Math Refresher
University of Santo Tomas
Differential Equations and Advanced Eng’g Math
1.
2.
Determine Determine the the solution solution of of the differ differentia entiall equation equation y’ + 5 y = 0 A. y = 5 x + C C. y = Ce5 x –5 x B. y = Ce * D. y = 5e5 x + C What is the the general general soluti solution on of the the differen differential tial equation
B=
2
d y + 4 y = 0 dx A. B. C. D. 3.
4.
5.
6.
7.
8.
9.
A curve curve passes passes through through the point point (1, 1). 1). Determin Determinee the absolute value of the slope of the curve at x = 25 if the differential equation of the curve is the exact equation y2dx + 2 xydy = 0 A. 1/250 C. 1/(50sqrt5) B. 1/125 D. 1/sqrt(125) * Determine Determine the constan constantt of integr integratio ation n for the separable differential equation xdx + 6 y5dy = 0 if x = 0 when y = 2. A. 12 C. 24 B. 16 D. 64 * What What is the the Laplac Laplacee transf transform orm of of e–6t? e–6t? A. 1/(s + 6) * C. e–6 + s B. 1/(s – 6) D. e6 + s Solve Solve the the compl complex ex equa equatio tion n (1 + j2)(–2 – j3) = a + jb A. a = 4, b = 7 C. a = –4, b = –7 B. a = –4, b = 7 D. a = 4, b = –7 * y – j3 x) = 2 + j3 Solv Solvee the the equa equati tion on ( x – j2 y) + ( y A. x = 7, y = 9 C. x = 7, y = –9 B. x = –7, y = 9 * D. x = 9, y = 7
A. 6. 6.785 B. 6. 6 .786
6
2
1
2
1
3
1
2
2
0
1
3
2
3
2
0
20. 20. Solve Solve y” – y’ – 2y 2y = 0 A. y = c1e x + c2e–2 x C. y = c1e –x + c2e2 x * x 2 x B. y = c1e + c2e D. y = c1e –x + c2e–2 x
and
21. 21. Solv Solvee y” + 4 y’ + 5 y = 0. A. y = e2 x(c1 cos x + c2 sin x) B. y = e – 2 x(c1 cos x + c2 sin x) * C. y = e x(c1 cos x + c2 sin x) D. y = e– x(c1 cos x + c2 sin x)
, solve for the
|
4
−3
−5
2
6
−1
−4
2
C. 32 D. –22 *
15. Determine Determine the value value of of
|
|
j2 1− j
1 + j
3
1
j
0
j4
5
A. 2 + j30 * B. 3 – j40
26. Solve Solve the differentia differentiall equation equation A.
x
B.
3
–2/5
C.
y =2 ln x − x
y ′ = x 2 y 27. Solve Solve the the equation equation
D.
y =2 ln x − x 3
y = Ce x 3 3
C.
y = Ce3 x
3
B. 3
*
y = Ce x
3
D.
28. Find the orthogonal orthogonal trajectory trajectory of the the family of of curves x2 – y2 = c2
+c
29. Find the orthogonal orthogonal trajectories trajectories of the the family of of 3
+c
2
4 4
–2/5
y 2 + sin y = x 3 + c D.
A.
4
y =2 ln x − x
C.
y = Ce3 x
dy 3 =2− 4 x dx y =2 ln x − x
y 2 + sin y = 6 x 3 + c
y 2 + sin y = 3 x 3 + c
17. Obtain Obtain the general general solution solution of
3
12. What is the the principal principal argument argument of (–2 + j)1/4 A. 26.4 26.45 5° C. 38.36 ° * B. 36.3 36.38 8° D. 45.26 °
y´ −6 ´ y + 25 y = 0
y 2 + sin y = 2 x 3 + c
B.
11. Determine the the principal principal root of (–14 + j3) . A. 0.34 0.3449 49∠–67.16° C. 0.3449∠148.83° B. 0.34 0.3449 49∠76.83° D. 0.3449 ∠4.83° *
24. A series RL circuit with R = 12 Ωand L = 4 H is connected to a constant supply of 60 V. The switch is turned on at t = 0. Determine the equation of the current as a function of time. What is the limiting value of the current? A. 3 A C. 5 A * B. 4 A D. 6 A
2 dy = 6 x dx 2 y + cos y
C. 2 – j30 D. 3 + j40
4
10. The expres expression sion (–14 + j3) has how many roots? A. 1 C. 3 B. 2 D. 5 *
C. y = c1e – 2t + c2te – 2t * D. y = c1e – 2t + c2te2t
23. Find a solution of the initial-value initial-value problem y' = xy3, y(0) = 2.
25. Solve Solve the the equation equation
16. Which of the the following following is not a solution of the linear system of equations x+y+z=4 2x – 3y + 4z = 33 3x – 2y – 2z = 2 A. 2 C. 5 B. –3 D. 4 *
. C. 6.875 D. 6.986 *
|
1
A. – 18 B. 24
A.
π 8
14. Obtain Obtain the the value value of
x´ + 4 x´ + 4 x =0
A. y = c1e – 2t + c2e2t B. y = c1e2t + c2e – 2t
C. 30 D. 40 *
3
Dete Determ rmin inee the the modu modulu luss of
+3 ∠
3
determinant of A × B A. 10 B. 20
In the differe differential ntial equat equation ion with with an initial initial conditi condition on x(0) = 12, what is the value of x(2)? A. 3.35 3.35 × 10–4 C. 3.35 B. 4.0 4.03 × 10–3 * D. 6.04
π
0
22. 22. Solv Solvee
y = sin x + 2 tan x + C y = e x – 2e– x + C y = 2 x2 – x + C y = sin 2 x + cos 2 x + C *
4∠
[ ] [ ]
A =
13. If
1
In-House Review for ECE
2
x = ky 2 *
curves
, where k is is an arbitrary constant.
x 2 + +c
+c
18. Find the curve which which satisfies the equation equation xy = (1 + x2) y’ y’ and passes through the point (0,1). A. y = (1 + x)1/2 C. y = (1 + x2)1/2 * –1/2 B. y = (1 + x) D. y = (1 + x2)–1/2
y 2
2
=C
Ans: 30. A tank contains contains 20 kg kg of salt dissolved in in 5000 L of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour? Ans: 38.1 kg
dy xy 2 = xy dx 31. Solve Solve the differentia differentiall equation equation
19. Solve Solve the differential differential equati equation on y’ = 2 y cos x A. ln y = 2 sin x + c * B. ln y = –2 sin x + c C. ln y = ½ sin x + c D. ln y = – ½ sin x + c
y = Ans:
2
K − − x 2
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Math Refresher
University of Santo Tomas
32. Find an equation equation of the the curve that that passes through through the point (0, 1) and whose slope at ( x, y) is xy. 33. Find the equation equation of a curve going going through the point (3, 7) having a slope of 4 x2 – 3 at a ny point in the curve. 34. Find the the orthogonal orthogonal trajectories trajectories of the the family of curves x2 + 2 y2 = k 2. Ans: y = Cx2
In-House Review for ECE
thousand. What is the population growth rate after three years? Ans: 389 dear/year dear/year 46. The rate at which which the population population of a city grows grows is given by the differential equation
dP = 0.08P 1 − P ÷ P ( 0) = 100 1000 dt
36. A tank contains contains 1000 L of brine brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 20 minutes? Ans: 15e– t/100 , 12.3 kg 37. A tank contains contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank a t a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. How much salt is in the tank (a) after minutes and (b) after one hour? 38. A bacteria culture culture has an initial population population of of 500. After 4 hours the population population has grown to 1000. Assuming the culture grows at a rate proportional to the size of the population, find a function representing the population size after t hours and determine the size of the population after 6 hours. Ans: 1414
y ′ + 3 x y = 6 x 2
2
47. Solve Solve the differential differential equati equation on
y = 2 + Ce x
40. How old old is is a fossil fossil whose whose 14C/ 12C ratio is 10 percent of that found in the atmosphere today? Ans: 19,000 years
42. An Egyptian Egyptian papyrus is is discovered and it is found found that the ratio of 14C to 12C is 65 percent of the known ratio of 14C to 12C in the air today. The half-life of 14C is 5730 years. How old is the papyrus? Ans: 3561 years 43. According to Newton’s Newton’s law of cooling, cooling, the temperature of an object changes at a rate proportional to the difference in temperature between the object and the outside medium. If an object whose temperature is 70 Fahrenheit is placed in a medium whose temperature is 20 and is found to be 40 after 3 minutes, what will its temperature be after 6 minutes? Ans: 28 °F °
*
y = e− x + xe xe− x y = e− x + ( 3e) xe xe− x B.
C.
y = 2 + Ce x
y = e− x + ( 3e− 1) xe xe− x
y = 2 + Ce− x
3
3
B.
D.
C.
*
*
y = e− x + 1 xe− x ( − 3e) xe
48. Find the solution of the initial-value problem
D.
x 2 y ′ + xy xy = 1 x > 0 y ( 1) = 2 56. Determine the the particular particular solution of the equation
y =
ln x + 2
y =
x
A.
*
y =
+ y ′ − 2 y = x 2 y ′ +
ln 2 + 2
x
A. B.
C.
ln x + 2
y =
2
B.
ln 2 + x 2
D.
49. 49. A ser serie iess RL circuit with R = 12 Ωand L = 4 H is connected to a supply of E (t ) = 60sin30 t volts. volts. Find the current after t = = 0.5 s. Assume ze ro initial condition. Ans: 519 mA
½ x
2
+ ½ x + ¾ ½ x – ½ x + ¾ 2
C. –½ x2 – ½ x – ¾ * D. –½ x2 + ½ x – ¾
57. Find the the Laplace Laplace transform transform of of A. 3 + 2 x2 B. 5 sin sin 3 x – 17e–2 x C. 2 sin sin x + 3 cos x D. xe4 x E. e–2 x sin 5 x 58. Find the the inverse Laplace Laplace of the following following:: s+ 1
′′ + + y ′ − y = 0 3 y
s2 − 9 A.
s
+ 12 y ′ + 9 y = 0 4 y ′ +
2
( s− 2) + 9
51. Solve Solve the the equati equation on
B.
s+ 4 s2 + 4 s+ 8
y ′ + + y ′ − 6 y = 0
A.
C.
y = c1e−2 x + c2e−3 x B.
C.
y = c1e2 x + c2e−3 x
y = c1e2 x + c2 e3 x
*
y = c1e−2 x + c2 e3 x
59. Solve Solve the differentia differentiall equation equation
0 ) = 1 y ′ + y = sin x y ( 0
D.
− 6 y ′ +13 y = 0 y ′ −
60. 60. Evalu Evaluate ate (1 + 2i)4
53. Solve Solve the the equati equation on
y = e3 x ( c1 cos2 x − c2 sin2 x )
3
y = e3 x ( c1 cos2 x + c2 sin2 x ) B.
62. 62. Evalu Evaluate ate (2 + 3i)i *
y = e ( c1 cos3 x + c2 sin3 x ) 2 x
C.
4 + 3i
61. 61. Eval Evalua uate te
A.
°
°
0 ) = 1 , y ( 1) = 3 y ′′ + + 2 y ′ + y = 0 , y ( 0
2
52. Solve Solve the the equati equation on 41. How long long will it take take a 100-mg sample sample of 14C to decay to 90 mg? Ans: 871 years
D.
A.
50. Find the solution of the differential equation 39. A radioacti radioactive ve substance substance has a mass of 100 mg. mg. After 10 years it has decayed to a mass of 75 mg. What will the mass of the substance be after another 10 years? Ans: 56.25 mg
y = 3cos x − 2sin x
.
y = 2 + Ce− x
2
A.
C.
55. Solve Solve the boundary boundary value value problem problem
. At what time t does does the population reach 900? Ans: 55
35. Find the the orthogonal orthogonal trajectories trajectories of the the family of curves xy = k . Ans: x2 – y2 = C
y = 2cos x − 3sin x
63. 63. Evalu Evaluate ate ln ln (1 – i) 64 Simpl Simplify ify (4 + 3 i)2 + i
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Math Refresher C = 5i + 2k A. 0 B. 15
University of Santo Tomas C. 20 * D. 25
67. What is the exponenti exponential al form of the complex complex number 3 + 4i? A. 5ei0.6435 C. 5ei0.6345 i0.9273 B. 5e * D. 5ei0.9237
In-House Review for ECE
68. What is the rationalized rationalized value of the complex quotient (6 + 2.5 i)/(3 + 4 i) A. 1.1 B. 28
−¿ −¿
0.66i C. –0.32 + 0.66 i 16.5i *
D. 0. 0.32 – 0.66i
x2 z – 2 xy) a y 69. Given Given the vect vector or D = (2 xyz – y2) a x + ( x 2 + x y a z, determine ∇ D
70. Given Given the vector vector field F = xy2 z3 a x + x3 yz2 a y + x2 y3 z a z, find ∇
D
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