Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`

Find the points at which the function f given by f (x) = (x – 2)⁴ (x + 1)³ has

1. local maxima
2. local minima
3. point of inflexion

Find the absolute maximum and minimum values of the function f given by f (x) = cos² x + sin x, x ∈ [0, π].

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.`

Integrate the functions:

`(2x)/(1 + x^2)`

Integrate the functions:

`(log x)^2/x`

Integrate the functions:

`1/(x + x log x)`

Integrate the functions:

sin x ⋅ sin (cos x)

Integrate the functions:

sin (ax + b) cos (ax + b)

Integrate the functions:

`sqrt(ax + b)`

Integrate the functions:

`xsqrt(x + 2)`

Integrate the functions:

`xsqrt(1+ 2x^2)`

Integrate the functions:

(4x + 2) `sqrt(x^2 + x +1)`

Integrate the functions:

`1/(x-sqrtx)`

Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 

Integrate the functions:

`(x^3 - 1)^(1/3) x^5`

Integrate the functions:

`x^2/(2+ 3x^3)^3`

Integrate the functions:

`1/(x(log x)^m),  x > 0, m ne 1`

Integrate the functions:

`x/(9 - 4x^2)`