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

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

Prove that if E and F are independent events, then the events E and F' are also independent. 

Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O

Solve the following L.P.P. graphically: 

Minimise Z = 5x + 10y

Subject to x + 2y ≤ 120

Constraints x + y ≥ 60

x – 2y ≥ 0 and x, y ≥ 0