HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions

If y = sin^–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0

If x = cos θ and y = sin³θ, show that `y(d^2y)/(dx^2) + (dy/dx)^2` = 3sin²θ(5cos²θ – 1)

Check whether the vectors `2hati + 2hatj+ 3hatk`, `-3hati + 3hatj +2hatk` and `3hati + 4hatk` form a triangle or not.

Check whether the vectors `2hati + 2hatj + 3hatk , -3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.

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

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

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

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

Rewrite the following statement without using if ...... then.

If a man is a judge then he is honest.

Rewrite the following statement without using if ...... then.

It 2 is a rational number then `sqrt2` is irrational number.

Rewrite the following statement without using if ...... then.

It f(2) = 0 then f(x) is divisible by (x – 2).

Without using truth table prove that:

(p ∨ q) ∧ (p ∨ ∼ q) ≡ p

Without using truth table prove that:

(p ∧ q) ∨ (∼ p ∧ q) ∨ (p ∧ ∼ q) ≡ p ∨ q

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