Chapter 5: Number System and Digital Logic (Set-8)
While converting a negative decimal number into binary for storage in CPU registers, which representation is most commonly used because it supports direct addition and subtraction?
A 1’s complement
B 2’s complement
C Sign-magnitude
D BCD code
Explanation: 2’s complement is standard for signed integers because subtraction can be done as addition of the complement. It has one zero and simple overflow rules, making hardware arithmetic efficient.
For an n-bit 2’s complement number, which formula correctly gives the representable value range used by most processors?
A 0 to 2ⁿ−1
B −2ⁿ to +2ⁿ
C −2ⁿ⁻¹ to +2ⁿ⁻¹−1
D −(2ⁿ−1) to +(2ⁿ−1)
Explanation: In n-bit 2’s complement, MSB is sign. Minimum is −2^(n−1) and maximum is +2^(n−1)−1. This asymmetric range is why −128 exists but +128 doesn’t in 8-bit.
In 8-bit 2’s complement, which binary pattern represents the most negative number available in that range?
A 01111111
B 11111111
C 00000001
D 10000000
Explanation: The most negative value is −128 in 8-bit 2’s complement. It is represented by 10000000₂, where MSB is 1 and all other bits are 0.
When adding two 8-bit signed numbers in 2’s complement, which condition is a correct sign-based indicator of overflow?
A Same signs change
B Different signs add
C Carry always present
D MSB always 1
Explanation: Overflow occurs when adding two positives gives a negative, or two negatives gives a positive. That means operands share sign but result sign differs, indicating out-of-range.
A common hardware method to detect 2’s complement overflow uses the relationship between which two carry signals?
A LSB carry and MSB
B Input carry and sum
C Carry into MSB and carry out
D Parity carry and borrow
Explanation: In 2’s complement addition, overflow happens when carry into the sign bit differs from carry out of the sign bit. XOR of these carries provides a reliable overflow flag.
When a binary number is converted to hexadecimal, which grouping ensures correct mapping without changing value?
A 4-bit groups
B 2-bit groups
C 3-bit groups
D 5-bit groups
Explanation: One hexadecimal digit represents 16 values, which equals 2⁴. So binary must be grouped into 4-bit chunks (nibbles) for direct binary-to-hex conversion.
A 6-bit 2’s complement representation is used in a device. What is the maximum positive value it can store?
A +32
B +31
C +63
D +15
Explanation: For n-bit 2’s complement, maximum positive is 2^(n−1)−1. With 6 bits, max is 2⁵−1 = 31. Range is −32 to +31.
In 6-bit 2’s complement, the binary 100000 represents which decimal value?
A −31
B +32
C −32
D +0
Explanation: In n-bit 2’s complement, the pattern 1000…0 represents the most negative value −2^(n−1). For 6 bits, that is −32.
In 1’s complement subtraction, after adding the complement, an extra step is needed if a carry occurs. What is that step?
A Ignore carry
B Flip MSB only
C Shift right
D Add carry back
Explanation: 1’s complement uses end-around carry. If there is a carry out of MSB, it must be added back to the LSB to obtain the correct final result.
In 2’s complement subtraction using addition, what is typically done with any carry out of the MSB?
A Ignore it
B Add back to LSB
C Store as sign
D Treat as borrow
Explanation: In 2’s complement, subtraction is A + (2’s complement of B). Any final carry out is discarded. This keeps arithmetic simple and consistent.
While converting decimal to binary using repeated division, the produced remainders form bits from which side first?
A MSB first
B Middle first
C LSB first
D Random order
Explanation: Each division remainder gives the least significant bit at that stage. So remainders are generated from LSB to MSB and must be read in reverse order to get the final binary.
A binary number is often converted to octal quickly by grouping bits into sets of three. Why is three bits correct for one octal digit?
A 2² equals 8
B 2³ equals 8
C 2⁴ equals 8
D 8 equals 3
Explanation: Octal base is 8. Since 8 = 2³, three binary bits can represent values 0–7. That is why each octal digit maps exactly to 3 bits.
In an 8-bit unsigned system, what happens when 11111111₂ is incremented by 1 and stored in 8 bits?
A Becomes 00000000
B Becomes 11111110
C Becomes 10000000
D Becomes 00000001
Explanation: 255 + 1 = 256, which needs 9 bits. In 8-bit unsigned, the carry out is discarded, so result wraps to 00000000₂, showing overflow.
A 4-bit BCD digit must represent only decimal 0–9. Which 4-bit pattern is invalid in BCD?
A 1001
B 0110
C 1010
D 0000
Explanation: BCD allows 0000 to 1001 for digits 0–9. Patterns 1010 to 1111 are invalid in standard BCD and require correction in BCD arithmetic.
In BCD addition, if a 4-bit sum is greater than 1001 or a carry occurs, what correction is added to the digit?
A Add 0110
B Add 0011
C Add 0101
D Add 1001
Explanation: Adding 6 (0110₂) corrects an invalid BCD result back into a valid digit range and creates proper carry to the next decimal digit.
A 3-variable truth table must list how many rows to cover all input combinations?
A 4 rows
B 6 rows
C 8 rows
D 16 rows
Explanation: For n inputs, combinations are 2ⁿ. With 3 inputs, 2³ = 8 possible input sets exist, so 8 rows are required.
Which pair of gates are called universal because any Boolean function can be implemented using only one of them?
A AND and OR
B XOR and XNOR
C NOT and XOR
D NAND and NOR
Explanation: NAND-only or NOR-only implementations can create NOT, AND, OR, and thus any Boolean function. That is why both NAND and NOR are called universal gates.
If a NAND gate inputs are tied together to the same signal A, the output performs which operation on A?
A A AND A
B NOT A
C A OR A
D A XOR A
Explanation: NAND(A,A) = (A·A)’ = A’. So tying inputs makes NAND act as an inverter, useful when building circuits using a single gate type.
A logic circuit that has outputs depending only on current inputs and not on past values is categorized as which type?
A Sequential logic
B Memory logic
C Combinational logic
D Clocked storage
Explanation: Combinational circuits contain no memory elements. Therefore output depends only on current input values, examples are adders, multiplexers, encoders, and decoders.
XOR gate is widely used in parity circuits because XOR outputs 1 when the number of 1s is what type?
A Odd count
B Even count
C Prime count
D Zero count
Explanation: XOR of multiple bits becomes 1 when there is an odd number of ones. This property lets XOR generate even or odd parity bits for error detection.
XNOR gate is useful for equality checking between two bits because it outputs 1 when inputs are
A Always different
B Both are 1 only
C Both are 0 only
D Always same
Explanation: XNOR produces 1 for 00 and 11, meaning both bits match. Combining XNOR outputs across bits helps compare two binary words for equality.
The Boolean expression A + A·B simplifies to A by which law that removes redundant terms?
A Identity law
B Null law
C Absorption law
D Complement law
Explanation: Absorption states A + A·B = A. If A is true, output is true regardless of B. If A is false, both terms are false.
The expression A·(A + B) simplifies to A mainly by using which rule?
A Absorption law
B Distributive law
C Commutative law
D Idempotent law
Explanation: A·(A+B) = A. Multiplying by A forces the output to depend only on A. This rule reduces logic complexity and gate count.
In Boolean algebra, commutative law for AND indicates which property?
A (A·B)’ = A’+B’
B A·B = B·A
C A·1 = A
D A+A = A
Explanation: Commutative law means order does not matter. For AND, A·B equals B·A, allowing rearrangement of terms without changing logic function.
Associative law allows regrouping of terms. Which is a correct associative form for OR?
A A+B=B+A
B A·(B+C)=AB+AC
C A+(B+C)=(A+B)+C
D A+A’=1
Explanation: Associative law states grouping doesn’t affect result. For OR, you can group terms in any way and output remains the same, helping in algebraic simplification.
A canonical SOP expression is formed by OR-ing which standard terms that include all variables once?
A Minterms
B Maxterms
C Parity terms
D Gray terms
Explanation: Canonical Sum of Products uses minterms. Each minterm is an AND of all variables, either complemented or not, matching exactly one input combination where output is 1.
A canonical POS expression is formed by AND-ing which standard terms that include all variables once?
A Minterms
B XOR terms
C Maxterms
D NOT terms
Explanation: Canonical Product of Sums uses maxterms. Each maxterm is an OR of all variables, complemented or not, corresponding to an input combination where output is 0.
In a Karnaugh map, grouping 4 adjacent 1s eliminates how many variables from the resulting simplified term?
A Eliminates 1
B Eliminates 3
C Eliminates 4
D Eliminates 2
Explanation: A group of 4 equals 2² cells, so two variables change across the group and cancel out. Remaining variables stay constant and form the simplified product term.
A half adder adds two bits A and B. Which expression correctly represents its carry output?
A A ⊕ B
B A · B
C A + B
D (A+B)’
Explanation: Carry occurs only when both bits are 1. That condition is exactly AND. So carry = A·B. Sum is A ⊕ B.
A full adder sum output can be written as A ⊕ B ⊕ Cin. What does this indicate about when Sum becomes 1?
A Odd ones
B Even ones
C All zeros
D Always ones
Explanation: XOR of three bits becomes 1 when an odd number of inputs are 1 (either 1 or 3 ones). This matches the binary addition sum behavior.
A full adder carry-out becomes 1 when at least how many of its three inputs are 1?
A At least one
B At least three
C At least two
D None required
Explanation: Carry-out is produced when two or more inputs are 1. This is expressed by AB + ACin + BCin, covering all combinations with at least two ones.
A 4-to-1 multiplexer requires how many select lines to choose one input out of four?
A Two select
B One select
C Three select
D Four select
Explanation: Select lines needed are log₂(N). For N=4, log₂(4)=2. Two select bits create four combinations to select one input.
A 1-to-8 demultiplexer distributes one input to one of eight outputs. How many select lines are required?
A Two select
B Three select
C Four select
D Eight select
Explanation: Eight outputs require 3 select bits because 2³ = 8. The select lines decide which single output line receives the input signal.
A 3-to-8 decoder has how many input lines and how many outputs in standard form?
A 8 input, 3 output
B 2 input, 4 output
C 4 input, 16 output
D 3 input, 8 output
Explanation: A 3-bit input can represent 8 combinations, so the decoder provides 8 output lines, typically one-hot, where exactly one output becomes active.
A priority encoder is mainly needed because it can handle which condition properly?
A No input active
B Only one output
C Multiple inputs active
D Always zero input
Explanation: A normal encoder becomes ambiguous if more than one input is 1. A priority encoder resolves this by selecting the highest-priority active input and outputs its code.
A latch is called level-sensitive because output may change as long as enable is
A High active
B Low only
C Clock absent
D Power off
Explanation: In a typical active-high latch, output follows input while enable is high. When enable goes low, it holds the last value, showing level-sensitive behavior.
A T flip-flop toggles its output on clock edges when the T input is
A 0 always
B Floating high
C Low pulse
D 1 always
Explanation: When T=1, each clock edge causes the output to toggle (0↔1). When T=0, it holds its previous state, making it useful for counters.
Standard ASCII is considered a 7-bit code. This means it can represent how many different characters?
A 64 chars
B 96 chars
C 128 chars
D 256 chars
Explanation: 7 bits provide 2⁷ = 128 unique patterns. ASCII assigns these codes to letters, digits, symbols, and control characters like newline and tab.
In BCD, the decimal digit 9 is represented as which 4-bit pattern?
A 1010
B 1001
C 1111
D 0111
Explanation: BCD encodes decimal 0–9 as binary 0000 to 1001. Thus 9 is 1001. Values 1010–1111 are invalid for a single BCD digit.
A parity bit method is called even parity when total number of 1s in data plus parity is
A Always odd
B Always zero
C Always even
D Always prime
Explanation: Even parity sets the parity bit so total ones become even. Receiver checks the total ones; if it becomes odd, an error is detected in many cases.
A single-bit error can be detected by parity because it changes the total number of 1s from even to odd or vice versa. This is because parity detects which type of error count?
A Odd errors only
B Even errors only
C No errors
D Prime errors
Explanation: Parity detects an odd number of bit flips. One flipped bit changes parity, so it is detected. Two flips may keep parity same and can be missed.
A circuit that stores one bit of information and changes state only on clock edge is generally called a
A Decoder
B Multiplexer
C Half adder
D Flip-flop
Explanation: Flip-flops are edge-triggered storage elements. They hold one bit and update only on clock edges, forming registers, counters, and other sequential systems.
A simple method to convert binary to decimal is to multiply each bit by its positional weight. What are the weights based on?
A Powers of 8
B Powers of 2
C Powers of 10
D Powers of 16
Explanation: Binary is base-2, so each position represents a power of 2: 1,2,4,8,16, etc. Summing weights where bits are 1 gives the decimal value.
In an unsigned 8-bit system, the binary number 00000000 represents 0. In 2’s complement signed system, the same pattern represents
A −0 only
B −1
C +0 only
D −128
Explanation: 2’s complement has only one zero representation: all bits 0. So 00000000 means 0 in both signed and unsigned systems, unlike 1’s complement.
In 2’s complement, the number −1 is represented by which 8-bit pattern?
A 11111111
B 10000001
C 00000001
D 01111111
Explanation: −1 is all ones in 2’s complement because complementing 00000001 gives 11111110 and adding 1 gives 11111111. This pattern adds nicely in arithmetic.
In binary arithmetic, adding 1 to 01111111 in 8-bit 2’s complement results in 10000000. What does this indicate?
A No change
B Parity error
C Signed overflow
D Decimal carry
Explanation: 01111111 is +127, the maximum positive. Adding 1 produces 10000000, which is −128. That sign flip shows overflow because +128 is not representable in 8 bits.
Which gate output is 1 only when at least one input is 1, making it useful for combining conditions?
A AND gate
B OR gate
C XOR gate
D NOR gate
Explanation: OR outputs 1 when any input is 1. It is used to combine conditions so that if one or more signals are true, the output becomes true.
In digital logic, De Morgan’s laws are especially useful when converting circuits into which single-gate implementations?
A XOR-only circuits
B BCD-only circuits
C ASCII-only circuits
D NAND-only or NOR-only
Explanation: De Morgan’s laws allow moving inversion bubbles across AND/OR gates. This helps convert mixed-gate designs into NAND-only or NOR-only forms for uniform hardware.
A MUX can implement a Boolean function by connecting select lines to variables and choosing data inputs as constants or variables. This makes MUX behave like a
A Code detector
B Voltage divider
C Logic generator
D Memory cell
Explanation: By wiring select lines as variables, a multiplexer can output values matching a truth table. This allows implementing many Boolean functions using one structured MUX block.
If a Boolean expression is written as an OR of multiple AND terms, that expression is in which standard form used for gate design?
A Sum of products
B Product of sums
C Exclusive OR form
D Complement form
Explanation: SOP form is OR of AND terms. It is commonly implemented using AND gates feeding an OR gate and often derived from minterms where output is 1.