operands; it is not always HIGH.Output forcing (like MSB LOW) is unrelated to Cin behavior. Common Pitfalls: Confusing Cin with Cout or assuming it only matters when cascading. It always influences the computed sum. Final Answer: a one will be added to the final result

Q: Bit slicing / indexing in vectors Given B[7..0] = 10100101 (bit 7 is MSB), what is the bit slice B[6..2]?

Introduction / Context: Bit slicing is used constantly in HDL design and low-level programming to extract fields from registers. This question tests careful indexing from a declared bit range. Given Data / Assumptions: B[7..0] = 10100101, where bit 7 is the most significant bit (leftmost). We are asked for B[6..2], i.e., bits 6, 5, 4, 3, and 2 in that order. Concept / Approach: Write the bit positions explicitly and read off the requested slice without reordering. Ensure you do not reverse endianness or miscount indices. Step-by-Step Solution: Label bits: b7 b6 b5 b4 b3 b2 b1 b0 = 1 0 1 0 0 1 0 1.Extract b6..b2 → 0 1 0 0 1.Therefore, B[6..2] = 01001. Verification / Alternative check: Cross-check by masking and shifting: ((B >> 2) & 0b11111) yields 01001 for B = 0b10100101. Why Other Options Are Wrong: Each incorrect choice corresponds to a common indexing or reversal mistake (e.g., reading from the LSB end or omitting a bit). Common Pitfalls: Confusing bit numbering conventions (MSB..LSB vs. LSB..MSB) and accidentally reversing the slice. Final Answer: 01001

Question 1

In combinational logic, what is the key difference between a half-adder and a full-adder when adding single-bit operands?

Accepted Answer

Introduction / Context: Adders are the building blocks of arithmetic logic units (ALUs). Understanding the structural difference between a half-adder and a full-adder helps explain how multi-bit adders are assembled in ripple-carry and look-ahead designs. Given Data / Assumptions: Half-adder inputs: A and B; outputs: SUM and CARRY. Full-adder inputs: A, B, and Cin (carry-in); outputs: SUM and Cout (carry-out). Single-bit operands are being added in each stage. Concept / Approach: A full-adder extends the half-adder by accepting a third input (Cin), enabling correct multi-bit addition by propagating carry from the less significant stage. The half-adder cannot incorporate an incoming carry, so it is generally used only in the least significant stage or in educational contexts. Step-by-Step Solution: 1) Identify half-adder limitation: only two inputs (A,B) → cannot add a prior carry.2) Identify full-adder enhancement: adds Cin so the total is A + B + Cin.3) Therefore, the essential difference is the presence of a carry-in in the full-adder.4) Hence the correct statement: Full-adder has a carry-in. Verification / Alternative check: Examine ripple-carry adder diagrams: only the least significant stage could be a half-adder; all higher stages require a full-adder to accept the carry from the previous stage. Why Other Options Are Wrong: Half-adder has a carry-in: false by definition.Half-adder lacks carry-out: incorrect; it does output carry when A and B are both 1.Full-adder lacks carry-out: false; Cout is essential for cascading. Common Pitfalls: Confusing SUM and CARRY roles or assuming both adders accept Cin. Only the full-adder incorporates the incoming carry for proper multi-bit addition. Final Answer: Full-adder has a carry-in.

Question 2

In VHDL hardware description, what is a GENERATE statement used for, and why is it important in structural design?

Accepted Answer

Introduction / Context: VHDL is a hardware description language for modeling digital systems. The GENERATE statement is a structural construct that allows parameterized replication or conditional instantiation of components, signals, and statements. This is crucial for scalable, reusable designs such as wide adders, multiplexers, or register arrays. Given Data / Assumptions: We are describing synthesis-friendly VHDL (IEEE standards). Designs often need multiple identical instances of a module (e.g., N-bit structures). Clarity and parameterization are desirable goals in RTL code. Concept / Approach: GENERATE comes in two primary forms: for-generate for iterated replication across an index range, and if-generate for conditional inclusion based on generics or constants. Both forms produce multiple copies or alternatives of concurrent statements, enabling concise, scalable hardware descriptions. Step-by-Step Solution: 1) Identify repeated hardware pattern (e.g., 32 slices of a 1-bit adder).2) Write a for-generate loop to instantiate that slice 32 times with indexed connections.3) Optionally use if-generate to select between alternative architectures (e.g., enable parity logic only when a generic is true).4) Result: fewer code lines, fewer copy/paste errors, and easier parameterization. Verification / Alternative check: Compare manual instantiation of many components with a generate loop. The synthesized netlist is equivalent, but the generate version is clearer and less error-prone. Simulation waveforms will match when connectivity is correct. Why Other Options Are Wrong: Start statement: VHDL does not have a single 'start' statement.Not used in VHDL: incorrect; GENERATE is standard.Auto-generate a program from a schematic: not the purpose of GENERATE.Forcing optimization: not a synthesis directive for optimization. Common Pitfalls: Confusing sequential loops with generate loops. Remember that GENERATE creates concurrent hardware instances; it does not 'iterate at runtime.' Also ensure each instance has unique labels and correctly indexed signals. Final Answer: A way to tell the compiler to replicate several components

Question 3

In 8-bit two's-complement representation used by microcontrollers and CPUs, what is the valid decimal range of values that can be stored?

Accepted Answer

Introduction / Context: Two's-complement is the dominant signed-integer representation in digital systems because it makes addition and subtraction uniform for positive and negative numbers. Understanding the numeric range for a given bit width is essential for preventing overflow and designing robust software/hardware interfaces. Given Data / Assumptions: Word size: 8 bits. Signed representation: two's-complement. MSB is the sign bit (0 for nonnegative, 1 for negative). Concept / Approach: For an n-bit two's-complement number, the range is from −2^(n−1) to +2^(n−1)−1. This asymmetry occurs because there is one more negative number than positive number in the encoding (zero consumes one nonnegative code word). Step-by-Step Solution: 1) Set n = 8.2) Compute negative bound: −2^(8−1) = −2^7 = −128.3) Compute positive bound: +2^(8−1) − 1 = 2^7 − 1 = 128 − 1 = +127.4) Therefore the representable range is −128 to +127. Verification / Alternative check: Check special codes: 1000 0000 represents −128 (unique, with no positive counterpart), and 0111 1111 represents +127. Zero is 0000 0000, which fits between them. This matches the stated range. Why Other Options Are Wrong: Ranges including +128 or excluding −128 are invalid in 8-bit two's complement.Symmetric ranges like −127 to +127 do not match two's-complement capacity. Common Pitfalls: Assuming symmetry about zero, or believing +128 is encodable. Attempting to negate −128 in 8 bits causes overflow, a frequent bug in low-level code. Final Answer: –128 to +127

Question 4

When multiplying 13 × 11 in binary (13 = 1101, 11 = 1011), what is the third partial product generated during the standard shift-and-add multiplication process (counting from the least significant multiplier bit upward)?

Accepted Answer

Introduction / Context: Binary multiplication by hand mirrors decimal long multiplication: for each '1' in the multiplier, you write a shifted copy of the multiplicand and then sum the partial products. This question asks specifically for the third partial product when multiplying 13 by 11 in binary. Given Data / Assumptions: Multiplicand: 13 = 1101. Multiplier: 11 = 1011 (bits b0=1, b1=1, b2=0, b3=1 from LSB to MSB). Partial products are ordered from LSB upward. Concept / Approach: For each multiplier bit b, the partial product equals (multiplicand << i) if b = 1; otherwise it is all zeros for that position. The 'third partial product' corresponds to i = 2 (the third bit from LSB, b2). Step-by-Step Solution: 1) b0 = 1 → P0 = 1101.2) b1 = 1 → P1 = 1101 shifted left 1 = 11010.3) b2 = 0 → P2 = 00000000 (all zeros for that row).4) b3 = 1 → P3 = 1101 shifted left 3 = 1101000. Verification / Alternative check: Compute the final product to validate the steps: 13 × 11 = 143 decimal. Binary sum of partials P0 + P1 + P2 + P3 equals 10001111, which is indeed 143. Hence P2 is zero as expected. Why Other Options Are Wrong: 1011: that is the unshifted multiplier or a possible P0 for a different setup, not the third partial product here.100000 or 100001: look like shifted patterns but do not correspond to b2 = 0.00110100: unrelated shifted value for this step. Common Pitfalls: Counting partial products from the MSB instead of the LSB, or confusing multiplicand/multiplier roles. Always index partials by multiplier bit position from least significant upward in standard long multiplication. Final Answer: 00000000

Question 5

Add the following BCD-encoded numbers (grouped in 4-bit nibbles): 0110 0111 1001 0101 1000 1000. Provide the correctly BCD-adjusted sum as 6 nibbles.

Accepted Answer

Introduction / Context: BCD addition requires normal binary addition per digit plus a correction step (adding 0110) whenever a 4-bit sum exceeds 1001 or a carry out of the nibble occurs. Here you are asked to add three BCD numbers given as consecutive nibbles and return the BCD-correct result. Given Data / Assumptions: Nibbles: 0110 0111 1001 0101 1000 1000. These correspond to BCD digits: 6 7 9 5 8 8 → decimal 67, 95, and 88. We perform 67 + 95 + 88 in BCD with proper digit-wise correction. Concept / Approach: Add the numbers as ordinary decimal: 67 + 95 + 88 = 250. Then encode the result 250 in BCD as digits 2, 5, 0 → 0010 0101 0000. Alternatively, you can carry out full BCD nibble-by-nibble addition with correction (add 0110 when a nibble sum exceeds 1001 or generates a carry) to reach the same result. Step-by-Step Solution: 1) Convert BCD to decimal: 67, 95, 88.2) Add in decimal: 67 + 95 = 162; 162 + 88 = 250.3) Convert 250 to BCD: 2 → 0010, 5 → 0101, 0 → 0000.4) Final BCD result: 0010 0101 0000. Verification / Alternative check: Perform digit-wise BCD addition: units digit 7 + 5 + 8 = 20 → write 0 (0000) and carry 2; tens digit 6 + 9 + 8 + carry 2 = 25 → write 5 (0101) and carry 2 to hundreds; hundreds digit carry only → 2 (0010). Matches 0010 0101 0000. Why Other Options Are Wrong: Long strings with 1011 (B) or 1111 (F) are not valid BCD digits (greater than 9).Appended zeros beyond 6 nibbles change the encoding length unnecessarily.Other patterns do not encode 250 correctly in BCD. Common Pitfalls: Forgetting to apply the BCD correction (add 0110) when a nibble exceeds 1001 or when carries propagate. Also, mixing up the digit boundaries when reading consecutive nibbles. Final Answer: 0010 0101 0000

Question 6

Digital logic design – parity generation using adders Could the SUM output (S) of a full-adder be used to generate the parity bit for two data inputs?

Accepted Answer

Introduction / Context: Parity generators are common in digital systems for simple error detection. A full-adder’s SUM output implements exclusive-OR behavior among its three inputs. This question explores whether the SUM output can directly act as a parity generator for two bits. Given Data / Assumptions: Full-adder inputs: A, B, and Cin. Full-adder SUM output S = A XOR B XOR Cin. Two-bit parity requires S = A XOR B when Cin = 0 for odd parity. Concept / Approach: The parity bit for two data inputs is the XOR of those inputs. Because a full-adder SUM equals XOR of all three inputs, tying Cin to 0 makes S = A XOR B. That is exactly an odd-parity result for two bits. If even parity is required, Cin can be tied HIGH to invert the parity. Step-by-Step Solution: Define parity: For two bits, odd parity = A XOR B.Use full-adder: S = A XOR B XOR Cin.Tie Cin = 0: S = A XOR B → odd parity bit for two inputs.Tie Cin = 1: S = NOT(A XOR B) → even parity bit for two inputs. Verification / Alternative check: Truth-table check for A,B in {00,01,10,11} confirms S toggles exactly as a 2-bit parity output when Cin is fixed. Why Other Options Are Wrong: No / conditional answers: They overlook that selecting Cin = 0 or Cin = 1 sets odd or even parity respectively. Common Pitfalls: Confusing SUM with addition; remember that SUM is bitwise XOR with Cin, not a majority function. Final Answer: Yes

Question 7

Hex arithmetic practice Compute ACE₁₆ − 999₁₆ and report the result in hexadecimal.

Accepted Answer

Introduction / Context: Hexadecimal subtraction is routine in digital design for addresses and register values. This exercise reinforces place-value borrowing in base-16 arithmetic and rapid mental conversion skills. Given Data / Assumptions: Minuend: ACE₁₆. Subtrahend: 999₁₆. We need the hexadecimal result. Concept / Approach: Subtract directly in hex columns or convert to decimal for a quick check. Hex digits A–F represent 10–15. Standard borrowing rules apply in base 16. Step-by-Step Solution: Convert to decimal for a check: ACE₁₆ = 1016^2 + 1216 + 14 = 2560 + 192 + 14 = 2766.999₁₆ = 916^2 + 916 + 9 = 2304 + 144 + 9 = 2457.Subtract: 2766 − 2457 = 309.Convert 309 to hex: 309 / 16 = 19 remainder 5; 19 / 16 = 1 remainder 3; 1 / 16 = 0 remainder 1 → 0x135. Verification / Alternative check: Column subtraction in hex: E−9 → borrow → 5, C−9 with borrow → 3, A−9 with borrow → 1, yielding 0x135. Matches the decimal cross-check. Why Other Options Are Wrong: 23516, 03516, 33516, 12516: Each differs by one nibble or digit; none equal 0x135 (decimal 309). Common Pitfalls: Forgetting that borrowing in hex subtracts 16 (not 10) and misreading A=10, C=12, E=14. Final Answer: 13516

Question 8

Multiword binary addition Add the 24-bit value 0010 0110 0011 1011 0011 1100 to the 24-bit value formed by concatenating 0101 0101 0001 1110 0001 1111. Report the 24-bit sum.

Accepted Answer

Introduction / Context: Adding wide binary words is common for addresses, checksums, and data processing. Grouping bits into nibbles (4-bit chunks) lets us use hexadecimal to streamline arithmetic and avoid carry mistakes. Given Data / Assumptions: Operand 1 (24-bit): 0010 0110 0011 1011 0011 1100. Operand 2 (24-bit): built by concatenation 0101 0101 0001 1110 0001 1111. Report the full 24-bit sum. Concept / Approach: Convert each 24-bit operand to hexadecimal (1 hex digit per nibble), perform hex addition with proper carries, then convert the result back to binary nibble-by-nibble. Step-by-Step Solution: Operand 1: 0010 0110 0011 1011 0011 1100 → 0x26 3B 3C → 0x263B3C.Operand 2: 0101 0101 0001 1110 0001 1111 → 0x55 1E 1F → 0x551E1F.Hex addition: 0x263B3C + 0x551E1F = 0x7B595B.Convert back: 0x7B 59 5B → 0111 1011 0101 1001 0101 1011. Verification / Alternative check: Decimal cross-check: 0x263B3C = 2,501,820; 0x551E1F = 5,578,527; sum = 8,080,347 = 0x7B595B. The binary matches this hex. Why Other Options Are Wrong: Each incorrect option differs in one or more nibbles (for example 0x7B415B instead of 0x7B595B), indicating carry or conversion errors. Common Pitfalls: Mixing independent 8-bit additions without propagating carry across bytes and misreading the concatenation as separate adds rather than a single 24-bit operand. Final Answer: 0111 1011 0101 1001 0101 1011

Question 9

Hex-by-hex subtraction by bytes Compute the three independent byte differences: 47 − 25, 34 − 1C, FA − 2F. Present the result as three hex bytes separated by spaces.

Accepted Answer

Introduction / Context: Byte-wise subtraction is frequent in low-level programming and protocol parsing. Treating each pair as an independent eight-bit subtraction with base-16 arithmetic sharpens comfort with hex digits and borrows. Given Data / Assumptions: Operate on bytes: (47 − 25), (34 − 1C), (FA − 2F). All values are hexadecimal (base 16). Each subtraction is performed independently. Concept / Approach: Subtract nibble-by-nibble with proper borrowing in base 16, where A=10 through F=15. No cross-byte borrowing occurs because each pair is separate. Step-by-Step Solution: 47 − 25: 0x47 − 0x25 = (71 − 37)₁₀ = 34₁₀ = 0x22.34 − 1C: 0x34 − 0x1C = (52 − 28)₁₀ = 24₁₀ = 0x18.FA − 2F: 0xFA − 0x2F = (250 − 47)₁₀ = 203₁₀ = 0xCB. Verification / Alternative check: Perform column-wise hex subtraction with borrows: A−F requires borrow; confirms 0xCB for the last byte, while the first two differences are straightforward without long borrow chains. Why Other Options Are Wrong: Any variation such as 22 17 CB, 22 19 CB, 22 18 CC, or 21 18 CB results from nibble borrow mistakes or decimal-hex confusion. Common Pitfalls: Treating a borrow as 10 instead of 16 in hex, or confusing C=12 and F=15 when subtracting. Final Answer: 22 18 CB

Question 10

4-bit parallel adder behavior If the carry-in (Cin) of a 4-bit parallel adder is tied to logic HIGH, what is the effect on the final sum output?

Accepted Answer

Introduction / Context: Carry-in is used to cascade adders or to implement two’s-complement subtraction. Understanding its effect on the sum is crucial in ALU and arithmetic block designs. Given Data / Assumptions: A standard 4-bit parallel adder with inputs A[3..0], B[3..0], and Cin. The carry-in Cin is tied HIGH (logic 1). Concept / Approach: The adder computes S = A + B + Cin. Thus, Cin = 1 simply adds an extra 1 to the least significant stage, which propagates as needed, making the overall sum equal to A + B + 1. Step-by-Step Solution: General formula: Sum = A + B + Cin.With Cin = 1: Sum = A + B + 1.This is sometimes used to add the +1 term in two’s-complement subtraction (A + B' + 1). Verification / Alternative check: Example: A=0101 (5), B=0011 (3): normal sum 1000 (8). With Cin=1 the sum is 1001 (9), confirming the +1 effect. Why Other Options Are Wrong: Cin is not ignored; it directly enters the LSB adder cell.Carry-out depends on operands; it is not always HIGH.Output forcing (like MSB LOW) is unrelated to Cin behavior. Common Pitfalls: Confusing Cin with Cout or assuming it only matters when cascading. It always influences the computed sum. Final Answer: a one will be added to the final result

Question 11

Bit slicing / indexing in vectors Given B[7..0] = 10100101 (bit 7 is MSB), what is the bit slice B[6..2]?

Accepted Answer

Introduction / Context: Bit slicing is used constantly in HDL design and low-level programming to extract fields from registers. This question tests careful indexing from a declared bit range. Given Data / Assumptions: B[7..0] = 10100101, where bit 7 is the most significant bit (leftmost). We are asked for B[6..2], i.e., bits 6, 5, 4, 3, and 2 in that order. Concept / Approach: Write the bit positions explicitly and read off the requested slice without reordering. Ensure you do not reverse endianness or miscount indices. Step-by-Step Solution: Label bits: b7 b6 b5 b4 b3 b2 b1 b0 = 1 0 1 0 0 1 0 1.Extract b6..b2 → 0 1 0 0 1.Therefore, B[6..2] = 01001. Verification / Alternative check: Cross-check by masking and shifting: ((B >> 2) & 0b11111) yields 01001 for B = 0b10100101. Why Other Options Are Wrong: Each incorrect choice corresponds to a common indexing or reversal mistake (e.g., reading from the LSB end or omitting a bit). Common Pitfalls: Confusing bit numbering conventions (MSB..LSB vs. LSB..MSB) and accidentally reversing the slice. Final Answer: 01001

Question 12

BCD addition with carry correction Adding 9₁₀ (1001) and 7₁₀ (0111) in BCD generates a carry from the lowest digit. What is the correct correction procedure?

Accepted Answer

Introduction / Context: Binary-Coded Decimal (BCD) adds decimal digits using 4-bit codes. When a digit-sum exceeds 9 or produces a carry, a correction of 0110 (decimal 6) is applied to restore a valid BCD digit and propagate a proper carry. Given Data / Assumptions: Digits: 9 (1001) and 7 (0111). Sum of these low-order digits generates a carry. We need the standard BCD correction procedure. Concept / Approach: BCD digit addition rules: If the digit result is greater than 1001 or if a carry out of the digit occurs, add 0110 to that digit and propagate the carry to the next higher decimal digit (as 0001 in the next nibble). Step-by-Step Solution: Compute raw sum: 1001 + 0111 = 1 0000 (decimal 16) → carry = 1; low nibble = 0000.Apply correction (since carry occurred): add 0110 to the low nibble → 0000 + 0110 = 0110, and the carry becomes the high-order nibble's +1 (prefixed as 0001 at the next digit).Thus the procedure is: correct the lowest digit with +0110 and prefix the propagated carry with three zeros at the next digit. Verification / Alternative check: Decimal check: 9 + 7 = 16 → BCD result should be 0001 0110 (16). The correction method yields exactly that. Why Other Options Are Wrong: Ignoring the digit or the carry, or adding 0110 to both digits, violates BCD rules and produces invalid codes. Common Pitfalls: Forgetting that carry alone (even if nibble is ≤1001) still mandates adding 0110 to the lower nibble for a valid BCD result. Final Answer: Add 0110 to the lowest order digit (since a carry occurred) and prefix the propagated carry with three zeros

Question 13

Bitwise operations and field extraction Given [A] = 1011 1010 and [B] = 0011 0110 (both 8-bit), let [C] = [A] • [B] where • denotes bitwise AND. What is the decimal value of the field C[4..2]?

Accepted Answer

Introduction / Context: Bitwise Boolean operations are ubiquitous for masking and extracting fields within registers. This problem combines a mask via AND and a subsequent field extraction from the result. Given Data / Assumptions: [A] = 1011 1010 (0xBA). [B] = 0011 0110 (0x36). [C] = [A] AND [B]; extract bits C[4..2]. Concept / Approach: Compute C with bitwise AND. Then read bits at positions 4 down to 2 (with bit 7 as MSB) to form a 3-bit number. Convert that 3-bit binary to decimal. Step-by-Step Solution: Compute AND: 1011 1010 AND 0011 0110 = 0011 0010 (0x32).Label bits of C (bit7..bit0): 0 0 1 1 0 0 1 0.Extract C[4..2] = bits 4,3,2 → 0 1 0 → binary 010 = decimal 2. Verification / Alternative check: Masking method: ((C >> 2) & 0b111) = 0b010 = 2 confirms the field value. Why Other Options Are Wrong: Values 1, 3, 4, 6 correspond to reading the wrong bit positions or reversing MSB/LSB order. Common Pitfalls: Confusing bit numbering or inadvertently using OR instead of AND; always define MSB and LSB conventions explicitly. Final Answer: 2

Question 14

Two’s-complement range check In an 8-bit two’s-complement system, what is the range of representable positive integers (including zero)?

Accepted Answer

Introduction / Context: Two’s-complement is the dominant signed number representation in computers. Knowing the numeric range prevents overflow and helps interpret register contents correctly. Given Data / Assumptions: Word size: 8 bits. Two’s-complement encoding. Concept / Approach: For n-bit two’s-complement, the range is from −2^(n−1) to +2^(n−1) − 1. Positive numbers include zero through the maximum positive. Step-by-Step Solution: Compute upper bound: +2^(8−1) − 1 = 2^7 − 1 = 128 − 1 = 127.Compute lower bound: −2^7 = −128 (not part of the positive range).Thus the nonnegative (positive including zero) range is 0 to 127. Verification / Alternative check: Binary view: 0111 1111 is 127; 1000 0000 represents −128, confirming symmetry around zero except for the additional negative value. Why Other Options Are Wrong: 0–64 and 0–100 understate the upper limit; 0–256 confuses unsigned range; 1–127 improperly excludes zero. Common Pitfalls: Mixing signed and unsigned ranges or forgetting that positive max is one less than 2^(n−1). Final Answer: 0 to 127

Question 15

Two’s-complement transformation Determine the two’s-complement of each 5-bit binary number: 00110, 00011, 11101. Provide all three results in order.

Accepted Answer

Introduction / Context: Two’s-complement conversion is fundamental for signed arithmetic. Applying it correctly requires inverting bits and adding one, and doing so at a fixed bit-width (here, 5 bits). Given Data / Assumptions: Bit-width: 5 bits for each number. Numbers: 00110, 00011, 11101. Find each number’s two’s complement. Concept / Approach: Two’s-complement of an n-bit value X is defined as (~X + 1) with operations constrained to n bits. For a positive X, this yields its negative; for an already negative pattern, it returns the positive magnitude. Step-by-Step Solution: For 00110: invert → 11001; add 1 → 11010.For 00011: invert → 11100; add 1 → 11101.For 11101: invert → 00010; add 1 → 00011. Verification / Alternative check: Add each original to its two’s complement (mod 5-bit) → result should be 00000 with a discarded carry, confirming correctness. Why Other Options Are Wrong: Other sequences reflect missing the +1 step, using one’s complement instead, or mixing bit widths. Common Pitfalls: Forgetting to limit to 5 bits after addition or performing complement on fewer/more bits than specified. Final Answer: 11010 11101 00011

Question 16

Adder performance and timing in digital design: In a 4-bit ripple-carry adder constructed from full-adder stages, how does carry propagation delay affect overall operation speed?

Accepted Answer

Introduction / Context: Ripple-carry adders are built by cascading full-adder (FA) stages. Each stage must wait for the carry from the previous stage before it can finalize its own sum and carry results. This question examines how that carry propagation limits arithmetic speed. Given Data / Assumptions: Implementation is a 4-bit ripple-carry adder made from discrete full-adder cells. Each full-adder has nonzero gate delays for carry generation and propagation. Sum output of a stage is valid only after its incoming carry has settled. Concept / Approach: In a ripple structure, the least-significant carry must propagate sequentially through every stage until it reaches the most significant stage. Total delay equals the per-stage carry delay multiplied by the number of stages, plus additional sum-path delay at the final stage. Step-by-Step Solution: Let t_carry be carry-out delay per FA stage.For an n-bit ripple adder, worst-case carry chain length is n stages.Worst-case delay ≈ n * t_carry + t_sum(msb).Therefore, delay grows with bit-width, directly limiting maximum clock rate. Verification / Alternative check: Designers often replace ripple adders with look-ahead or prefix adders (carry-look-ahead, carry-select, Kogge-Stone) precisely because ripple delay is cumulative and becomes the critical path. Why Other Options Are Wrong: “Not a consideration” is incorrect—adder delay is frequently the ALU critical path.“Decreases with number of stages” contradicts cascade timing.“Increases but not limiting” is false; it is limiting.“Remains constant” is not true for ripple structures. Common Pitfalls: Assuming nanosecond-scale delays are negligible; in modern high-speed designs even tens of picoseconds matter across many bits. Final Answer: is cumulative for each stage and limits the speed at which arithmetic operations are performed

Question 17

Understanding full adders in arithmetic logic design: Which statement about full adders is correct in the context of building a parallel adder?

Accepted Answer

Introduction / Context: Binary adders are foundational to arithmetic/logic units. A half adder sums two bits without a carry-in. A full adder sums two bits plus a carry-in, making it the standard cell in multi-bit adders. This item checks practical implementation details of a parallel adder chain. Given Data / Assumptions: We are building a multi-bit parallel (ripple) adder. The least significant bit (LSB) often has Cin = 0. Higher-order bits must accept the carry from the previous stage. Concept / Approach: If the LSB stage never receives a carry-in, a half adder suffices at that position. All remaining stages must be full adders because they require a carry input from the prior stage. Step-by-Step Solution: Identify LSB Cin: for plain addition, Cin(0) = 0.LSB hardware: half adder can be used, reducing gate count.All other bits: full adders required to accept propagated carry.Hence the correct statement is that the first stage may be a half adder. Verification / Alternative check: Many textbooks present a “1 half adder + (n−1) full adders” implementation for an n-bit ripple adder when there is no external carry-in. Why Other Options Are Wrong: Decimal addition requires BCD adders, not simple binary full adders.Full adders are not used to make half adders (it is the other way around conceptually).A full adder has three inputs (A, B, Cin), not “limited to two.”“No carry input” contradicts the full-adder definition. Common Pitfalls: Forgetting that certain systems include an external carry-in, in which case even the LSB must be a full adder. Final Answer: In a parallel full adder, the first stage may be a half adder.

Question 18

Hexadecimal arithmetic practice: Compute the sum of the hexadecimal numbers 0110₁₆ and 10010₁₆. Choose the correct hexadecimal result.

Accepted Answer

Introduction / Context: Hexadecimal (base-16) arithmetic is widely used for addressing, memory dumps, and low-level debugging. Adding hex numbers follows the same column-wise carry rules as decimal addition, with digits 0–9 and A–F representing values 10–15. Given Data / Assumptions: Add 0110₁₆ (which equals 0x110) and 10010₁₆ (which equals 0x10010). Perform standard base-16 addition. Concept / Approach: Align the least significant digits and add column by column, carrying whenever a column sum is 16 or more. Alternatively, convert to decimal, add, and convert back to hex. Step-by-Step Solution: Write operands aligned: 10010₁₆ + 0110₁₆.Add lower nibbles: 0 + 0 = 0; next: 1 + 1 = 2; next: 0 + 1 = 1; next: 0 + 1 = 1; highest: 1 + 0 = 1.Result reads 10120₁₆.Check by decimal: 0x10010 = 65552, 0x0110 = 272; sum = 65824 = 0x10120. Verification / Alternative check: Use positional weights: (1×16^4) + (0×16^3) + (1×16^1) = 65536 + 16 = 65552; (1×16^2) + (1×16^4? no) → for 0x110: 1×256 + 1×16 = 272. Sum converts back to 1×16^4 + 0×16^3 + 1×16^2 + 2×16 + 0 → 0x10120. Why Other Options Are Wrong: 10020₁₆ and 11120₁₆ differ in the middle digits, not matching correct column carries.00120₁₆ ignores the higher-order 16^4 term.11020₁₆ introduces an incorrect extra 16^3 term. Common Pitfalls: Misaligning digits, dropping leading zeros incorrectly, or treating hex digits as decimal. Final Answer: 10120₁₆

Question 19

Binary subtraction fundamentals: For the single-bit operation 0 − 0 in binary, what are the resulting difference and borrow?

Accepted Answer

Introduction / Context: Understanding one-bit subtraction is foundational for building half-subtractors and full-subtractors, and for interpreting multi-bit subtraction with borrows in arithmetic logic units. Given Data / Assumptions: Operands: A = 0, B = 0. No incoming borrow from a less significant bit. Binary subtraction is performed bitwise. Concept / Approach: For single-bit subtraction A − B with no borrow-in, the truth cases are simple: 0−0, 0−1, 1−0, and 1−1. The difference bit is the result in the current position; the borrow indicates that the next higher bit must be reduced by one. Step-by-Step Solution: Compute 0 − 0: since the minuend equals the subtrahend, the difference is 0.No borrowing is required because we are not subtracting a larger bit from a smaller one.Hence, difference = 0 and borrow = 0. Verification / Alternative check: Truth table for a half-subtractor confirms: A=0, B=0 → D=0, Borrow=0; A=0, B=1 → D=1, Borrow=1; A=1, B=0 → D=1, Borrow=0; A=1, B=1 → D=0, Borrow=0. Why Other Options Are Wrong: Difference = 1 (options b and c) contradicts equal operands.Borrow = 1 (options c and d) is only needed when subtracting 1 from 0. Common Pitfalls: Confusing “borrow” with “carry,” or assuming a default borrow-in of 1. Final Answer: difference = 0 borrow = 0

Question 20

Full-adder inputs: How many input signals must a full adder accept to produce a sum and carry output?

Accepted Answer

Introduction / Context: The full adder (FA) is the fundamental 1-bit arithmetic cell inside multi-bit adders. It extends the half adder by considering a carry-in from a less significant position, enabling chained (ripple or look-ahead) operation. Given Data / Assumptions: Outputs: Sum (S) and Carry-out (Cout). We are discussing a standard full adder block. Concept / Approach: A full adder must combine two operand bits along with a carry-in bit. Thus, its three inputs are traditionally labeled A, B, and Cin. Step-by-Step Solution: Inputs required: A, B (the bits to add) and Cin (carry from the previous, less significant stage).Outputs produced: Sum S and carry out Cout to the next more significant stage.Therefore, the required number of inputs is 3. Verification / Alternative check: Canonical equations: S = A ⊕ B ⊕ Cin and Cout = AB + ACin + BCin, confirming dependence on three inputs. Why Other Options Are Wrong: 2 inputs describes a half adder, not a full adder.4 or more inputs would be nonstandard and unnecessary for 1-bit addition with carry-in. Common Pitfalls: Confusing the half adder (two inputs) with the full adder, or assuming Cin is optional in a chained adder. Final Answer: 3

Digital Arithmetic Operations and Circuits Questions