HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Divide the number 30 into two parts such that their product is maximum.

Divide the number 20 into two parts such that sum of their squares is minimum.

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 ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t². Find the maximum height it can reach.

Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.

An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.

The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?

A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?

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.