Computational Mathematics and Computer Architecture – Set 3
Question 1: Number Systems (8 Marks)
Convert the following decimal numbers to their respective binary and hexadecimal
representations:
a) 23 (4 marks – Binary 2 marks/Hex 2 marks)
b) 78 (4 marks – Binary 2 marks/Hex 2 marks)
Solution: Question 1 - Number Systems
a) Convert 23 to binary and hexadecimal:
Step 1: Divide 23 by 2 for binary:
= 11 remainder 1
= 5 remainder 1
= 2 remainder 1
= 1 remainder 0
= 0 remainder 1
Binary representation is 10111.
Step 2: Convert 23 to hexadecimal:
= 1 remainder 7
Hexadecimal representation is 17.
b) Convert 78 to binary and hexadecimal:
Step 1: Divide 78 by 2 for binary:
= 39 remainder 0
= 19 remainder 1
= 9 remainder 1
= 4 remainder 1
= 2 remainder 0
= 1 remainder 0
= 0 remainder 1
Binary representation is 1001110.
Step 2: Convert 78 to hexadecimal:
= 4 remainder 14 (E in hexadecimal)
Hexadecimal representation is 4E.
, Question 2: Boolean Algebra (10 Marks)
Simplify the following Boolean expression using Boolean algebra laws:
a) A + AB + A′C (5 marks)
b) Create a truth table for the original and simplified expressions to verify the equivalence.
(3 marks)
c) Draw the logic circuit for the simplified expression. (2 marks)
Solution: Question 2 - Boolean Algebra
a) Simplify A + AB+A′C :
Step 1: Apply the Absorption Law:
From A+AB=A (Absorption Law), simplify:
A+A′C
Step 2: Factorize A+A′C
Using the Distributive Law:
A+C(A′)
Step 3: Simplify further
From C(A′)=A′C:
A+C
Final Simplified Expression:
A+C
b. Create a Truth Table
The truth table confirms equivalence between the original and simplified expressions.
c. Logic Circuit Design for the simplified expressions.
Question 1: Number Systems (8 Marks)
Convert the following decimal numbers to their respective binary and hexadecimal
representations:
a) 23 (4 marks – Binary 2 marks/Hex 2 marks)
b) 78 (4 marks – Binary 2 marks/Hex 2 marks)
Solution: Question 1 - Number Systems
a) Convert 23 to binary and hexadecimal:
Step 1: Divide 23 by 2 for binary:
= 11 remainder 1
= 5 remainder 1
= 2 remainder 1
= 1 remainder 0
= 0 remainder 1
Binary representation is 10111.
Step 2: Convert 23 to hexadecimal:
= 1 remainder 7
Hexadecimal representation is 17.
b) Convert 78 to binary and hexadecimal:
Step 1: Divide 78 by 2 for binary:
= 39 remainder 0
= 19 remainder 1
= 9 remainder 1
= 4 remainder 1
= 2 remainder 0
= 1 remainder 0
= 0 remainder 1
Binary representation is 1001110.
Step 2: Convert 78 to hexadecimal:
= 4 remainder 14 (E in hexadecimal)
Hexadecimal representation is 4E.
, Question 2: Boolean Algebra (10 Marks)
Simplify the following Boolean expression using Boolean algebra laws:
a) A + AB + A′C (5 marks)
b) Create a truth table for the original and simplified expressions to verify the equivalence.
(3 marks)
c) Draw the logic circuit for the simplified expression. (2 marks)
Solution: Question 2 - Boolean Algebra
a) Simplify A + AB+A′C :
Step 1: Apply the Absorption Law:
From A+AB=A (Absorption Law), simplify:
A+A′C
Step 2: Factorize A+A′C
Using the Distributive Law:
A+C(A′)
Step 3: Simplify further
From C(A′)=A′C:
A+C
Final Simplified Expression:
A+C
b. Create a Truth Table
The truth table confirms equivalence between the original and simplified expressions.
c. Logic Circuit Design for the simplified expressions.
