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

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

A car is moving in such a way that the distance it covers, is given by the equation s = 4t² + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

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