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CHAPTER 6 APPLICATIONS OF DERIVATIVES Exercise 1: NCERT Based Topic-wise MCQs 6.1 INTRODUCTION 1. One of the given options is related to chapter "Application of Derivatives" is NCERT Page-194/N-147 (a) Rate of change of quantities (b) Monotonic functions (c) Maxima and minima (d) All of these 6.2 RATE OF CHANGE OF QUANTITIES 2. The altitude of a cone is 20cm and its semi-vertical angle is 30∘ . If the semi-vertical angle is increasing at the rate of 2 ∘ per second, then the radius of the base is increasing at the rate of- NCERT Page-236/N-148 (a) 30cm/sec. (b) 160 3 cm/sec. (c) 10cm/sec. (d) 160cm/sec. 3. Water is dripping out from a conical funnel of semi-vertical angle π 4 at the uniform rate of 2cm2 /s is the surface area through a tiny hole at the vertex of the bottom. When the slant height of cone is 4cm, then rate of decrease of the slant height of water is NCERT Page-195/N-149 (a) √2 3π cm/s (b) √2 π cm/s (c) √2 4π cm/s (d) None of these 4. A kite is moving horizontally at a height of 151.5 . If the speed of kite is 10m/s, then the rate at which the string is being let out; when the kite is 250m away from the boy who is flying the kite and the height of the boy is 1.5m, is NCERT Page-197/N-148 (a) 4m/s (b) 6m/s (c) 7m/s (d) 8m/s 5. The cost of running a bus from A to B, is ₹ (av + b v ), where vkm/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be ₹75 while at 40km/h, it is ₹65. Then the most
economical speed (in km/h ) of the bus is : NCERT Page-196/N-148 (a) 45 (b) 50 (c) 60 (d) 40 6. Water is being filled at the rate of 1cm3 /sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14cm. When the height of the water level is 10cm, the rate (in cm2 /sec ) at which the wet conical surface area of the vessel increases is NCERT Page-196/N-148 (a) 5 (b) √21 5 (c) √26 5 (d) √26 10 7. The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is : (a) 9 (b) 10 (c) 11 (d) 12 8. A point on the parabola y 2 = 18x at which the ordinate increases at twice the rate of the abscissa is NCERT Page-196/N-147 (a) ( 9 8 , 9 2 ) (b) (2, −4) (c) ( −9 8 , 9 2 ) (d) (2,4) 9. A stone is dropped into a quiet lake and waves moves in circles at the speed of 5cm/s. If at a instant, the radius of the circular wave is 8cm, then the rate at which enclosed area is increasing, is (a) 20πcm2 /s NCERT Page-196/N-148 (b) 40πcm2 /s (c) 60πcm2 /s (d) 80πcm2 /s 10. A particle moves along the curve 6y = x 3 + 2. The point ' P ' on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate, are (4,11) and (−4, − 31 3 ). NCERT Page-196/N-148 (a) x-coordinates at the point P are ±4 (b) y-coordinates at the point P are 11 and −31 3 (c) Both (a) and (b) (d) None of the above 11. A ball is dropped from a platform 19.6m high. Its position function is - (a) x = −4.9t 2 + 19.6(0 ≤ t ≤ 1) NCERT Page-197/N-149 (b) x = −4.9t 2 + 19.6(0 ≤ t ≤ 2) (c) x = −9.8t 2 + 19.6(0 ≤ t ≤ 2)
(d) x = −4.9t 2 − 19.6(0 ≤ t ≤ 2) 12. A lizard, at an initial distance of 21cm behind an insect, moves from rest with an acceleration of 2cm/s 2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20cm/s. Then the lizard will catch the insect after NCERT Page-197/N-149 (a) 20s (b) 1s (c) 21s (d) 24s 13. The radius of a cylinder is increasing at the rate of 3m/s and its altitude is decreasing at the rate of 4m/s. The rate of change of volume when radius is 4m and altitude is 6m, is (a) 20πm3 /s (b) 40πm3 /s (c) 60πm3 /s (d) None of these 14. The radius of a sphere initially at zero increases at the rate of 5cm/sec. Then its volume after 1sec is increasing at the rate of : NCERT Page-196/N-149 (a) 50π (b) 5π (c) 500π (d) None of these 15. If a circular plate is heated uniformly, its area expands 3c times as fast as its radius, then the value of c when the radius is 6 units, is NCERT Page-197/N-149 (a) 4π (b) 2π (c) 6π (d) 3π 16. A man is moving away from a tower 41.6m high at a rate of 2m/s. If the eye level of the man is 1.6m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30m from the foot of the tower, is (a) − 4 125 rad/s (b) − 2 25 rad/s (c) − 1 625 rad/s (d) None of these 17. A ladder is resting with the wall at an angle of 30∘ . A man is ascending the ladder at the rate of 3ft/sec. His rate of approaching the wall is NCERT Page-236/N-148 (a) 3ft/sec (b) 3 2 ft/sec (c) 3 4 ft/sec (d) 3 √2 ft/sec 18. The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be NCERT Page-236/N-148