HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

For the differential equation, find the particular solution (x – y²x) dx – (y + x²y) dy = 0 when x = 2, y = 0

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).

The probability mass function for X = number of major defects in a randomly selected
appliance of a certain type is 

Find the expected value and variance of X.

Given X ~ B(n, p) if p = 0.6 and E(X) = 6, find n and Var(X). 

If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = ______.

The probability that certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

If X ~ B (n, p) and E(X) = 6 and Var (X) = 4.2, then find n and p.

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

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.

[(p→q) ∧ q]→p

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

Write the converse and contrapositive of the statement -
“If two triangles are congruent, then their areas are equal.”

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

[~(~p ∧ ~q)] v q

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Prove that the following statement pattern is equivalent :

(p ∨ q) →  r and (p → r) ∧ (q → r)

If p ˄ q = F, p → q = F, then the truth value of p and q is ______.

Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).

Construct the simplified circuit for the following circuit: