2.2 Combinational and arithmetic circuits
MULTIPLEXER
1. A multiplexer is a combinational circuit that:
(a) Converts one digital signal into multiple signals.
(b) Converts analog signals into digital signals.
(c) Selects one of several digital input signals for transmission on a single output line.
(d) Stores binary data.
Answer: (c)
Explanation: A multiplexer, also known as a MUX, is a digital circuit that allows you to choose one out of many data inputs and transmit it on a single output line. The selection of the input is controlled by additional select lines.
2. The number of selection lines in a 2^n to 1 multiplexer is:
(a) n
(b) n + 1
(c) 2n
(d) n^2
Answer: (a)
Explanation: The number of selection lines determines the number of input lines that can be addressed. With n selection lines, you can address 2^n input lines (2 raised to the power of n).
3. In a 4-to-1 multiplexer, how many AND gates are typically required for its implementation?
(a) 2
(b) 4
(c) 8
(d) 16
Answer: (c)
Explanation: A 4-to-1 MUX has 4 data inputs. To select one of these inputs, we typically use AND gates with the selection lines controlling each gate. Therefore, 8 AND gates (2^3) are needed for a 4-to-1 MUX.
4. A demultiplexer performs the opposite function of a multiplexer. It takes:
(a) One digital signal and transmits it on multiple lines.
(b) Multiple digital signals and transmits them on a single line.
(c) Analog signals and converts them to digital.
(d) Digital data and stores it.
Answer: (a)
Explanation: A demultiplexer takes one input signal and distributes it on multiple output lines based on the selection lines. It's the opposite operation of a multiplexer.
5. What is the advantage of using multiplexers in digital circuits?
(a) They amplify digital signals.
(b) They reduce the number of data lines required.
(c) They convert AC signals to DC.
(d) They perform complex mathematical operations.
Answer: (b)
Explanation: Multiplexers allow for transmitting multiple data streams on a single line, reducing the number of wires needed for communication. This is particularly beneficial in applications where space is limited or reducing wire usage is crucial.
6. A 2x1 MUX can be used as a:
(a) NOT gate
(b) Buffer
(c) OR gate
(d) Flip-flop
Answer: (b)
Explanation: A 2x1 MUX with one select line set to 0 will always output the data on the first input line, essentially acting as a buffer.
7. In a multiplexer, what determines which data input is selected for the output?
(a) The data on the input lines themselves.
(b) A separate set of control lines called select lines.
(c) An internal clock signal.
(d) The power supply voltage.
Answer: (b)
Explanation: The select lines determine which data input is routed to the output by enabling the corresponding AND gate in the MUX circuit.
8. How many data inputs does an 8-to-1 multiplexer have?
(a) 2
(b) 4
(c) 8
(d) 16
Answer: (d)
Explanation: An 8-to-1 MUX can select one out of eight data inputs. Therefore, it has 8 data input lines.
9. Multiplexers are commonly used in applications like:
(a) Power supplies
(b) Data acquisition systems
(c) Audio amplifiers
(d) Memory circuits
Answer: (b)
Explanation: Multiplexers are valuable in data acquisition systems where multiple sensors share a single communication line. They allow for efficient transmission of data from various sources.
10. Multiplexers are essential components in:
(a) Microprocessors
(b) Analog-to-Digital Converters (ADCs)
(c) Digital-to-Analog Converters (DACs)
(d) Both (a) and (b)
Answer: (d)
Explanation: Multiplexers play a crucial role in microprocessors for routing data between various internal components. Additionally, they are vital in ADCs for selecting signals from multiple channels before conversion.
DECODER
1. A decoder is a combinational circuit that:
(a) Selects one of several data signals for transmission.
(b) Converts analog signals into digital signals.
(c) Takes one digital input signal and distributes it.
(d) Converts a coded digital input into individual output lines.
Answer: (d)
Explanation: A decoder interprets a digital code (typically a group of binary bits) and activates a specific output line based on the code value.
2. The number of output lines in a decoder with n input lines is:
(a) n
(b) n + 1
(c) 2^n
(d) n^2
Answer: (c)
Explanation: The number of output lines in a decoder is determined by 2 raised to the power of n (2^n), where n is the number of input lines. Each unique code combination can activate a specific output line.
3. A 3-to-8 decoder has how many unique code combinations?
(a) 2
(b) 3
(c) 8
(d) 2^3
Answer: (d)
Explanation: With 3 input lines (n = 3), there are 2^3 = 8 possible unique code combinations (000, 001, 010, etc.) that the decoder can interpret.
4. Decoders are commonly used in applications like:
(a) Data amplification
(b) Memory address decoding
(c) Analog-to-Digital Conversion (ADC)
(d) Signal filtering
Answer: (b)
Explanation: Decoders play a vital role in memory address decoding. The address lines from the CPU are decoded to activate the specific memory location corresponding to the binary address value.
5. How do decoders achieve their functionality?
(a) They use comparators to compare the input with reference voltages.
(b) They employ a series of AND gates with the input lines connected.
(c) They utilize a truth table to map codes to output lines.
(d) They rely on a complex clock signal for operation.
Answer: (b)
Explanation: Decoders typically use a combination of AND gates with the input lines connected in a specific way to generate a unique output for each code combination.
6. A 2-to-4 decoder with the input set to "10" will activate which output line?
(a) The first output line.
(b) The second output line.
(c) The third output line.
(d) The fourth output line.
Answer: (c)
Explanation: Assuming a typical decoder implementation, "10" on the inputs (second bit high) would activate the third output line based on the binary value it represents.
7. Decoders can be cascaded to create a larger decoder with more input lines and output lines. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Decoders can be cascaded to expand the number of input lines and output lines. The lower-order decoder outputs are used as inputs to the higher-order decoder, allowing for decoding of larger binary codes.
8. What is the main difference between an encoder and a decoder?
(a) Decoders amplify signals, while encoders attenuate them.
(b) Decoders have multiple inputs and single outputs, while encoders have single inputs and multiple outputs.
(c) Decoders convert analog to digital, while encoders convert digital to analog.
(d) Decoders require an external clock, while encoders don't.
Answer: (b)
Explanation: The key distinction lies in the data flow. Encoders take a single code and convert it into a multi-bit digital representation, while decoders interpret a multi-bit code and activate a specific output line.
9. Decoders can be used for display driver circuits. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Decoders are valuable components in display driver circuits. They can decode the binary data representing the image or text to activate specific segments or pixels on the display.
10. Decoders are essential components in Programmable Logic Devices (PLDs). True or False?
(a) True
(b) False
Answer: (a)
Explanation: PLDs are programmable logic circuits, and decoders are fundamental building blocks used within them.
ENCODER
1. An encoder is a combinational circuit that:
(a) Selects one of several data signals for transmission.
(b) Converts analog signals into digital signals.
(c) Takes one digital input signal and distributes it.
(d) Converts a digital code (typically a single input) into a multi-bit digital representation.
Answer: (d)
Explanation: An encoder interprets a single digital input or a set of input signals (like sensor readings) and generates a corresponding multi-bit digital code representing the input state.
2. The number of output lines in a priority encoder with n input lines is:
(a) n
(b) n + 1
(c) log2(n) (This is logarithm base 2 of n)
(d) 2^n
Answer: (c)
Explanation: In a priority encoder, the number of output lines is determined by the logarithm base 2 of the number of input lines (log2(n)). This is because the code needs to represent the highest priority active input.
3. A 4-to-2 priority encoder has how many possible unique output codes?
(a) 2
(b) 4
(c) 8
(d) 16
Answer: (a)
Explanation: With a 4-to-2 priority encoder (log2(4) = 2), there are only 2 possible unique output codes (00, 01, 10, or 11) to represent the active input with the highest priority.
4. Encoders are commonly used in applications like:
(a) Data transmission over long distances
(b) Keyboard interfaces
(c) Memory address decoding
(d) Digital-to-Analog Conversion (DAC)
Answer: (b)
Explanation: Encoders are crucial in keyboard interfaces. They convert the key pressed (single input) into a binary code representing the row and column information, allowing the computer to identify the specific key.
5. How do priority encoders determine which input to encode?
(a) They compare the input voltages with a reference value.
(b) They utilize a series of OR gates with priority levels assigned to inputs.
(c) They employ a lookup table to map inputs to codes.
(d) They rely on a specific clock signal for operation.
Answer: (b)
Explanation: Priority encoders typically use a network of OR gates with priority levels assigned. The active input with the highest priority level gets encoded first, even if other lower-priority inputs are also active.
6. A 3-to-8 encoder with only the second input active will generate which output code?
(a) All 0s (representing no active input).
(b) A code with only the first bit high.
(c) A code with only the second bit high.
(d) A random 3-bit code.
Answer: (c)
Answer: Assuming a standard encoder, with only the second input active, the output code should have only the second bit high, representing the active input.
7. Encoders can be cascaded to create an encoder with a larger number of input lines. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Encoders, particularly non-priority encoders, can be cascaded to handle a larger number of input lines. The lower-order encoder outputs are connected to the inputs of the higher-order encoder, allowing for encoding of a wider range of input combinations.
8. What is the difference between a priority encoder and a non-priority encoder?
(a) Priority encoders amplify signals, while non-priority encoders attenuate them.
(b) Priority encoders encode the highest priority active input, while non-priority encoders encode any active input.
(c) Priority encoders require an external clock, while non-priority encoders don't.
(d) Priority encoders have more output lines than non-priority encoders.
Answer: (b)
Explanation: The key distinction lies in how they handle multiple active inputs. Priority encoders prioritize the highest active input for encoding, while non-priority encoders simply encode any active input they detect.
BINARY ADDITION AND BINARY SUBSTRACTION
1. How do you add 1 and 1 in binary?
(a) 0 (with a carry of 1)
(b) 10
(c) 11
(d) 2
Answer: (a)
Explanation: In binary addition, 1 + 1 = 10 (with a carry of 1) because the sum exceeds the single-digit binary value (0 or 1). The carry bit (1) is carried over to the next column if addition involves multiple digits.
2. What is the binary sum of 101 and 110?
(a) 1001
(b) 111
(c) 10001
(d) None of the above
Answer: (c)
Explanation: Perform the addition bit by bit, considering carry overs:
1 (carry from previous sum)
101
+ 110
-------
10001
3. How do you perform binary addition with multiple columns and carry overs?
(a) Add each bit column directly.
(b) Start from the least significant bit (LSB) and add, carrying over 1s if the sum exceeds 1.
(c) Binary addition requires a special calculator.
(d) It's the same as decimal addition.
Answer: (b)
Explanation: Binary addition follows the same principles as decimal addition, but with only two digits (0 and 1). Start from the rightmost column (LSB) and add the bits. If the sum is 2 (represented by 10 in binary), carry over a 1 to the next column.
4. Binary addition with a 0 in both columns always results in:
(a) 1
(b) 0
(c) Depends on the previous carry
(d) An error
Answer: (b)
Explanation: Adding two 0s in binary always results in 0 (no carry over).
5. When adding binary numbers with different numbers of digits, you can add leading zeros to make the lengths equal. True or False?
(a) True
(b) False
Answer: (a)
Explanation: To simplify addition, you can add leading zeros to the shorter number to make both numbers have the same number of digits. These leading zeros don't affect the final sum.
Binary Subtraction
6. How do you subtract 0 from 1 in binary?
(a) 1
(b) 0
(c) Error, subtraction is not possible
(d) Requires borrowing
Answer: (a)
Explanation: Subtracting 0 from any binary number simply results in the original number itself (1).
7. What is the binary difference between 101 and 010?
(a) 11
(b) 100
(c) 011
(d) None of the above
Answer: (c)**
Explanation: In binary subtraction, we can't directly subtract a larger digit from a smaller one. We borrow a 1 from the next higher-order bit (considered as 10) and adjust the subsequent calculations accordingly. Here's the breakdown:
(Borrow 1 from next bit, becomes 0)
101
- 010
-------
011
8. How do you handle borrowing in binary subtraction?
(a) Borrow 1 from the next bit and subtract 2 from the current bit.
(b) Borrow 1 from the next bit and add 1 to the current bit.
(c) Borrowing is not necessary in binary subtraction.
(d) Borrow 1 from the next bit and subtract 1 from the current bit.
Answer: (d)
Explanation: When subtracting a larger digit from a smaller one in a binary column, borrow a 1 from the next higher-order bit (considered as 10). This borrowed 1 becomes 1 in the current column, and you subtract 1 from the current digit.
10. Binary subtraction can be performed by converting the subtrahend (number being subtracted) to its two's complement and adding it to the minuend (number being subtracted from). True or False?
(a) True
(b) False
Answer: (a)
Explanation: Two's complement is a method for representing negative numbers in binary. Subtracting a binary number can be achieved by converting it to its two's complement and adding it to the minuend.
11. How is the two's complement of a binary number obtained?
(a) Invert all bits and add 1.
(b) Invert all bits and subtract 1.
(c) Double the number and add 1.
(d) No two's complement exists for positive numbers.
Answer: (a)
Explanation: To find the two's complement of a binary number, invert all the bits (0 becomes 1 and vice versa) and add 1 to the least significant bit (LSB).
12. What is the advantage of using two's complement for binary subtraction?
(a) It simplifies the subtraction process by converting it to addition.
(b) It allows for a wider range of negative numbers to be represented.
(c) It reduces the number of bits required for binary numbers.
(d) All of the above
Answer: (a)
Explanation: Two's complement allows us to perform subtraction using binary addition logic, simplifying the process and eliminating the need for separate subtraction circuits.
13. When subtracting binary numbers with different lengths and using two's complement, what should be done?
(a) Ignore the leading zeros in the shorter number.
(b) Add leading ones to the shorter number to match the length of the longer number.
(c) The subtraction is not possible.
(d) Convert the longer number to two's complement.
Answer: (b)
Explanation: When using two's complement for subtraction with unequal lengths, add leading zeros (considered as 0s in binary) to the shorter number to make both operands have the same number of digits.
14. Overflow occurs in binary addition when the sum exceeds the maximum representable value for the given number of bits. True or False?
Answer: (a)
Explanation: Overflow happens in binary addition when the sum of two binary numbers exceeds the maximum value that can be represented in the given number of bits. This results in a loss of the most significant bit (MSB).
15. How can overflow be detected in binary addition?
(a) Check if the carry bit from the most significant bit (MSB) column is 1.
(b) Verify if any bits are flipped during the addition process.
(c) Overflow cannot be detected during addition.
(d) It requires a special overflow flag in the circuit.
Answer: (a)
Explanation: In binary addition, if the carry bit from the MSB column overflows (becomes 1), it indicates that the sum exceeds the representable range, signaling an overflow condition.
OPERATION ON UNSIGNED AND SIGNED BINARY NUMBERS
- Unsigned binary numbers can represent both positive and negative values. True or False?
(a) True
(b) False
Answer: (b)
Explanation: Unsigned binary numbers can only represent non-negative values (0 and positive integers). They use all available bits to represent the magnitude of the number.
- Signed binary numbers use a specific method to represent negative values. Which method is commonly used?
(a) Sign-magnitude
(b) Two's complement
(c) Excess-127
(d) All of the above
Answer: (b)
Explanation: Two's complement is the most widely used method for representing signed binary numbers. It allows for efficient arithmetic operations by converting negative numbers to a binary format suitable for addition and subtraction using the same logic as unsigned numbers.
Addition
- How do you add two unsigned binary numbers?
(a) Same as decimal addition, without considering signs.
(b) Requires special signed addition circuits.
(c) Similar to unsigned addition, but check for overflow.
(d) Not possible without converting them to signed numbers.
Answer: (c)
Explanation: Unsigned binary addition is similar to decimal addition, but you only deal with 0s and 1s. However, be mindful of overflow if the sum exceeds the maximum representable value for the number of bits.
- What happens when adding two positive signed binary numbers with two's complement representation?
(a) The result is always positive.
(b) The result can be positive, negative, or zero depending on the operands. (c) Requires special handling of signs before addition.
(d) Not possible without converting them to unsigned numbers.
Answer: (b)
Explanation: Adding two positive signed binary numbers using two's complement behaves similarly to unsigned addition. The result can be positive, negative (if the sum overflows), or zero.
- How can overflow be detected during addition of signed binary numbers?
(a) Check the sign bit of the result.
(b) Verify if the carry bit from the MSB column flips from 0 to 1 or vice versa.
(c) Overflow cannot occur in signed binary addition.
(d) Requires a complex overflow detection circuit.
Answer: (b)
Explanation: In signed binary addition with two's complement, overflow occurs when the carry bit from the MSB column flips from 0 to 1 or vice versa. This indicates the sum exceeds the representable range for the given number of bits.
Subtraction
- How can you subtract two unsigned binary numbers?
(a) Direct subtraction (larger number minus smaller number).
(b) Requires converting them to signed numbers first.
(c) Use two's complement of the subtrahend and perform addition.
(d) Not possible without special circuits.
Answer: (d)
Explanation: Unsigned binary subtraction is not directly supported because it can lead to negative results beyond the representable range. You might need to convert them to signed numbers or use other techniques like two's complement for subtraction.
- How do you subtract two signed binary numbers using two's complement?
(a) Convert the subtrahend to two's complement and add it to the minuend. (b) Subtract the binary representations directly based on signs.
(c) Requires special signed subtraction circuits.
(d) Not possible without converting them to unsigned numbers.
Answer: (a)
Explanation: Two's complement allows us to perform subtraction of signed binary numbers by converting the subtrahend to its two's complement and adding it to the minuend. This leverages the same logic as addition for signed numbers.
- What is the result of subtracting a positive number from a negative number in signed binary using two's complement?
(a) Always a negative number.
(b) The result can be positive, negative, or zero depending on the operands. (c) Requires special handling of signs before subtraction.
(d) Not possible without converting them to unsigned numbers.
Answer: (b)
Explanation: Subtracting a positive number from a negative number using two's complement in signed binary results in a negative number with a larger magnitude (more negative) than the original negative number. However, the result can also be positive (overflow) or zero depending on the operands.
9. How do you compare the magnitudes of two unsigned binary numbers?
(a) The number with more 1s is larger.
(b) The number with fewer leading zeros is larger.
(c) You need to convert them to signed numbers first.
(d) Both (a) and (b) are valid methods.
Answer: (d)**
Explanation: You can compare the magnitudes of unsigned binary numbers in two ways:
Number of 1s: The number with more 1s has a larger magnitude.
Leading zeros: The number with fewer leading zeros is considered larger.
10. How do you compare the magnitudes of two signed binary numbers using two's complement representation?
(a) It's the same as unsigned comparison, considering signs.
(b) Requires complex sign manipulation before comparison.
(c) Compare the binary representations directly based on signs and magnitude.
(d) Not possible without converting them to unsigned numbers.
Answer: (c)
Explanation: Comparing signed binary numbers with two's complement involves considering both the sign bit and the magnitude:
If signs are different, the number with the positive sign has a larger magnitude.
If signs are the same, compare the magnitudes of the binary representations as with unsigned numbers (more 1s or fewer leading zeros indicates a larger magnitude).
11. Multiplication of unsigned binary numbers follows the same principles as decimal multiplication. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Unsigned binary multiplication is similar to decimal multiplication, but you only deal with 0s and 1s. Shift bits and perform partial products based on the multiplicand and multiplier bits.
12. Division of unsigned binary numbers can be achieved using a similar approach to long division in decimal. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Unsigned binary division resembles long division in decimal. You can perform it using a process of repeated shifting and subtraction.
13. Signed binary multiplication and division are more complex than unsigned operations due to sign handling. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Signed binary multiplication and division require additional steps to handle signs based on the two's complement representation. You need to consider the signs of operands and adjust the results accordingly.
14. Overflow can also occur during multiplication of binary numbers. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Overflow can happen in both signed and unsigned binary multiplication if the product exceeds the maximum representable value for the given number of bits.
15. When performing operations on binary numbers, it's crucial to understand whether they are signed or unsigned to interpret the results correctly. True or False?
(a) True
(b) False
Answer: (a)
Explanation: Understanding whether binary numbers are signed or unsigned is essential. Signed numbers use specific methods like two's complement to represent negative values, impacting how you perform arithmetic operations and interpret the results.