BEC198 Questions. Differential Equations

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  BEC198-DIFFERENTIAL EQUATIONS 1 POINT 1. Solve (cos x cos y - cot x) dx - sin x sin y dy = 0. Select one: a. sin x cos y = -ln(C cos) b. sin x cos y = ln (C cos x) c. sin x cos y = ln (C sin x) d. sin x cos y = -ln (C sin x) 2. The Laplace transform of t is Select one: a. 1/s-1 b. 1/s 2   c. 2/s2 d. 1/s 3. Determine the general solution of xdy + ydx = 0. Select one: a. x + y = c b. ln x + ln y = c c. xy = c d. ln (xy) = c 4. A cylindrical tank is 12ft. In diameter and 8=9 ft high. Water flows into the tank at the rate of /10 cuft/sec. It has a hole radius 1/2 inch at the bottom. The time the tank will be full if initially it is empty is? Select one: a. 56 min b. 65 min c. 76 min d. 50 min 5. The differential equation (x2 +4xy+y2) dx  –  xy dy = 0 is? Select one: a. homogenous b. variable separable c. linear differential equation d. exact 6. What is the order of the differential equation (4 + y )  1/3  = e 2x . Select one: a. three b. one-third c. two d. one 7. Find the differential equations of the family of lines through the srcin. Select one: a. ydx - xdy = 0 b. xdx + ydy = 0 c. xdy - ydx = 0 d. ydx + xdy = 0 8. A mothball loses mass by evaporation at rate that is proportional to the surface area. If half that mass is lost in 100 days, how long will it take the radius to decreases to half its initial value? Select one: a. 234 days b. 255 days c. 275 days d. 243 days 9. The differential equation dv = (y 2  - 3vy) dy is said to be? Select one: a. linear in y b. non linear in V c. linear in V d. non linear in x 10. The differential equation given is correctly described by which one of the following choices: d2y/dx2 + bxy dy/dx = f(x). Select one: a. linear,second order homogenous b. non linear, second order, homogenous c. linear. Second order, non homogenous d. non-linear, second order, non homogenous   11. A spherical tank whose inner diameter is 2 meters is filled with water (density 1 g/cc). If a tank has a hole 1 cm in diameter at the bottom, the time the tank will be totally empty is? Select one: a. 2.41 hrs. b. 4.21 hrs. c. 3.61 hrs. d. 6.31 hrs. 12. A cylindrical tank is 12ft. In diameter and 8=9 ft high. Water flows into the tank at the rate of /10 cuft/sec. It has a hole radius 1/2 inch at the bottom. The time the tank will be full if initially it is empty is? Select one: a. 50 min b. 65 min c. 76 min d. 56 min  13. Calculate the time in hrs, that it will take to reach the fatal conc. Of 40% methane in a kitchen measuring 15 ft x 12.5 ft x8 ft for a leaking stove. The rate of leak is 15 cuft of 100% methane/hr. Assume no fresh air is coming in. The gas rate is measured at the rate conditions prevailing in the kitchen. Select one: a. 40 hrs. b. 50 hrs. c. 30 hrs. d. 45 hrs. 14. Solve the differential equation : x(y - x = 1,determine y when x = 2. Select one: a. 1.48 b. 1.8 c. 1.63 d. 1.55 15. A water container whose circular cross section is 6 ft in diameter and whose height is 8 ft. is filled with water. It has a hole at the bottom of radius 1 inch. The time it will take if the tank rests on support so that its 8 ft height is in a horizontal direction and the hole in its bottom is? Select one: a. 25.46 min b. 24.95 min c. 28.95 min d. 29.4 min 16. A new water pump has a capacity of 60 cu m/day. If its capacity goes down by 15% every year, in how many years will the capacity go down to 20 cu m/day? Select one: a. 4.72 yrs. b. 8.6 yrs. c. 3.72 yrs. d. 7.32 yrs. 17. The order of the differential equation is? a. 3 b. 4 c. 6 d. 2 18. The equation y 2  = cx is the general solution of? Select one: a. y' = 2y/x b. y' = y/2x c. y' = 2x/y d. y' = x/2y 19. A new water pump has a capacity of 60 cu m/day. If its capacity goes down by 15% every year, in how many years will the capacity go down to 20 cu m/day? Select one: a. 4.72 yrs. b. 7.32 yrs. c. 8.6 yrs. d. 3.72 yrs. 20. Solve the equation y +6y+9y=0subject to the conditions y(0) = -4 and y(0) = 5. Select one: a. y = (-11x-4) e 3x  b. y = (-7x-4) e 3x  c. y = (11x-4) e- 3x   d. y = (-7x-4) e -3x   21. What is the order of the differential equation (4 + y ) 1/3  = e 2x ? Select one: a. one-third b. three c. one d. two 22. Find the general solution of y' = ysec x. Select one: a. y = C (sec x + tan x) b. y = C sec x tan x c. y = C (sec x - tan x) d. y = C (sec2 x - tan y) 23. The order of the differential equation is? a. 3 b. 2 c. 1 d. 4 24. A 50 lb iron ball is heated to 200oF and then plunged immediately into a vessel containing 100b lbs of water whose temperature is 40oF. The specific heat of iron is 0.11 Btu/lb o F. The common temperature, approached by the iron and water as time approaches infinity is? Select one: a. 38.43 o F b. 58.4 o F c. 48.34 o F d. 68.5 o F  25. The Laplace transform of e t  is? Select one: a. 1/(s-1) 2   b. 1/(s-1) c. 1/s d. 1/(s+1) 2POINTS 26. The population of a certain municipality increases at a rate to the square root of the population. If the present population is 90,000, how long will it take for the population to reach 160,000? Select one: a. 200 years b. 150 years c. 180 years d. 210 years 27. A substance containing 10 lbs. of moisture is placed in a scaled room whose vol.is 2000 cu.ft and which when saturated can hold 0.015 lbs.moisture/cu.ft. Initially the air contains 30% of the moisture which it can hold when saturated. If the substance loses 4 lbs. of moisture content? Assume the substance loses moisture content and the different between the moisture at a rate that is proportional to the moisture content and the difference between the moisture content of the saturated air and the moisture content of the air. Select one: a. 8.3 hrs b. 3.8 hrs c. 5.6 hrs d. 4.76 hrs. 28. A tank and its contents weigh 100 lbs. The average heat capacity of the system is 0.5 Btu/ lb.F. The liquid in the tank is heated by an immersion heater which delivers 100 Btu/min. Heat is lost from the system at a rate proportional to the difference between the temperature of the system (assumed uniform throughout at any instant) and the temperature of the surrounding air, the proportionality constant being 2 Btu/min o F. If the air temperature remains constant at 70oF and if the initial temperature of the tank and its contents is 55oF, the temperature of the tank as a function of the is? Select one: a. T=-120+65e -t/25 b. T=120+65e t/25  c. T=120-65e -t/25  d. T=12-6.5e -t/25 29. The general solution of the ordinary different equation with c = constant is? Select one: a. - In(1 - 2 y) = x 22 + c b. 2 y = 1 + ce -x2   c. - 1 In(1 - 2 y) = x22 + c d. In(1 - 2y) = x2 + c 30. Find the equation of the orthogonal trajectories of the system of parabolas y2=2x+C. Select one: a. y = C e -2x   b. y = C e -x   c. y = C e 2x  d. y = C e x   31. A 1000 ft3 storage tank is filled with natural gas at 80oF and 1 atm pressure. The tank is flushed out with nitrogen gas at 80oF and 1 atm pressure, at a constant rate of 300 cfm. The flushing process is carried out at constant temperature and pressure, under conditions of perfect mixing in the tank at all times. The time required to reach a gas composition of 95 vol. % nitrogen in the tank is nearest yo. Select one: a. 5 min b. 10 min c. 7 min d. 3 min 32. The differential equation is?  a. exact b. linear but not homogenous c. variable separable d. linear and homogenous 33. A certain substance increases at a rate proportional to the square of the instantaneous amount. After 5 days the amount is doubled. Determine the time before the amount is tripled. Select one: a. 20/3 b. 45/3 c. 40/3 d. 25/3 34. If dy = x 2 dx, what is the equation of y in terms of x if the curve passes through (1, 1)? Select one: a. x 3  - 3y + 2 = 0 b. x 2  - 3y + 3 = 0 c. x 3 + 3y + 2 = 0 d. 2y + x3 + 2 = 0 35. Solve xy'(2y -1) = y(1-x). Select one: a. ln (xy) = 2y - x + C b. ln (xy) = x + 2y + C c. ln (xy) = x - 2y + C d. ln (xy) = 2 (x - y) + C  36. A certain substance increases at a rate proportional to the square of the instantaneous amount. After 5 days the amount is doubled. Determine the time before the amount is tripled. Select one: a. 20/3 b. 25/3 c. 45/3 d. 40/3 37. A certain quantity increases at a rate proportional to q itself. If q = 25 when t = 0 and q = 75 when t =2, find q when t = 6. Select one: a. 576 b. 657 c. 756 d. 675 38. Find the equation of the curve at every point at which the tangent line has a slope of 2x. Select one: a. x = -y 2  + C b. y = -x 2  + C c. x = y 2  + C d. y = x 2  + C 39. Solve the differential equation dy - xdx = 0, if the curve passes through (1, 0). Select one: a. x 2  - 2y -1 = 0 b. 2x 2  + 2y - 2 = 0 c. 2y + x2 - 1 = 0 d. 3x 2  + 2y - 3 = 0 40. What is the differential equation of a family of parabolas having their vertices at the srcin and their vertices on the x-axis? Select one: a. xdy + ydx = 0 b. 2xdy - ydx = 0 c. dy/dx - x = 0 d. 2ydx - xdy = 0 41. A low radioactive material is used in biochemical process to induce biological mutation. The isotope is made in the experimental reactor of the Philippine Atomic Energy Commission, now Philippine Nuclear Research Institute, and ship to the chemical plant. It has a half-life of 8.06 days. The plant receive the shipment of the radioactive material which on arrival contain 1 gram of the radioactive material. The plant uses the material at the rate of 0.1 gram per week. The time it will take for the radioactivity to last is? Select one: a. 4.34 weeks b. 3.24 weeks c. 4.74 weeks d. 5.4 weeks 5 POINTS 42. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years? Select one: a. 88.6 b. 95.32 c. 90.72 d. 92.16 43. A body weighing 1960 N is pulled by a constant force of 492 N along a horizontal plane where in the coefficient of friction between the body and the plane id 0.20. Determine the velocity after 20 seconds. Select one: a. 9.06 m/s b. 8.25 m/s c. 13.1 m/s d. 10.57 m/s 44. A tank initially contains 400 liters of water. Salt solution, containing 1/8 kg of salt per liter of solution flows into the tank at the rate of 8 li/min and the solution, kept well-stirred, flows out of the tank at the rate of 4 li/min. find the amount of salt in the tank after 100 minutes. Select one: a. 85 kg b. 80 kg c. 75 kg d. 70 kg 45. Solve a. y= -x 5 +cx 6   b. y=-x 6 +cx 5  c. y=x 6 +cx 5  d. y=x 5 +cx 6   46. What is the general solution of (D 3  -3D 2  - 4D + 12) y = 0. Select one: a. y = C1e 2x  + C2e 3x  + C3e- 2x b. y = C1e -x  + C2e -4x  + C3e 3x  c. y = C1e 2x  + C2e -2x  + C3e -3x  d. y = C1e x  + C2e 2x  + C3e -6x 47. Solve the equation a. b. c. d.
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