# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1 - Mathematical Logic [Latest edition]

## Chapter 1: Mathematical Logic

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Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1 Mathematical Logic Q.1

#### MCQ

Q.1 | Q 1

Choose the correct alternative :

Which of the following statement is true?

• 3 + 7 = 4 or 3 – 7 = 4

• If Pune is in Maharashtra, then Hyderabad is in Kerala

• It is false that 12 is not divisible by 3

• The square of any odd integer is even

Q.1 | Q 2

Choose the correct alternative :

Which of the following is not a statement?

• 2 + 2 = 4

• 2 is the only even prime number

• Come here

• Mumbai is not in Maharashtra

Q.1 | Q 3

Choose the correct alternative:

If p is any statement, then (p ˅ ~p) is a

• Contingency

• Tautology

• None of these

Q.1 | Q 4

Choose the correct alternative :

If p and q are two statements then (p → q) ↔ (∼ q → ∼ p) is

• Tautology

• Neither (i) not (ii)

• None of the these

Q.1 | Q 5

Choose the correct alternative:

Negation of p → (p ˅ ~q) is

• ~p → (~p ˅ q)

• p ˄ (~p ˄ q)

• ~p ˅ (~p ˅ ~q)

• ~p → (~p → q)

Q.1 | Q 6

Choose the correct alternative :

If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

• ∼ p ∨ ∼ p

• ∼ (p ∧ q)

• ∼ (p ∨ q)

• p ∨ ∼ q

Q.1 | Q 7

Choose the correct alternative:

A biconditional statement is the conjunction of two ______ statements

• Negative

• Compound

• Connective

• Conditional

Q.1 | Q 8

Choose the correct alternative:

If p → q is an implication, then the implication ~q → ~p is called its

• Converse

• Contrapositive

• Inverse

• Alternative

Q.1 | Q 9

Choose the correct alternative:

The dual of the statement (p ˅ q) ˄ ( r ˅ s) is

• (p ˄ q) ˄ ( r ˄ s)

• (p ˄ q) ˅ (r ˄ s)

• (p ˅ q) ˅ ( r ˅ s)

• (p ˅ q) ˄ (r ˅ s)

Q.1 | Q 10

Choose the correct alternative:

The false statement in the following is

• p ˄ (∼ p) is contradiction

• (p → q) ↔ (∼ q → ∼ p) is a contradiction

• ∼ (∼ p) ↔ p is a tautology

• p ˅ (∼ p) ↔ p is a tautology

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1 Mathematical Logic Q.2

Q.2 | Q 1

State whether the following statement is True or False:

The negation of 10 + 20 = 30 is, it is false that 10 + 20 ≠ 30.

• True

• False

Q.2 | Q 2

State whether the following statement is True or False :

x2 = 25 is true statement.

• True

• False

Q.2 | Q 3

State whether the following statement is True or False:

p → q is equivalent to p → ~ q

• True

• False

Q.2 | Q 4

State whether the following statement is True or False:

Truth value of sqrt(3) is not an irrational number is F

• True

• False

Q.2 | Q 5

State whether the following statement is True or False:

(p ˅ q) ˄ ~ p is a contradiction

• True

• False

Q.2 | Q 6

State whether the following statement is True or False:

p ↔ q is false when p and q have different truth values

• True

• False

Q.2 | Q 7

State whether the following statement is True or False:

The dual of (p ˄ q) ˅ ~ q is (p ˅ q) ˄ ~ q

• True

• False

Q.2 | Q 8

State whether the following statement is True or False:

Mathematical identities are true statements

• True

• False

Q.2 | Q 9

State whether the following statement is True or False:

p ˅ ~ p ≡ ~ c

• True

• False

Q.2 | Q 10

State whether the following statement is True or False:

The converse of inverse of ~ p → q is q → ~ p

• True

• False

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1 Mathematical Logic Q.3

#### Fill in the following blanks

Q.3 | Q 1

Fill in the blanks :

Conjunction of two statement p and q is symbolically written as ______.

Q.3 | Q 2

Negation of “Some men are animal “ is ______

Q.3 | Q 3

The truth value of negation of “London is in England” is ______

Q.3 | Q 4

The truth value of the statement “Neither 27 is a prime number nor divisible by 4” is ______

Q.3 | Q 5

The contrapositive of p → ~ q is ______

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1 Mathematical Logic Q.4

Q.4 | Q 1

Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°”

Q.4 | Q 2. a)

Write the following statement in symbolic form.

Milk is white if and only if the sky is not blue.

Q.4 | Q 2. b)

Write the following statements in symbolic form.

If Qutub – Minar is in Delhi then Taj-Mahal is in Agra

Q.4 | Q 2. c)

Write the following statement in symbolic form.

Even though it is not cloudy, it is still raining.

Q.4 | Q 3. a)

Use quantifiers to convert the given open sentence defined on N into a true statement

n2 ≥ 1

Q.4 | Q 3. b)

Use quantifiers to convert the given open sentence defined on N into a true statement.

3x – 4 < 9

Q.4 | Q 3. c)

Use quantifiers to convert the given open sentence defined on N into a true statement.

Y + 4 > 6

Q.4 | Q 4

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

Q.4 | Q 5

Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p

Q.4 | Q 6. a)

Write the dual of the following.

13 is prime number and India is a democratic country

Q.4 | Q 6. b)

Write the dual of the following

(p ˄ ∼q) ˅ (∼p ˄ q) ≡ (p ˅ q) ˄ ∼(p ˄ q)

Q.4 | Q 7

Write the converse, inverse, and contrapositive of the following statement.

"If it snows, then they do not drive the car"

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1 Mathematical Logic Q.5

Q.5 | Q 1

Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

Q.5 | Q 2

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Q.5 | Q 3

Without using truth table show that

(p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ ( ~ p ∧ q)

Q.5 | Q 4

With proper justification, state the negation of the following.

(p ↔ q) v (~ q → ~r)

Q.5 | Q 5

Prepare truth table for (p ˄ q) ˅ ~ r

(p ∧ q) ∨ ~ r

Q.5 | Q 6. a)

If p ↔ q and p → q both are true, then find truth values of the following with the help of activity

p ˅ q

 p ↔ q and p → q both are true if p and q has truth value square, square or square, square. p ˅ q i. If both p and q are true, then p ˅ q = square ˅ square = square ii. If both p and q are false, then p ˅ q = square ˅ square = square
Q.5 | Q 6. b)

If p ↔ q and p → q both are true, then find truth values of the following with the help of activity

p ˄ q

 p ↔ q and p → q both are true if p and q has truth value square, square or square, square p ˄ q i. If both p and q are true, then p ˄ q = square ˄ square = square ii. If both p and q are false, then p ˄ q = square ˄ square = square
Q.5 | Q 7

Given following statements
p: 9 × 5 = 45
q: Pune is in Maharashtra
r: 3 is the smallest prime number

Write truth values by activity

 i) (p ˄ q) ˄ r = (square ˄ square) ˄ square = square ˄ square = square ii) ~ ( p ˄ r ) = ~(square ˄ square) = ~ square = square iii) p → q = square → square = square
Q.5 | Q 8

Complete the truth table.

 p q r q → r r → p (q → r) ˅ (r → p) T T T T square T T T F F square square T F T T square T T F F T square square F T T square F T F T F square T square F F T square F T F F F square T square

The given statement pattern is a square

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## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1 - Mathematical Logic

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1 (Mathematical Logic) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1 Mathematical Logic are Truth Value of Statement, Quantifier and Quantified Statements in Logic, Statement Patterns and Logical Equivalence, Algebra of Statements, Venn Diagrams, Logical Connective, Simple and Compound Statements.

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