HSC Commerce: Marketing and Salesmanship इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions

Without using truth table prove that:

∼ [(p ∨ ∼ q) → (p ∧ ∼ q)] ≡ (p ∨ ∼ q) ∧ (∼ p ∨ q)

Using rules in logic, prove the following:

p ↔ q ≡ ∼(p ∧ ∼q) ∧ ∼(q ∧ ∼p)

Using rules in logic, prove the following:

∼p ∧ q ≡ (p ∨ q) ∧ ∼p

Using rules in logic, prove the following:

∼ (p ∨ q) ∨ (∼p ∧ q) ≡ ∼p

Using the rules in logic, write the negation of the following:

(p ∨ q) ∧ (q ∨ ∼r)

Using the rules in logic, write the negation of the following:

p ∧ (q ∨ r)

Using the rules in logic, write the negation of the following:

(p → q) ∧ r

Using the rules in logic, write the negation of the following:

(∼p ∧ q) ∨ (p ∧ ∼q)

Solve the following differential equation:

`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`

Solve the following differential equation:

`(1 + 2"e"^("x"/"y")) + 2"e"^("x"/"y")(1 - "x"/"y") "dy"/"dx" = 0`

Solve the following differential equation:

y² dx + (xy + x²)dy = 0

Solve the following differential equation:

`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`

Solve the following differential equation:

`x * dy/dx - y + x * sin(y/x) = 0`

Solve the following differential equation:

`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`

Solve the following differential equation:

`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`

Solve the following differential equation:

`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`

Solve the following differential equation:

x dx + 2y dx = 0, when x = 2, y = 1

Solve the following differential equation:

`x^2.  dy/dx = x^2 + xy + y^2`

Solve the following differential equation:

(9x + 5y) dy + (15x + 11y)dx = 0

Solve the following differential equation:

(x² + 3xy + y2)dx - x² dy = 0