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CBSE CLASS XII

Differentiability Applications (6 Mark)

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21. If the lengths of three sides of trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.
22. Given the sum of the perimeter of a square and a circle, show that the sum of their areas is least when the side of the square is equal to diameter of circle.
23. Find a point on the parabola y² = 4x which is nearest to the point (2,-8)
24. Show that the right circular cone of least curved surface and given volume has an altitude equal to√2 times the radius of the base.
25. A given quantity of metal is to be cast into a solid half circular cylinder( i.e. with rectangular base and semicircular ends).Show that the surface area may be minimum, if the ratio of the length of the cylinder to the diameter of its circular ends is π : π + 2.
26. A window is in the form of a rectangle surmounted by a semi circular opening . The total perimeter of the window is 10m. Find dimensions of the window to admit maximum light through the whole opening.
27. An open box with a square base is to be made out of a given quantity of sheet of area a² . Show that the maximum volume of the box is a³ / 6√3.
28. An open tank with square base and vertical sides is to be constructed from a metal sheet so as to hold given quantity of water. Show that the cost of the material will be least when the depth of the tank is half of its width.
29. Show that the height of a closed circular cylinder of given total surface are and maximum volume is equal to the diameter its base.
30. A Cylindrical container with a capacity of 20 cubic feet is to be produced. The top and bottom of the container are to be made of a material that costs Rs.6 per squares foot while the side of the container is made of material costing Rs.3 per squares foot. Find the dimension that will minimize the total cost.
31. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle 30⁰ is 4/81 πh³.
32. Find the largest possible area of a right angled triangle whose hypotenuse is 5cm long.
33. Find two positive numbers whose sum is 16 and sum of whose cube is minimum.
34. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius a is square of side √2 a.
35. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is 8m³. If building of tank cost Rs. 70 per square meters for the base and Rs. 45 per square meters for sides. What is the cost of least expensive tank?
36. Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
37. Prove that :
38. Prove that : tan x > x for x ϵ (0,π/2).
39. Find the intervals on which the following functions are
    (a) strictly increasing and
    (b) strictly decreasing: f(x) = (x+2)e^-x.
40. f(x) = x⁴ – 4x³ + 4x² + 15.

Submitted By Mrs. E.Praveen
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