HSC Science (Computer Science) १२ वीं कक्षा - Maharashtra State Board Important Questions

A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which
(i) his shadow is lengthening
(ii) the tip of the shadow is moving

Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`

Prove that:

`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`

Evaluate the following:

`int x tan^-1 x . dx`

`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`

`int sec^2x sqrt(tan^2x + tanx - 7)  "d"x`

`int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x`

Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x)  "d"x`

Evaluate:

`int_0^(pi/2) cos^3x  dx`

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 = ______.