Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

An open box with square base is to be made of a given quantity of cardboard of area c². Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.

If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?

AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.

A metal box with a square base and vertical sides is to contain 1024 cm³. The material for the top and bottom costs Rs 5/cm² and the material for the sides costs Rs 2.50/cm². Find the least cost of the box.

The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.

If x is real, the minimum value of x² – 8x + 17 is ______.

The smallest value of the polynomial x³ – 18x² + 96x in [0, 9] is ______.

The function f(x) = 2x³ – 3x² – 12x + 4, has ______.

The maximum value of sin x . cos x is ______.

Maximum slope of the curve y = –x³ + 3x² + 9x – 27 is ______.

The maximum value of `(1/x)^x` is ______.

The curves y = 4x² + 2x – 8 and y = x³ – x + 13 touch each other at the point ______.

If A, B be two square matrices such that |AB| = O, then ____________.

A square matrix A is invertible if det A is equal to ____________.

Which of the following is a second order differential equation?

Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 is ______.

The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.

If A `= [(0,-1,2),(1,0,3),(-2,-3,0)],` then A + 2A^T equals

Find the adjoint of the matrix A `= [(1,2),(3,4)].`