HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

Express the truth of each of the following statements using Venn diagrams: 

(a) No circles are polygons

(b) Some quadratic equations have equal roots 

Evaluate: ∫ x . log x dx

If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧  ∼ r ) ∧ ( ∼ q ∧ s )

Using the rules of negation, write the negatlon of the following: 

(a) p ∧ (q → r)

(b)  ~P ∨ ~q

If   p : It is raining
     q : It is humid

Write the following statements in symbolic form:

(a) It is raining or humid.
(b) If it is raining then it is humid.
(c) It is raining but not humid. 

Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]

Express the truth of the following statements with the help of Venn diagram:
(a) No circles are polygon
(b) If a quadrilateral is rhombus , then it is a parallelogram .

Write the Truth Value of the Negation of the Following Statement :

The Sun sets in the East. 

Show that the following statement pattern in contingency : 

(~p v q) → [p ∧ (q v ~ q)] 

Use the quantifiers to convert the following open sentence defined on N into true statement
5x - 3 < 10

Use the quantifiers to convert the following open sentence defined on N into true statement:
x² ≥ 1

Using the Venn diagram, examine the logical equivalence of the following statements: 

(a) Some politicians are actors.

(b) There are politicians who are actors.

(c) There are politicians who are not actors.

Write the negation of the Following Statement :
∀ y ∈  N, y² + 3 ≤ 7

Write the negation of the following statement : 
If the lines are parallel then their slopes are equal.

State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.

State if the following sentence is a statement. In case of a statement, write down the truth value :
√-4 is a rational number.

Write the truth value of the negation of the following statement : 

cos² θ + sin² θ = 1, for all θ ∈ R 

Express the truth of each of the following statements using Venn diagram.
(1) All teachers are scholars and scholars are teachers.
(2) If a quadrilateral is a rhombus then it is a parallelogram..

Write converse and inverse of the following statement :
"If Ravi is good in logic then Ravi is good in Mathematics."

Write converse, inverse contrapositive of the statement "If two triangles are not congruent then their areas are not equal.