HSC Science (General) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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.

If the points (1, 1, λ) and (–3, 0, 1) are equidistant from the plane `barr*(3hati + 4hatj - 12hatk) + 13` = 0, find the value of λ.

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

A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.

Find the equations of the planes parallel to the plane x – 2y + 2z – 4 = 0 which is a unit distance from the point (1, 2, 3).