The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.

Also prepare marginal distributions :

(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

Concept: Probability Distribution > Distribution Function of a Continuous Random Variable

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:

(a) the mean and

(b) P(X = 0). [Given: e^{-2} = 0.1353)

Concept: Probability Distribution > Distribution Function of a Continuous Random Variable

The negation of p ∧ (q → r) is ______________.

Concept: Mathematical Logic > Algebra of Statements

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

[(p → q) ∧ q] → p

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

**Write the converse and contrapositive of the statement -**

“If two triangles are congruent, then their areas are equal.”

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Prove that the following statement pattern is equivalent :

(p ∨ q) → r and (p → r) ∧ (q → r)

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers.

Concept: Venn Diagrams

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.

(B) Price increases and demand falls

Concept: Mathematical Logic > Logical Connectives

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

Concept: Mathematical Logic > Algebra of Statements

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Write the dual of the following statements: (p ∨ q) ∧ T

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Concept: Mathematical Logic > Algebra of Statements

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Draw a Venn diagram for the truth of the following statement :

All rational number are real numbers.

Concept: Venn Diagrams

Draw Venn diagram for the truth of the following statements :

Some rectangles are squares.

Concept: Venn Diagrams

Write the truth values of the following statements:

(1) 2 is a rational number and`sqrt2` is an irrational number. .

(2) 2 + 3 = 5 or `sqrt 2 +sqrt3 = sqrt5`

Concept: Mathematical Logic > Statements - Introduction in Logic

Find the symbolic form of the given switching circuit. Construct its switching table and interpret yout result. diagram

Concept: Mathematical Logic > Statements - Introduction in Logic

Write converse and inverse of the following statement:

“If a man is a bachelor then he is unhappy.”

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )

Concept: Mathematical Logic > Statement Patterns and Logical Equivalence

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