Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.

Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is  `Sin^(-1) (1/3).`

The point on the curve x² = 2y which is nearest to the point (0, 5) is ______.

For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.

The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.

Find the maximum area of an isosceles triangle inscribed in the ellipse  `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.

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.`

Find `int (cos theta)/((4 + sin^2 theta)(5 - 4 cos^2 theta)) d theta`

Maximise Z = x + 2y subject to the constraints

`x + 2y >= 100`

`2x - y <= 0`

`2x + y <= 200`

Solve the above LPP graphically

Determine the product `[(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)]` and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

Solve the following linear programming problem graphically :

Maximise Z = 7x + 10y subject to the constraints

4x + 6y ≤ 240

6x + 3y ≤ 240

x ≥ 10

x ≥ 0, y ≥ 0