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

Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`

Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively 

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`

Solve the differential equation `y - x dy/dx = 0`

Solve the differential equation sec²y tan x dy + sec²x tan y dx = 0

Find the differential equation of family of all ellipse whose major axis is twice the minor axis

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

Find the expected value, variance and standard deviation of random variable X whose probability mass function (p.m.f.) is given below: 

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.

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

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 ______________.

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

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]

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

Construct the simplified circuit for the following circuit:

Using truth table, prove the following logical equivalence:

(p ∧ q) → r ≡ p → (q → r)

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls