HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Prove that:

`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Write converse and inverse of the following statement: 
“If a man is a bachelor then he is unhappy.” 

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)]

Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4

Examine whether the following statement (p ∧ q) ∨ (∼p ∨ ∼q) is a tautology or contradiction or neither of them.

Using the truth table prove the following logical equivalence.

∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p

. Show that the lines represented by 3x² - 4xy - 3y² = 0 are perpendicular to each other.

Show that the lines represented by x² + 6xy + 9y² = 0 are coincident.

Using the truth table prove the following logical equivalence.

p ↔ q ≡ ∼ [(p ∨ q) ∧ ∼ (p ∧ q)]

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

Using the truth table prove the following logical equivalence.

p → (q → p) ≡ ∼ p → (p → q)

Using the truth table prove the following logical equivalence.

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

Using the truth table prove the following logical equivalence.

p → (q ∧ r) ≡ (p → q) ∧ (p → r)

Using the truth table, prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Find the measure of the acute angle between the line represented by:

2x² + 7xy + 3y² = 0

Find the measure of the acute angle between the line represented by:

4x² + 5xy + y² = 0