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CBSE CLASS XII

Differentiability Applications (6 Mark)

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1. Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12cm is 16cm.
2. Find the equation of line through the point (3, 4) which cuts the 1st quadrant a triangle of minimum area.
3. Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.
4. Show that the semivertical angle of a cone of maximum volume and given slant height is tan^-1√2.
5. A wire of length 28m 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.
6. A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.
7. An open box with a square base is to be made out of a given iron sheet of area 27 sq.m. Show that the maximum volume of the box is 13.5 m³
8. A point on hypotenuse of right angled triangle is at a distance ‘a’ and ‘b’ from the sides Show that the length of hypotenuse is at least (a^2/3 + b^2/3)^3/2
9. A square piece of tin of side 48 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off, so that the volume of the box is the maximum possible? Also find the maximum volume.
10. 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.
11. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is Sin^-1(1/3)
12. Awindow is in the form of rectangle surmounted by semi circular opening.The total perimeter of the window is 'p'.c.m.Show that the window will allow the maximum possible light only when the redius of semi circle is
13. A rectangle is inscribed in a semi circle or radius a with one of its sides on the diameter of the semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area .
14. Show that the surface Area of closed cuboid with square base & given volume is maximum when it is a cube.
15. Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius 5√3cm if 500∏cm³.
16. A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m find the dimensions of the rectangle that will produce the largest area of the window.
17. Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone.
18. A wire of length 36cm is cut into two pieces. One of them is turned into square and other is into equilateral triangle. Find the length of each piece so that the sum of the areas of two is minimum
19. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3
20. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Submitted By Mrs. E.Praveen
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