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

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.

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

A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

The negation of p ^ (q → r) is ______.

The maximum value of the function f(x) = `logx/x` is ______.

Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.

If x = f(t) and y = g(t) are differentiable functions of t, then prove that:

`dy/dx = ((dy//dt))/((dx//dt))`, if `dx/dt ≠ 0`

Hence, find `dy/dx` if x = a cot θ, y = b cosec θ.

Find the derivative of 7^x w.r.t.x⁷

Suppose y = f(x) is differentiable function of x and y is one-one onto, `dy/dx ≠ 0`. Also, if x = f^–1(y) is differentiable, then prove that `dx/dy = 1/((dy/dx))`, where `dy/dx ≠ 0`

Hence, find `d/dx(tan^-1x)`.

Find the maximum and the minimum values of the function f(x) = x²e^x.