UP Board Solutions Class 10 Computer Science Chapter 4 Discrete Mathematics

Discrete Mathematics Long Answer Type Questions (8 Marks)

 

Question 1. How are characters created by Binary Numbers? What are its different codes? Explain with examples. Answer: Computer Code: Computer codes are used to convert data into binary form to make the computer understand it. Apart from this, they are responsible for error-free signal flow in the computer. Three popular computer codes are:
BCD (Binary Coded Decimal): It is one of the earliest developed memory codes. In this, every digit is converted into binary form separately: e.g.,
\[(43)_{10}= \begin{array}{cc} 0100 & 0011 \\ 4 & 3 \end{array}\] \((43)_{10}=(0100 \ 0011)_2\)
But four-bit code can handle only \(2^4 = 16\) different characters that are why it is extended to 6-bit code and It can handle \(2^6 = 64\) different characters. It is 6-bit code and divided into two parts i.e., zone bit and character code. Zone bit consists of Z bits and character zone consists of 4 bits. To understand more look at the table given below:

BCD Codes
EBCDIC (Extended Binary Coded Decimal Interchange Code): BCD can convert only 64 characters but we use more than 64 characters in computer to represent data. To overcome this problem, 2 more bits have been added to the zone bit to develop new 8-bit code and, that is why it is known as extended binary coded decimal interchange code. EBCDIC is 8- bit code which can encode \(2^8 = 256\) different characters. It is similar to BCD in working but it has 4 bits in bit zone. To understand more table is given below:
EBCDIC Codes
ASCII (American Standard Code for Information Interchange): This code is the most popular and widely accepted computer code. It is the standard code for computers, developed by the American National Standards Institute in the year 1963 for encoding different characters in the computer. It is used by almost every manufacturing company. ASCII codes are of two types:
(a) 7-bit Code: To encode \(2^7 = 128\) characters with 3 bits in zone bit and four in character zone.
ASCII-7 Codes
(b) 8-bit code: To encode \(2^8 = 256\) characters with 4 bits in bit zone and 4 bits in character zone.
ASCII-8 Codes
In simple words: Binary numbers represent data in computers using different codes like BCD, EBCDIC, and ASCII. Each code uses a specific number of bits to create unique combinations for characters, allowing computers to understand and process information.

🎯 Exam Tip: Clearly defining the purpose and structure of each coding scheme (BCD, EBCDIC, ASCII) with correct bit lengths and character capacities is crucial for scoring well.

 

What is 5/8 as a decimal you ask? Converting the fraction 5/8 into a decimal is very easy.
Answer: 6.5.

 

Question 2. Define character representation in computers. Or What is meant by “character representation”? Explain one such code in detail. Or What is meant by character coding? Explain one coding methods in detail. Or What is Character Representation? Answer: Character Representation: Physical devices used to store and process data in computers are two-state devices. A switch, for example, is a two-state device. It can be either ON or OFF. Electronic devices such as transistors used in computers must function reliably when operated as switches. Thus, all data to be stored and processed in computers are transformed or coded as strings of two symbols, one symbol to represent each state.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक साधारण स्विचिंग सर्किट को दर्शाता है, जिसमें 'स्टेट 1' ऑन (ON) स्थिति को और 'स्टेट 0' ऑफ (OFF) स्थिति को प्रदर्शित करता है। इसमें दो स्विच, 'स्विच 1' और 'स्विच 2' हैं, जो यह बताते हैं कि डेटा को कंप्यूटर में कैसे दो-अवस्था वाले उपकरणों (ON/OFF) के माध्यम से दर्शाया जाता है।
Coding of characters has been standardized to facilitate the exchange of recorded data between computers. The most popular standard is known as ASCII. Each letter is a unique combination of Binary Digits (BITS). That is, each letter is a group of charged and uncharged transistors and it is grouped in such a way that a particular combination represents a specific character. A group of 8 BITS which is used to represent a character is called a byte. The length of 1 word is called word length which ranges from 1 byte to 64 bytes.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र 'HARSH' स्ट्रिंग के आंतरिक कोड प्रतिनिधित्व को दिखाता है, जहाँ प्रत्येक अक्षर को बाइनरी बाइट्स के एक समूह के रूप में दर्शाया गया है। इसमें पांच बाइट्स (H, A, R, S, H) एक 'वर्ड' बनाने के लिए संयोजित होते हैं, जो यह स्पष्ट करता है कि कंप्यूटर डेटा को आंतरिक रूप से कैसे स्टोर और प्रोसेस करता है।
In simple words: Character representation defines how symbols (letters, numbers) are converted into binary format for a computer to understand. It uses standardized codes like ASCII, where each character is assigned a unique binary sequence or "byte" for digital processing.

🎯 Exam Tip: When defining character representation, emphasize the two-state nature of computer devices and how codes like ASCII standardize this process. Including a brief mention of bits and bytes will enhance your answer.

 

Question 3. Explain the Number System. Answer: Number System: Number systems are very important to understand because the design and organization of a computer system depend on it. Number systems are basically of two types:
1. Non-positional Number System: In this number system, each symbol represents the same value regardless of its position in the number and the symbols are simply added to find out the value of a particular number. Since it is very difficult to perform arithmetical operations with such a number system, positional number systems have been developed.
2. Positional Number System: In a positional number system, there are only a few symbols called digits, and these symbols represent different values depending on the position they occupy in the number.
The value of each digit in such a number is determined by three considerations :
1. The digit itself Face Value: The face value of a digit always remains the same regardless of its position in the number, e.g., the face value of 4 in 554, 40567 etc. is 4.
2. The position of the digit in the number. Place Value: Place value of a digit changes due to change in its position, e.g., place value of 2 in 4210 is 2 hundred, in 32,450 it is 2 thousand, etc.
3. The base of the number system (where the base is defined as the total number of digits available in the number system), e.g., Decimal number system has base 10 since it includes only 10 digits 0, 1, 2, ....., 9, to represent any number.
The various positional systems in use are:
1. Binary number system
2. Octal number system
3. Decimal number system
4. Hexadecimal number system.
Convert fraction 6 and 1/2 to decimal. What is 6 1/2 as a decimal?
Answer: 6.5.
In simple words: A number system is a method of representing numbers. There are two main types: non-positional, where symbol value is fixed, and positional, where a digit's value depends on its position and the system's base.

🎯 Exam Tip: Define both non-positional and positional number systems clearly, providing examples for face value, place value, and base. Listing the different positional systems is also a key scoring point.

 

Question 4. What are the logical operators? What are their different types? Explain operators making their Truth Table. Answer: Logical Operators: AND, OR, and NOT are logical operators. Since these operators are operated on logical values 0 and 1, that is why these operators are called logical operators.
AND Operator: An AND operator is represented by the symbol '.'. Basically AND operator is used to performing logical multiplication. A, B, and C are three logical variables, where A, B, are input variables and C is the output variable. We can define the AND operator by listing all possible combinations of A and B and the resulting value of C in the operation A.B = C.
It may be noted that since the variables A and B can have only two possible values (0 or 1) so only four (\(2^2\)) combinations of inputs are possible as shown in the following table. The resulting output values for each of the four input combinations are given in the table. Such a table is known as the truth table. Thus, the table is the truth table for the logical AND operator.

Table : Truth Table for Logical AND (.) Operator
As we can observe from the truth table that in AND operation, the output will be 1 when all inputs are 1 else output will be 0.
OR Operator: An OR operator is represented by the symbol V. Basically an OR operator is used to perform logical addition. As in the previous example, A and B are input variables and C its output variable. We can define the OR operator by listing all possible combinations of A and B and the resulting value of C in the equation A + B = C. The truth table for Logical OR operator is shown in the following Table:
Table : Truth Table for Logical OR (+) Operator
As we can observe from the truth table that in logical addition, the output will be 1 when any one input is 1. It means if all inputs are 0 then the output will be 0.
NOT Operator: The two operators (AND and OR) are binary operators because they operate on two variables. NOT operator denoted by is a Unary operator because it operates on a single variable. NOT operator is also known as complementation operator or inverse operator. Thus, complement of A is \(\bar { A }\). Complement of (A + B) is \(\bar { (A+B) }\). If value of \(\bar { A }\) is 0 then value of A is 1 and if value of A is 1 then value of \(\bar { A }\) is 0. The truth table for logical NOT operator is shown in table.
Table : Truth Table for Logical NOT (-) Operator
In simple words: Logical operators (AND, OR, NOT) are fundamental to Boolean algebra, working with binary values (0 and 1) to perform logical operations. Truth tables illustrate all possible input combinations and their corresponding outputs for each operator.

🎯 Exam Tip: For logical operators, always provide clear definitions for AND, OR, and NOT, including their symbols and truth tables. Explain how each operator produces its output based on the input values (0s and 1s).

 

Question 5. Write about the postulates of Boolean Algebra. Answer: Postulates of Boolean Algebra: Boolean Algebra is an algebraic structure defined on a set of elements B together with two binary operators + and . provided the following postulates are satisfied:
(1) (a) Closure with respect to the operator +
(b) Closure with respect to the operator.
(2) (a) An identity element with respect to +, designated by 0 : X + 0 = 0 + X = X.
(b) An identify element with respect to designated by 1 : X . 1 = 1 . X = X.
(3) (a) Commutative with respect to + : X + Y = Y + X
(b) Commutative with respect to . : X .Y = Y. X
(4) (a) . is distributive over : X . (Y + Z) = (X . Y) + (X . Z)
(b) + is distributive over . : X + (Y . Z) = (X + Y) . (X + Z)
(5) For every element X \(\in\) B, there exists an element \(\bar { X }\) \(\in\) B such that:
(a) X \(\times\) \(\bar { X }\) = 1
(b) X . \(\bar { X }\) = 0 The postulates listed above are called Huntington Postulates and need no proof. They are used to prove the theorems of Boolean Algebra.
In simple words: Boolean Algebra postulates are fundamental rules that define its structure, covering properties like closure, identity, commutativity, distributivity, and the existence of complements for its elements.

🎯 Exam Tip: When listing Boolean Algebra postulates, ensure you explicitly state each property (closure, identity, commutativity, distributivity, complement) along with its corresponding mathematical expression for both addition and multiplication.

 

Question 6. What are the different Gates in Boolean Algebra? Or What is 'Truth Table'? How is it helpful in understanding GATES? Or Explain the working of a NAND Gate. Give its two application. Or How can you show that NAND is a Universal Gate? Explain with diagrams and truth tables. Answer: Truth Table: A table that shows all the input-output possibilities of a logic circuit is called a truth table. There are several types of the truth table. AND, OR, NOT, NAND, NOR, XOR, XNOR Gates are described below:
(a) AND Gate: In English language, Input A is ANDed with Input B to get output Y.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक AND गेट के लॉजिक सिंबल को दर्शाता है, जिसमें दो इनपुट A और B हैं तथा एक आउटपुट Y है। इसका कार्य यह है कि Y का मान A और B के लॉजिकल गुणन (A.B) के बराबर होता है, जो यह दर्शाता है कि आउटपुट केवल तभी 'हाई' होगा जब दोनों इनपुट 'हाई' हों।

Table : AND Gate Truth Table
The truth table illustrates four ways to express the logical ANDing of A and B. The AND Gate works on the principle that output will be high when all the inputs are high otherwise output will be low.
(b) OR Gate: In OR Gate, input A is ORed with input B to get output Y.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक OR गेट के लॉजिक सिंबल को दर्शाता है, जिसमें दो इनपुट A और B हैं तथा एक आउटपुट Y है। इसका कार्य यह है कि Y का मान A और B के लॉजिकल योग (A+B) के बराबर होता है, जो यह दर्शाता है कि आउटपुट तभी 'हाई' होगा जब दोनों में से कोई एक या दोनों इनपुट 'हाई' हों।

Table : OR Gate Truth Table
The OR Gate works on the principle that, if anyone input is high, the output will be high. Thus, the only case when output will be low is when all inputs are low i.e., 0.
(c) NOT Gate: The NOT Gate is an electronic circuit that generates an output signal which is the reverse of the input signal. A NOT gate is also known as an inverter because it inverts the input.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक NOT गेट के लॉजिक सिंबल को दर्शाता है, जिसमें एक इनपुट A और एक आउटपुट Y है। NOT गेट इनपुट को उलटा करता है, यानी यदि इनपुट A 'हाई' है तो आउटपुट Y 'लो' होगा (\(Y = \bar{A}\)) और इसके विपरीत।

Table : NOT Gate Truth Table
(d) NAND Gate: A NAND Gate is a complemented AND gate. That is, the output of NAND Gate will be 1 if anyone of the inputs is 0 and will be 0 when all the inputs are 1.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र NAND गेट के लॉजिक सिंबल को दर्शाता है, जिसमें इनपुट A और B तथा आउटपुट Y है। NAND गेट AND गेट का उलटा होता है, यानी इसका आउटपुट A और B के लॉजिकल गुणन के पूरक (\(Y = \overline{A.B}\)) के बराबर होता है।

Table : NAND Gate Truth Table
(e) NOR Gate: A NOR Gate is a complemented OR Gate. That is, the output of a NOR Gate will be 1 only when all inputs are 0 and will be 0 if any input represents a 1.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र NOR गेट के लॉजिक सिंबल को दर्शाता है, जिसमें इनपुट A और B तथा आउटपुट Y है। NOR गेट OR गेट का उलटा होता है, यानी इसका आउटपुट A और B के लॉजिकल योग के पूरक (\(Y = \overline{A+B}\)) के बराबर होता है।

Table: NOR Gate Truth Table
(f) XOR Gate (Exclusive-OR Gate): XOR Gate is a combination of AND, OR, and NOT Gates. symbol denotes XOR operation. This Gate works on the principle that if an odd number of inputs are 1, the output will be 1 otherwise output will be 0.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र (a) एक XOR गेट के कार्यान्वयन को AND/OR/NOT गेट्स का उपयोग करके और (b) XOR गेट के मानक प्रतीक को दर्शाता है। यह गेट तभी 'हाई' आउटपुट देता है जब इसके इनपुट 'विषम' संख्या में 'हाई' हों (यानी एक इनपुट 'हाई' और दूसरा 'लो' हो)।

Table: XOR Gate Truth Table
As observed from the truth table, the output is 1 when odd numbers of inputs are 1.
(g) XNOR Gate (Exclusive-NOR Gate): Similarly, XNOR gate is also formed with a combination of AND, OR, and NOT gates. symbol denotes XNOR operation. Since this gate is the inverse of XOR gate, the output will be 0 when odd numbers of inputs are 1 otherwise output will be 1.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र (a) एक XNOR गेट के कार्यान्वयन को AND/OR/NOT गेट्स का उपयोग करके और (b) XNOR गेट के मानक प्रतीक को दर्शाता है। यह गेट XOR गेट का उलटा है, जिसका अर्थ है कि यह तभी 'हाई' आउटपुट देता है जब इसके इनपुट 'समान' हों (दोनों 'हाई' या दोनों 'लो' हों)।

Table: XNOR Gate Truth Table
In simple words: Logic gates are basic building blocks of digital circuits that perform logical operations on binary inputs, producing a single binary output. A truth table is a comprehensive list showing all possible input combinations and their corresponding outputs for a logic gate.

🎯 Exam Tip: For each logic gate (AND, OR, NOT, NAND, NOR, XOR, XNOR), draw its symbol and provide its truth table. Clearly explain the logic behind each gate's operation, especially how NAND and NOR gates are universal.

 

Question 7. Explain the basic features of ASCII Code. Or Explain in detail the features of the ASCII character code. Answer: ASCII: Binary numbers are coded to represent characters in the computer memory. Several codes are used for this purpose. One most commonly used code is the American Standard Code for Information Interchange (ASCII). ASCII has been adopted by several American computer manufacturers as their computer's internal code. This code is popular in data communications, is used almost exclusively to represent data internally in microcomputers, and is frequently found in the larger computers produced by some vendors.
ASCII is of two types: ASCII-7 and ASCII-8. ASCII-7 is a 7-bit code that represents \(128 (2^7)\) different characters. ASCII-8 is an extended version of ASCII-7. It is an 8-bit code that represents \(256 (2^8)\) different characters rather than 128. e.g. (i) A is given ASCII code 65.
Now if we convert 65 into Binary form we get \(01000001 \rightarrow 1\) byte In the same way, every character has its own ASCII value after converting into binary code stored on the computer.
In simple words: ASCII (American Standard Code for Information Interchange) is a widely used character encoding standard that assigns unique binary codes to letters, numbers, and symbols. It comes in 7-bit (128 characters) and 8-bit (256 characters) versions, making it fundamental for computer communication and data storage.

🎯 Exam Tip: When discussing ASCII, highlight its full form, the number of bits (7-bit and 8-bit), and the corresponding number of characters each can represent. Emphasize its role as a standard for data representation.

 

Question 8. Describe various Binary Arithmetic Operations. Or What is binary arithmetic? Explain with suitable example. Answer: Four basic arithmetic operations are performed inside a computer using binary numbers. These are addition, subtraction, multiplication, and division. Since binary numbers are made up of 0's and 1's, results of arithmetic operations are also in 0's and 1's only.
In simple words: Binary arithmetic involves performing basic mathematical operations-addition, subtraction, multiplication, and division-using only binary digits (0s and 1s), which is how computers process numerical data.

🎯 Exam Tip: To score well on binary arithmetic, list and briefly describe each of the four operations (addition, subtraction, multiplication, division). Providing simple, clear examples for each operation is essential for full marks.

Binary Addition: Binary addition is performed in the same manner as decimal addition. However the binary system has only two digits, the addition table for binary arithmetic is very simple, consisting of only four entries. The complete table for binary addition is as follows:
\(0 + 0 = 0\)
\(0 + 1 = 1\)
\(1 + 0 = 1\)
\(1 + 1 = 0\) Plus a carry of 1 to next higher column. Carryovers are performed in the same manner as in decimal arithmetic. Since 1 is the largest digit in the binary system, any sum greater than 1 requires that a digit be carried over. Example:

BINARY

Binary Subtraction: From the following table, it is clear that the lower digit is subtracted from the upper digit. If the lower digit is larger than the upper digit, it is necessary to borrow from the column to the left which equals to 2 (10).
\(0 - 0 = 0\)
\(1 - 0 = 1\)
\(1 - 1 = 0\)
\(0 - 1 = 1\) with borrow from the next column. Thus, the only case in which it is necessary to borrow is when 1 is subtracted from 0. Example

Binary Multiplication: Multiplication in the binary system also follows the same general rules as decimal multiplication. The table for binary multiplication is as follows:
\(0 \times 0 = 0\)
\(0 \times 1 = 0\)
\(1 \times 0 = 0\)
\(1 \times 1 = 1\)
Example:

Binary Division: Binary division is, again, very simple. As in the decimal system (or in any other system), division by zero is meaningless, here too. Hence, the complete table for the binary division is as follows:
\(0/1 = 0\)
\(1/1 = 1\)
The division process is performed in a manner similar to the decimal division. Example:

Discrete Mathematics Short Answer Type Questions (4 Marks)

 

Question 1. What is the order of precedence in Boolean Algebra? Answer: In a Boolean expression, many operators are used. The order in which they are operated is known as precedence. The precedence of Boolean operators is as follows:
1. The expression is scanned from left to right.
2. Expressions enclosed within parentheses are evaluated first.
3. All complement (NOT) operations are performed next.
4. All '.' (AND) operations are performed after that.
5. Finally, all '+' (OR) operations are performed in the end.
In simple words: Precedence in Boolean Algebra defines the order in which operators (NOT, AND, OR) are processed in an expression, similar to order of operations in arithmetic, often starting with parentheses, then NOT, AND, and finally OR.

🎯 Exam Tip: Understanding operator precedence is crucial for correctly evaluating Boolean expressions and designing logic circuits, impacting the final output.

 

Question 2. Define the Principal of Quality. Answer: The Huntington Postulates have been listed in two parts:
(a) and (b). One part may be obtained from the other if '+' is interchanged '+' with '.' and '0' in interchanged with '1' and vice-versa. This important property of Boolean Algebra is called Principle of Quality. This principle ensures that, if a theorem is proved using the postulates, then a dual theorem obtained interchanging '+' with '.' and '0' with '1' automatically holds and need not be proved separately. The table below lists theorems and their corresponding dual theorems.

In simple words: The Principle of Quality in Boolean Algebra states that if you swap addition with multiplication and 0s with 1s in any theorem, you get another valid theorem, saving proof effort.

🎯 Exam Tip: Explaining the interchangeability of operators and identities is key to demonstrating a full understanding of the Principle of Quality.

 

Question 3. Write a note on De Morgan's Theorems to prove it. Answer: Theorem(a): De Morgan's Theorems: \( (x + y)' = x' \cdot y' \) Proof: The truth table for proving this theorem is given below:

From the truth table, it is clear that both sides of the theorem are equal. Hence, the theorem is proved. Theorem (b): \( (x + y)' = x' + y' \) Proof: The truth table for proving this theorem is given below:

From the truth table, It is clear that both sides of the theorem are equal. Hence, the theorem is proved. Theorems 6(a) and 6(b) are very important and useful. They are known as De Morgan's theorems. They can be extended to n variables as given below: \( (X1 + X2 2 + X3 + .......... + X)' = X₁' \cdot X2' \cdot X3' ........ Xn \)
\( (X1 \cdot X2 \cdot X3 \cdot \dots \cdot Xn)' = X₁' + X’2 + X3' + \dots + Xn \)
In simple words: De Morgan's Theorems simplify complex Boolean expressions by showing how to negate a sum or a product of variables, converting a sum to a product of negations, or a product to a sum of negations.

🎯 Exam Tip: Be precise with the complement notation and truth table values; a single error can invalidate the proof of the theorem.

 

Question 4. Write AND and OR LAWS of Discrete mathematics. Answer: AND LAWS: AND LAWS are the laws which work on logical multiplication. They are:
1. \(x \cdot 1 = x\)
2. \(x \cdot x' = 0\)
3. \(x \cdot x = x\)
4. \(x \cdot 0 = 0\) “The tabular representations of truth values of a compound statement based on the truth values of the prime connective ness of statements is called TRUTH TABLE." Truth table consists of horizontal lines (rows) and vertical lines (columns). If a compound statement consists of N statements, the number of rows will be \(2^N\). The number of columns in a truth table depends upon the number of relationships between these statements.
In simple words: AND Laws describe how logical multiplication (AND operation) works with binary values (0s and 1s), similar to multiplication in arithmetic. A truth table is a comprehensive list showing all possible inputs and their corresponding outputs for a logical expression.

🎯 Exam Tip: Remember that AND laws simplify expressions and are fundamental to understanding how computers process logical decisions.

Discrete Mathematics Very Short Answer Type Questions (2 Marks)

 

Question 1. Discuss De-Morgan's Theorem. Answer: First Theorem: This theorem states that the complement of a sum of the binary variable is equal to the product of the complement of the binary variables. Second Theorem: The theorem states that the complement of a product of binary variable is equal to the sum of the complement of the binary variable
In simple words: De-Morgan's theorems explain how negation interacts with logical sums and products, essentially stating that negating an 'OR' operation is equivalent to 'AND'ing the negations, and negating an 'AND' operation is equivalent to 'OR'ing the negations.

🎯 Exam Tip: Focus on the two main theorems and their dual relationships to concisely explain De-Morgan's contribution to Boolean algebra.

 

Question 2. What is the full form of ASCII? Answer: The full form of ASGII is American Standard Code for Information Interchange.
In simple words: ASCII stands for American Standard Code for Information Interchange, a common character encoding standard used by computers.

🎯 Exam Tip: Accurately spelling out the full form is the primary scoring point for this question.

 

Question 3. If A = 0 and B = 1, then find the value of y from the following expression: Y = (A B) Answer: Y = \( \bar{(0 \cdot 1)} \) = \( \bar{(0)} \) = 1
In simple words: If A is 0 and B is 1, then the expression Y = (A AND B) negated becomes 1 because (0 AND 1) is 0, and the negation of 0 is 1.

🎯 Exam Tip: Show each step of the logical evaluation clearly, especially the AND operation and then the NOT operation.

 

Question 4. Give the name of the Boolean operators. Answer: AND Operator, Or operator and NOT operator.
In simple words: The basic Boolean operators used in logic are AND, OR, and NOT, which perform logical multiplication, addition, and inversion, respectively.

🎯 Exam Tip: List all three fundamental operators correctly; no detailed explanation is required here.

 

Question 5. Write a full form of EBCDIC. Answer: Extended Binay Coded Decimal Interchange Code.
In simple words: EBCDIC stands for Extended Binary Coded Decimal Interchange Code, another character encoding system, primarily used on IBM mainframe operating systems.

🎯 Exam Tip: Ensure correct spelling and sequence for each word in the acronym.

Discrete Mathematics Objective Type Questions (1 Marks)

There are four alternative answers for each part of the questions. Select the correct one and write in your answer book:

 

Question 1. Each letter is a unique combination of:
(a) Bits
(b) Bytes
(c) Word length
(d) Binary.
Answer:
(a) Bits
In simple words: Each character or letter in a computer system is fundamentally represented by a unique combination of bits, which are the smallest units of binary data.

🎯 Exam Tip: Remember that bits are the fundamental building blocks of digital data representation.

 

Question 2. A group of 8 bits which is used to represent a character is called :
(a) Bits
(b) Bytes
(c) Integer
(d) None of these.
Answer: (b) Bytes
In simple words: A byte is a standard unit of digital information, typically consisting of eight bits, used to encode a single character.

🎯 Exam Tip: Clearly distinguish between bits (single binary digits) and bytes (groups of bits, usually 8, forming a character).

 

Question 3. The most popular standard is known as:
(a) BCD
(b) ABC
(c) ASC
(d) ASCII
Answer: (d) ASCII
In simple words: ASCII (American Standard Code for Information Interchange) is the most widespread character encoding standard for computers and digital devices.

🎯 Exam Tip: Recognize ASCII as the predominant character encoding system in modern computing for general use.

 

Question 4. In the hexadecimal number system, the base :
(a) 8
(b) 10
(c) 16
(d) None of these.
Answer: (c) 16
In simple words: The hexadecimal number system uses 16 unique digits (0-9 and A-F) as its base.

🎯 Exam Tip: Associate "hexadecimal" directly with a base of 16 for quick recall.

 

Question 5. The binay equivalent of the number (15)10.
(a) (1101)
(b) (1110)2
(c) (1111)2
(d) (1000)2.
Answer:
(a) (1101)
In simple words: The binary equivalent of decimal 15 is 1111 (represented as (1111)₂), which means four 1s.

🎯 Exam Tip: Practice converting decimal numbers to their binary equivalents to quickly identify the correct option.

 

Question 6. Which logic gate has only one input and one output?
(a) NOT
(b) NOR
(c) OR
(d) AND.
Answer:
(a) NOT
In simple words: The NOT gate is unique among common logic gates because it takes only a single input and produces a single, inverted output.

🎯 Exam Tip: Remember that NOT is a unary operator, making it the only standard gate with just one input.

 

Question 7. The value of the binary number (1010)2 would be?
(a) (14)10
(b) (12)10
(c) (10)10
(d) (11)10
Answer: (c) (10)10
In simple words: To convert binary (1010)₂ to decimal, calculate \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\), which equals 8 + 0 + 2 + 0, resulting in 10 in decimal.

🎯 Exam Tip: Accurately apply positional weighting for each binary digit to convert it to its decimal equivalent.

 

Question 8. What is binary equivalent of [31]10.
(a) 10000
(b) 11111
(c) 100000
(d) 11110.
Answer: (b) 11111
In simple words: The binary representation of decimal 31 is 11111 (represented as (11111)₂), which consists of five 1s.

🎯 Exam Tip: Repeatedly divide the decimal number by 2 and note the remainders to find its binary equivalent, reading from bottom to top.

 

Question 9. Which of the following is a single input logic gate?
(a) NAND
(b) AND
(c) NOT
(d) NOR.
Answer: (c) NOT
In simple words: The NOT gate is the only gate among the choices that operates on a single input and produces one output, which is the inverse of the input.

🎯 Exam Tip: Focus on the number of inputs each gate requires to correctly identify the single-input gate.

UP Board Solutions For Class 10 Computer Science