Page 1 CBSE TEST PAPER-04 CLASS - XII MATHEMATICS (Calculus: Application of Derivatives) Topic: - application of derivatives 1. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900cm 3 /s. find the rate at which the radius of the balloon increase when the radius is 15cm. [4] 2. A circular disc of radius 3cm is being heated. Due to expansion, their radius increase at the rate of 0.05 cm/s. find the rate at which its area is increasing when radius is 3.2cm. [4] 3. Find the intervals in which the function f given by 4sin 2 cos ( ) 2 cos x x x x f x x - - = + is (i) increasing (ii) decreasing [4] 4. Find the interval in which the function f given by 3 3 1 ( ) , 0 f x x x x = + ? is (i) increasing (ii) decreasing. [4] 5. Find the equation of the normal to the curve 2 4 x y = which passes through the point (1, 2) [4] 6. Show that the normal at any point ? to the curve cos .sin , x a a ? ? ? = + sin .cos y a a ? ? ? = - is at a constant distance from origin. [4] 7. 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. [4] 8. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2 3 R Also find the maximum volume. [4] 9. Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and having semi-vertical angle a is one third that of the cone and the greatest volume of cylinder is 3 4 tan . 27 h p a [6] 10. 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. [6]Read More

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