HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.

Solution of the differential equation `(x + y dy/dx)(x^2 + y^2)` = 1, is ______.

If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m²y

Solve:

`1 + (dy)/(dx) = cosec (x + y)`; put x + y = u.

Examine the maxima and minima of the function f(x) = 2x³ - 21x² + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`

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]

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also, find the maximum volume.

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2  units from the point (1,1, 2)

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.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

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.

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .

 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. 

Write the negation of the following.

Tirupati is in Andhra Pradesh.

Write the negation of the following.

3 is not a root of the equation x² + 3x – 18 = 0

Write the negation of the following.

`sqrt2` is a rational number.

Write the negation of the following.

Polygon ABCDE is a pentagon.