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

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

Solve the following differential equation:

(x² – y²)dx + 2xy dy = 0

Let p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Then, this law is known as ______.

Without using truth table, show that

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Without using truth table, show that

p ∧ [(~ p ∨ q) ∨ ~ q] ≡ p

Without using truth table, show that

~ [(p ∧ q) → ~ q] ≡ p ∧ q

Without using truth table, show that

~r → ~ (p ∧ q) ≡ [~ (q → r)] → ~ p