HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Important Questions

If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0

Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`

Find `(dy)/(dx)`, if x³ + x2y + xy² + y³ = 81

If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... +  ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.

Find `dy/dx` at x = 0.

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

Find `dy/dx`, if y = (log x)^x.

Evaluate:

`int log x dx`

If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at², y = 2at.

Price P for demand D is given as P = 183 +120D - 3D² Find D for which the price is increasing

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

Find the approximate value of ` sqrt8.95 `

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

The function f (x) = x³ – 3x² + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

Find the approximate value of log₁₀ (1016), given that log₁₀e = 0⋅4343.

 A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle. 

Find the equation of the tangent to the curve at the point on it.

y = x² + 2e^x + 2 at (0, 4)