HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.

Show that among rectangles of given area, the square has least perimeter.

Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.

Choose the correct option from the given alternatives : 

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a². Show that the maximum volume of the box is `a^3/(6sqrt(3)`.

Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.

Solve the following :  A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.

Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

Solve the following:

A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.

Solve the following : 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)`.

Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.

Solve the following: 

Find the maximum and minimum values of the function f(x) = cos²x + sinx.

Examine the function for maxima and minima f(x) = x³ - 9x² + 24x

The perpendicular distance of the origin from the plane x − 3y + 4z = 6 is ______ 

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

The function f(x) = x log x is minimum at x = ______.

Find the local maximum and local minimum value of  f(x) = x³ − 3x² − 24x + 5