Computer Organizations- Question Paper- 2

Computer Organizations
Question Paper- 2

Computer Organization Question Bank on Binary Arithmetic (Addition, Overflow, Subtraction, Booth’s Multiplication, Division Algorithm)

1. Binary addition of 1 + 1 equals
A) 1
B) 10
C) 11
D) 0
Answer: B

2. Binary addition of 101 + 11 equals
A) 1000
B) 1100
C) 1110
D) 1001
Answer: A

3. Carry is generated in binary addition when
A) both bits are 0
B) both bits are 1
C) bits are different
D) one bit is 0
Answer: B

4. Overflow occurs when
A) carry = 0
B) result is zero
C) result exceeds bit capacity
D) subtraction occurs
Answer: C

5. In signed addition, overflow is detected when
A) carry into MSB ≠ carry out of MSB
B) carry = 1
C) result negative
D) bits unequal
Answer: A

6. 1101 + 0011 =
A) 1110
B) 10000
C) 1010
D) 1100
Answer: B

7. 2’s complement of 1010 is
A) 0101
B) 0110
C) 1011
D) 1100
Answer: B

8. Subtraction using complements is based on
A) addition
B) division
C) shifting
D) comparison
Answer: A

9. In 2’s complement subtraction, we
A) add complements
B) divide numbers
C) multiply bits
D) reverse order
Answer: A

10. Borrow in subtraction occurs when
A) minuend < subtrahend
B) minuend > subtrahend
C) bits equal
D) both zero
Answer: A

11. Booth’s algorithm is used for
A) division
B) addition
C) multiplication
D) subtraction
Answer: C

12. Booth’s algorithm reduces
A) shifts
B) additions
C) storage
D) bits
Answer: B

13. Booth examines bits in pairs of
A) Qn and Qn+1
B) Qn and Qn-1
C) MSB and LSB
D) sign bits
Answer: B

14. Pair 01 in Booth’s algorithm means
A) subtract multiplicand
B) add multiplicand
C) no operation
D) divide
Answer: B

15. Pair 10 in Booth’s algorithm means
A) add multiplicand
B) subtract multiplicand
C) shift only
D) stop
Answer: B

16. Pair 00 in Booth’s algorithm means
A) add
B) subtract
C) no operation
D) divide
Answer: C

17. Pair 11 in Booth’s algorithm means
A) subtract
B) add
C) no operation
D) complement
Answer: C

18. Arithmetic shift right preserves
A) LSB
B) MSB
C) middle bit
D) carry
Answer: B

19. Binary multiplication is based on
A) repeated subtraction
B) repeated addition
C) shifting only
D) division
Answer: B

20. In binary multiplication, each partial product is
A) shifted left
B) shifted right
C) divided
D) inverted
Answer: A

21. 101 × 10 equals
A) 1010
B) 1110
C) 1001
D) 1100
Answer: A

22. Restoring division restores
A) quotient
B) remainder
C) divisor
D) carry
Answer: B

23. Non-restoring division avoids
A) subtraction
B) restoration step
C) shifting
D) quotient bits
Answer: B

24. In division, divisor is aligned with
A) LSB
B) MSB
C) carry
D) quotient
Answer: B

25. Binary division is similar to
A) decimal division
B) subtraction only
C) multiplication
D) complement
Answer: A

26. 1000 − 0011 =
A) 0101
B) 0111
C) 1001
D) 0011
Answer: A

27. Overflow in signed numbers occurs when
A) signs differ and result same
B) signs same and result different
C) carry = 1
D) borrow = 1
Answer: B

28. Full adder adds
A) 1 bit
B) 2 bits
C) 3 bits
D) 4 bits
Answer: C

29. Half adder adds
A) 1 bit
B) 2 bits
C) 3 bits
D) 4 bits
Answer: B

30. Sum bit in half adder =
A) AND
B) OR
C) XOR
D) NOT
Answer: C

31. Carry bit in half adder =
A) XOR
B) AND
C) OR
D) NOT
Answer: B

32. Arithmetic right shift divides by
A) 10
B) 2
C) 4
D) 8
Answer: B

33. Left shift multiplies by
A) 2
B) 4
C) 8
D) 16
Answer: A

34. 111 + 1 =
A) 1000
B) 1111
C) 1100
D) 1010
Answer: A

35. 2’s complement subtraction ignores
A) final carry
B) MSB
C) LSB
D) divisor
Answer: A

36. Booth’s algorithm supports
A) signed multiplication
B) division
C) addition only
D) floating only
Answer: A

37. Division algorithm uses
A) shift & subtract
B) shift & add
C) XOR
D) complement only
Answer: A

38. Quotient bit becomes 1 when
A) subtraction successful
B) subtraction fails
C) overflow occurs
D) shift occurs
Answer: A

39. Quotient bit becomes 0 when
A) subtraction negative
B) subtraction positive
C) carry = 1
D) add occurs
Answer: A

40. Binary subtraction can be done using
A) 2’s complement
B) XOR only
C) shift only
D) OR
Answer: A

41. 1010 + 0101 =
A) 1111
B) 1101
C) 1001
D) 1011
Answer: A

42. MSB indicates
A) magnitude
B) sign
C) carry
D) shift
Answer: B

43. Overflow flag is used in
A) signed arithmetic
B) unsigned only
C) division only
D) shifting only
Answer: A

44. Booth algorithm uses
A) arithmetic shifts
B) logical shifts
C) rotate
D) divide
Answer: A

45. Multiplication hardware uses
A) adders
B) decoders
C) encoders
D) flip-flops only
Answer: A

46. Restoring division restores after
A) negative remainder
B) positive remainder
C) shift
D) add
Answer: A

47. Non-restoring division alternates
A) add & subtract
B) shift & rotate
C) XOR & AND
D) OR & NOT
Answer: A

48. Binary addition uses base
A) 2
B) 8
C) 10
D) 16
Answer: A

49. Subtraction A − B = A +
A) B
B) 1’s complement of B
C) 2’s complement of B
D) B/2
Answer: C

50. Binary arithmetic is used in
A) CPU
B) Printer
C) Scanner
D) Plotter
Answer: A

Part B — 50 Fill in the Blanks (with Answers)

Binary addition base is 2
1 + 1 in binary equals 10
Overflow occurs when result exceeds bit capacity
2’s complement = 1’s complement + 1
Subtraction is performed using complements
Booth’s algorithm is used for multiplication
Booth checks bit pair Qn and Qn-1
Pair 01 means add multiplicand
Pair 10 means subtract multiplicand
Pair 00 means no operation
Pair 11 means no operation
Arithmetic right shift preserves sign bit
Left shift multiplies by 2
Right shift divides by 2
Binary multiplication uses repeated addition
Division uses shift and subtract
Restoring division restores the remainder
Non-restoring avoids restoration step
Full adder adds three bits
Half adder adds two bits
XOR gives sum bit
AND gives carry bit
Signed overflow checks MSB carries
101 + 11 = 1000
100 − 1 = 11
2’s complement of 0001 = 1111
Borrow occurs when minuend is smaller
Quotient is produced in division
Partial products are shifted
Booth reduces number of additions
MSB stands for Most Significant Bit
LSB stands for Least Significant Bit
Negative numbers use 2’s complement
Subtraction converts to addition
Binary arithmetic is used in ALU
Overflow flag indicates error
Arithmetic shift keeps sign
Logical shift inserts zero
Division remainder is stored in A register
Multiplier stored in Q register
Multiplicand stored in M register
Booth uses arithmetic shift right
Binary subtraction may ignore final carry
111 + 1 = 1000
1000 − 1 = 111
Booth works for signed numbers
Division compares with divisor
Quotient bit set to 1 when subtraction is positive
Quotient bit set to 0 when subtraction is negative
Binary arithmetic operates inside the CPU

==================================================================

Topic :- Logic Gates & Boolean Algebra)

1. AND gate produces HIGH output only when
A) All inputs are HIGH
B) Any input is HIGH
C) All inputs are LOW
D) Only one input is HIGH
Answer: A

2. OR gate produces LOW output when
A) All inputs are HIGH
B) Any input is HIGH
C) All inputs are LOW
D) Only one input is LOW
Answer: C

3. NOT gate is also called
A) Buffer
B) Inverter
C) Comparator
D) Encoder
Answer: B

4. NAND gate is combination of
A) AND + NOT
B) OR + NOT
C) XOR + NOT
D) AND + OR
Answer: A

5. NOR gate is combination of
A) AND + NOT
B) OR + NOT
C) XOR + NOT
D) NAND + NOT
Answer: B

6. XOR gate outputs HIGH when
A) All inputs same
B) Inputs differ
C) All inputs HIGH
D) All inputs LOW
Answer: B

7. XNOR gate outputs HIGH when
A) Inputs differ
B) Inputs same
C) One input HIGH
D) One input LOW
Answer: B

8. Universal gates are
A) AND, OR
B) XOR, XNOR
C) NAND, NOR
D) NOT, Buffer
Answer: C

9. Boolean algebra was introduced by
A) Boole
B) Shannon
C) Turing
D) Pascal
Answer: A

10. A + 0 =
A) 0
B) A
C) 1
D) A’
Answer: B

11. A · 1 =
A) 1
B) 0
C) A
D) A’
Answer: C

12. A + A =
A) 0
B) A
C) 1
D) A²
Answer: B

13. A · A =
A) A
B) 1
C) 0
D) A’
Answer: A

14. A + A’ =
A) 0
B) A
C) 1
D) A²
Answer: C

15. A · A’ =
A) A
B) 1
C) 0
D) A’
Answer: C

16. Dual of AND is
A) OR
B) NOT
C) XOR
D) NAND
Answer: A

17. De Morgan’s law: (AB)’ =
A) A’B’
B) A’ + B’
C) AB’
D) A + B
Answer: B

18. De Morgan’s law: (A + B)’ =
A) A’ + B’
B) A’B’
C) AB
D) A + B
Answer: B

19. Number of inputs in NOT gate
A) 0
B) 1
C) 2
D) 3
Answer: B

20. XOR symbol is
A) +
B) ⊕
C) ·
D) —
Answer: B

21. Boolean variable can have values
A) 0 only
B) 1 only
C) 0 or 1
D) 0–9
Answer: C

22. OR gate is also called
A) Logical addition
B) Logical multiplication
C) Inversion
D) Comparison
Answer: A

23. AND gate is also called
A) Logical addition
B) Logical multiplication
C) Inversion
D) Comparison
Answer: B

24. Output of NAND is LOW when
A) All inputs HIGH
B) Any input HIGH
C) All inputs LOW
D) One input LOW
Answer: A

25. Output of NOR is HIGH when
A) All inputs HIGH
B) Any input HIGH
C) All inputs LOW
D) One input HIGH
Answer: C

26. Complement of 0 is
A) 0
B) 1
C) A
D) Undefined
Answer: B

27. Complement of 1 is
A) 1
B) 0
C) A
D) Undefined
Answer: B

28. Idempotent law is
A) A+A=A
B) A+0=A
C) A·1=A
D) A+A’=1
Answer: A

29. Absorption law: A + AB =
A) AB
B) A
C) B
D) 1
Answer: B

30. Commutative law example
A) A+B = B+A
B) A(B+C)
C) A+1
D) A·0
Answer: A

31. Associative law example
A) (A+B)+C = A+(B+C)
B) A+B=B+A
C) A·1=A
D) A+A=A
Answer: A

32. Distributive law
A) A(B+C)=AB+AC
B) A+B=B+A
C) A·A=A
D) A+1=1
Answer: A

33. A + 1 =
A) A
B) 0
C) 1
D) A’
Answer: C

34. A · 0 =
A) A
B) 1
C) 0
D) A’
Answer: C

35. Double complement A’’ =
A) A
B) 0
C) 1
D) A’
Answer: A

36. Gate with triangle and bubble is
A) AND
B) OR
C) NOT
D) XOR
Answer: C

37. NAND gate truth table is opposite of
A) OR
B) AND
C) NOT
D) XOR
Answer: B

38. NOR gate truth table is opposite of
A) OR
B) AND
C) NOT
D) XOR
Answer: A

39. Which gate checks equality?
A) XOR
B) XNOR
C) AND
D) OR
Answer: B

40. XOR output for inputs 1,1
A) 1
B) 0
C) A
D) Undefined
Answer: B

41. XNOR output for inputs 1,1
A) 1
B) 0
C) A
D) Undefined
Answer: A

42. A + AB =
A) AB
B) A
C) B
D) 0
Answer: B

43. A(A+B) =
A) A
B) B
C) AB
D) A+B
Answer: A

44. Boolean algebra deals with
A) Integers
B) Fractions
C) Logic values
D) Matrices
Answer: C

45. NAND using only NAND =
A) Possible
B) Not possible
C) Only for OR
D) Only for AND
Answer: A

46. NOR using only NOR =
A) Possible
B) Not possible
C) Only for NOT
D) Only for XOR
Answer: A

47. Number of rows in 2-input truth table
A) 2
B) 3
C) 4
D) 8
Answer: C

48. Truth table rows =
A) 2n
B) n²
C) n
D) 2+n
Answer: A

49. Boolean expression uses operators
A) +, ·, ’
B) −, ÷
C) %, $
D) ^ only
Answer: A

50. Output of NOT(1)
A) 1
B) 0
C) A
D) Undefined
Answer: B

PART B — 50 Fill in the Blanks (with Answers)

1. AND gate performs logical ______.
Answer: multiplication

2. OR gate performs logical ______.
Answer: addition

3. NOT gate is also called ______.
Answer: inverter

4. NAND = AND + ______.
Answer: NOT

5. NOR = OR + ______.
Answer: NOT

6. XOR means ______ OR.
Answer: Exclusive

7. Boolean algebra works on values ______ and ______.
Answer: 0, 1

8. A + 0 = ______.
Answer: A

9. A · 1 = ______.
Answer: A

10. A + 1 = ______.
Answer: 1

11. A · 0 = ______.
Answer: 0

12. A + A = ______.
Answer: A

13. A · A = ______.
Answer: A

14. A + A’ = ______.
Answer: 1

15. A · A’ = ______.
Answer: 0

16. Double complement of A is ______.
Answer: A

17. Universal gates are ______ and ______.
Answer: NAND, NOR

18. (A+B)’ = ______.
Answer: A’B’

19. (AB)’ = ______.
Answer: A’ + B’

20. XOR outputs HIGH when inputs are ______.
Answer: different

21. XNOR outputs HIGH when inputs are ______.
Answer: same

22. Idempotent law: A+A = ______.
Answer: A

23. Absorption law: A+AB = ______.
Answer: A

24. Commutative law: A+B = ______.
Answer: B+A

25. Associative: (A+B)+C = ______.
Answer: A+(B+C)

26. Distributive: A(B+C) = ______.
Answer: AB+AC

27. Boolean algebra was proposed by ______.
Answer: George Boole

28. NOT gate has ______ input.
Answer: one

29. Truth table rows = ______ⁿ.
Answer: 2

30. NAND output is LOW only when all inputs are ______.
Answer: HIGH

31. NOR output is HIGH only when all inputs are ______.
Answer: LOW

32. OR gate symbol is ______.
Answer: +

33. AND gate symbol is ______.
Answer: ·

34. Complement symbol is ______.
Answer: ’

35. XOR symbol is ______.
Answer: ⊕

36. Boolean variables are also called ______ variables.
Answer: logical

37. A + AB simplifies to ______.
Answer: A

38. A(A+B) simplifies to ______.
Answer: A

39. Complement of 0 is ______.
Answer: 1

40. Complement of 1 is ______.
Answer: 0

41. Logic gates are basic building blocks of ______ circuits.
Answer: digital

42. AND gate requires ______ condition true for HIGH output.
Answer: all

43. OR gate requires ______ condition true for HIGH output.
Answer: any

44. Boolean algebra is used in ______ design.
Answer: circuit

45. NAND is inverse of ______ gate.
Answer: AND

46. NOR is inverse of ______ gate.
Answer: OR

47. Equality detector gate is ______.
Answer: XNOR

48. Difference detector gate is ______.
Answer: XOR

49. Boolean expressions can be simplified using ______ laws.
Answer: algebraic

50. Logic gate outputs are always ______ or ______.
Answer: 0, 1

============================================================

Topic:- Map Simplification (K-Map)

1. K-Map is mainly used for:
A) Encoding
B) Simplifying Boolean expressions
C) Memory allocation
D) Programming
Answer: B

2. A 2-variable K-Map contains how many cells?
A) 2
B) 4
C) 8
D) 16
Answer: B

3. A 3-variable K-Map has:
A) 4 cells
B) 6 cells
C) 8 cells
D) 16 cells
Answer: C

4. A 4-variable K-Map contains:
A) 8 cells
B) 12 cells
C) 16 cells
D) 32 cells
Answer: C

5. Adjacent cells in K-Map differ by:
A) Two variables
B) One variable
C) Three variables
D) No variable
Answer: B

6. K-Map uses which code ordering?
A) Binary
B) BCD
C) Gray code
D) Excess-3
Answer: C

7. Maximum group size in K-Map must be a power of:
A) 3
B) 5
C) 2
D) 10
Answer: C

8. Valid K-Map grouping sizes are:
A) 3,5,6
B) 1,2,4,8,16
C) 7,9
D) Any number
Answer: B

9. Corner cells in K-Map are:
A) Not adjacent
B) Adjacent
C) Diagonal
D) Invalid
Answer: B

10. Wrapping around edges in K-Map is called:
A) Folding
B) Mapping
C) Adjacency
D) Overlapping
Answer: C

11. Grouping more cells gives:
A) Larger expression
B) Smaller expression
C) Same result
D) No simplification
Answer: B

12. Don’t care conditions are represented by:
A) 0
B) 1
C) X
D) D
Answer: C

13. Don’t care values are used to:
A) Increase terms
B) Help grouping
C) Remove variables
D) Store memory
Answer: B

14. Each cell in K-Map represents:
A) One variable
B) One minterm
C) One gate
D) One flip-flop
Answer: B

15. SOP form groups are made using:
A) 0s
B) 1s
C) X only
D) Variables
Answer: B

16. POS form groups are made using:
A) 1s
B) 0s
C) X only
D) All cells
Answer: B

17. Diagonal grouping is:
A) Allowed
B) Not allowed
C) Preferred
D) Mandatory
Answer: B

18. Overlapping groups are:
A) Not allowed
B) Allowed
C) Wrong
D) Optional only in POS
Answer: B

19. Prime implicant is:
A) Smallest group
B) Largest possible group
C) Single cell
D) Random cell
Answer: B

20. Essential prime implicant is one that:
A) Covers at least one unique minterm
B) Covers all cells
C) Covers zeros
D) Covers diagonals
Answer: A

21. A 5-variable K-Map has cells:
A) 16
B) 32
C) 64
D) 8
Answer: B

22. K-Map method is best for variables up to:
A) 3
B) 4
C) 5–6
D) 10
Answer: C

23. Group of 8 cells removes how many variables (in 4-var map)?
A) 1
B) 2
C) 3
D) 4
Answer: C

24. Group of 4 cells removes:
A) 1 variable
B) 2 variables
C) 3 variables
D) 0
Answer: B

25. Group of 2 cells removes:
A) 1 variable
B) 2 variables
C) 3 variables
D) None
Answer: A

26. Single cell group removes:
A) All variables
B) No variable
C) Two variables
D) One variable
Answer: B

27. K-Map simplification reduces:
A) Cost
B) Hardware
C) Gate count
D) All of these
Answer: D

28. SOP means:
A) Sum of Products
B) Series of Products
C) Sum of Powers
D) System of Products
Answer: A

29. POS means:
A) Product of Sums
B) Power of Sums
C) Product of Series
D) None
Answer: A

30. Largest grouping priority is given to:
A) Pairs
B) Quads
C) Octets
D) Singles
Answer: C

31. Adjacent grouping must be:
A) Horizontal/Vertical
B) Diagonal
C) Random
D) Circular only
Answer: A

32. Gray code ensures:
A) Two bit change
B) One bit change
C) No change
D) Three bit change
Answer: B

33. K-Map is graphical form of:
A) Boolean algebra
B) Assembly
C) C language
D) Registers
Answer: A

34. Minterms are marked by:
A) 1
B) 0
C) X
D) –
Answer: A

35. Maxterms are marked by:
A) 1
B) 0
C) X
D) M
Answer: B

36. Overlapping helps to:
A) Increase cost
B) Reduce literals
C) Add variables
D) None
Answer: B

37. K-Map is also called:
A) Veitch diagram
B) Flow map
C) Tree map
D) Logic tree
Answer: A

38. Group must be rectangular:
A) Yes
B) No
C) Only square
D) Only line
Answer: A

39. Group sizes cannot be:
A) 2
B) 4
C) 6
D) 8
Answer: C

40. K-Map reduces:
A) Propagation delay
B) Complexity
C) Gate count
D) All
Answer: D

41. Each K-Map cell corresponds to:
A) Binary number
B) Decimal number
C) Instruction
D) Address
Answer: A

42. Adjacent edges are treated as:
A) Separate
B) Connected
C) Ignored
D) Optional
Answer: B

43. Simplification result is always:
A) Unique
B) Not always unique
C) Same form
D) Random
Answer: B

44. K-Map grouping eliminates:
A) Changing variables
B) Constant variables
C) Unused variables
D) All variables
Answer: A

45. Don’t cares are used when output is:
A) Known
B) Unknown/irrelevant
C) Always 1
D) Always 0
Answer: B

46. Pair grouping size is:
A) 2
B) 3
C) 5
D) 7
Answer: A

47. Quad grouping size is:
A) 2
B) 4
C) 6
D) 8
Answer: B

48. Octet grouping size is:
A) 6
B) 8
C) 10
D) 12
Answer: B

49. K-Map is best suited for:
A) Manual simplification
B) Programming
C) Storage
D) Networking
Answer: A

50. Goal of K-Map is to obtain:
A) Canonical form
B) Minimal expression
C) Truth table
D) Machine code
Answer: B

Part B — 50 Fill in the Blanks (with Answers)

1. K-Map stands for ______ Map.
Answer: Karnaugh

2. K-Map is used to simplify ______ expressions.
Answer: Boolean

3. A 4-variable K-Map has ______ cells.
Answer: 16

4. A 3-variable K-Map has ______ cells.
Answer: 8

5. Cell ordering in K-Map follows ______ code.
Answer: Gray

6. Adjacent cells differ by ______ bit.
Answer: one

7. Valid group sizes are powers of ______.
Answer: 2

8. SOP grouping is done using ______ values.
Answer: 1

9. POS grouping is done using ______ values.
Answer: 0

10. Don’t care is represented by ______.
Answer: X

11. Each cell represents one ______.
Answer: minterm

12. Largest grouping gives ______ expression.
Answer: minimal

13. Diagonal grouping is ______ allowed.
Answer: not

14. Corner cells are considered ______.
Answer: adjacent

15. Edge wrapping is called ______ adjacency.
Answer: circular

16. Overlapping groups are ______.
Answer: allowed

17. A group of 2 removes ______ variable.
Answer: one

18. A group of 4 removes ______ variables.
Answer: two

19. A group of 8 removes ______ variables.
Answer: three

20. Single cell grouping removes ______ variables.
Answer: zero

21. Prime implicant is the ______ possible group.
Answer: largest

22. Essential implicant covers a ______ minterm.
Answer: unique

23. K-Map is a ______ method.
Answer: graphical

24. POS means Product of ______.
Answer: Sums

25. SOP means Sum of ______.
Answer: Products

26. K-Map reduces number of ______.
Answer: gates

27. K-Map reduces circuit ______.
Answer: complexity

28. Group must be ______ in shape.
Answer: rectangular

29. 5-variable K-Map contains ______ cells.
Answer: 32

30. Maxterms correspond to output value ______.
Answer: 0

31. Minterms correspond to output value ______.
Answer: 1

32. K-Map is also called ______ diagram.
Answer: Veitch

33. Adjacent grouping is horizontal or ______.
Answer: vertical

34. Grouping 16 cells removes ______ variables (4-var map).
Answer: four

35. Don’t care conditions help in ______ grouping.
Answer: larger

36. K-Map is practical for up to ______ variables.
Answer: six

37. Group sizes cannot be ______.
Answer: 3

38. Gray code changes only ______ bit at a time.
Answer: one

39. Simplification reduces hardware ______.
Answer: cost

40. Each K-Map row/column follows ______ sequence.
Answer: Gray

41. A pair is a group of ______ cells.
Answer: 2

42. A quad is a group of ______ cells.
Answer: 4

43. An octet is a group of ______ cells.
Answer: 8

44. K-Map solution may not be ______.
Answer: unique

45. Grouping eliminates ______ variables.
Answer: changing

46. Simplified form has fewer ______.
Answer: literals

47. K-Map is used in ______ logic design.
Answer: digital

48. Cells must be ______ to group.
Answer: adjacent

49. Main goal is ______ expression.
Answer: minimal

50. K-Map avoids complex Boolean ______.
Answer: algebra

Topic:- K-Map Numerical Problems with Solutions

Problem 1 — 3 Variable K-Map (SOP)

Simplify:
F(A,B,C) = Σm(1,3,5,7)

Step:

All minterms are odd → C = 1 in all.

Group all four cells → quad.

Answer:

F = C

Problem 2 — 3 Variable

F(A,B,C) = Σm(0,2,4,6)

All even → C = 0

Answer:

F = C′

Problem 3 — 3 Variable

F(A,B,C) = Σm(2,3,6,7)

Cells where B = 1

Group quad.

Answer:

F = B

Problem 4 — 3 Variable Mixed Grouping

F(A,B,C) = Σm(1,3,4,5)

Groups:

(1,3) → A′C
(4,5) → AB′

Answer:

F = A′C + AB′

Problem 5 — 3 Variable with Don’t Care

F(A,B,C) = Σm(1,3,7) + d(5)

Use don’t care (5) to make quad: (1,3,5,7)

Answer:

F = C

Variable K-Map Problems

Problem 6 — 4 Variable

F(A,B,C,D) = Σm(0,1,2,3)

All first row → A=0, B=0 constant

Answer:

F = A′B′

Problem 7 — 4 Variable

F = Σm(8,9,10,11)

Row where A=1, B=0

Answer:

F = AB′

Problem 8 — 4 Variable Pair + Quad

F = Σm(0,2,8,10)

These form a quad (column grouping wrap-around allowed)

Constants: B=0, D=0

Answer:

F = B′D′

Problem 9 — 4 Variable Complex

F = Σm(1,3,7,11,15)

Group:

(3,7,11,15) → CD
(1,3) → A′B′D

Answer:

F = CD + A′B′D

Problem 10 — 4 Variable with Don’t Care

F = Σm(2,3,6,7,10,11) + d(14,15)

Use don’t cares to form octet covering all where C=1, B=1 varies.

Group of 8 → C = 1

Answer:

F = C

Bonus Exam-Type Question

Problem 11 — POS Simplification

F = ΠM(0,2,5,7)

Group zeros:

(0,2) → (A + C)
(5,7) → (A′ + C)

Answer:

F = (A + C)(A′ + C)

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DR. PANKAJ DADHICH

Computer Organizations- Question Paper- 2