In-house Review (de, Advance Math And Discrete Math)

IN-HOUSE REVIEW DHVTSU Differential Equations 1. Obtain the differential equation of the family of straight lines passing through the origin. A. 𝑦 𝑑𝑥 + 𝑥 𝑑𝑦 = 0 C. 𝑦 2 𝑑𝑥 + 𝑥 2 𝑑𝑦 = 0 B. 𝑦 𝑑𝑥 − 𝑥 𝑑𝑦 = 0 * D. 𝑦 2 𝑑𝑥 − 𝑥 2 𝑑𝑦 = 0 2. What is the degree and order of the following differential equation: A. (2,2) B. (1,2)

𝑑𝑥 2

𝑑𝑦 2

+ ( ) − 2𝑦 = 4𝑥 ? 𝑑𝑥

C. (2,1) * D. (1,1)

3. The order and degree of the differential equation (1 + 3 A. (1,2/3) B. (3,1)

𝑑2𝑦

𝑑𝑦 2/3 𝑑𝑥

)

=4

𝑑3𝑦 𝑑𝑥 3

are

C. (3,3) * D. (1,2)

Consider the following differential equations in answering questions 4 and 5: 1 I. 𝑦 ′ + = 0 IV. 𝑦 ′′ + 3𝑦 ′ + cos 𝑦 = 3𝑥 𝑦

II.

𝑑𝑦 𝑑𝑥

+ 4𝑦 = sin 𝑥

III. (𝑦 ′′ )2 + 2𝑦 ′ + 2 = 0

V. 𝑥 ′′ + 2𝑥 ′ + 𝑥 = 0 VI.

𝑑2𝑦 𝑑𝑥 2

+ 2𝑦

𝑑𝑦 𝑑𝑥

+ 5𝑦 = 0

4. Which among the given differential equations is/are linear? A. II, IV and V C. V only B. I and II D. II and V * 5. Which among the given differential equations is/are homogenous? A. I, V and VI * C. I, III, V and VI B. II, IV and V D. II and IV 6. Solve the following differential equation: (1 + 𝑥 2 )

𝑑𝑦 + 𝑥𝑦 = 0 𝑑𝑥

A. 𝑥√1 + 𝑦 2 = 𝐶

C. 𝑦 2 √1 + 𝑥 = 𝐶

B. 𝑦√1 + 𝑥 2 = 𝐶 *

D. 𝑥 2 √1 + 𝑦 = 𝐶

7. Which among the following is an exact differential equation? A. 𝑦 2 𝑑𝑡 + 𝑡 2 𝑑𝑦 = 0 C. 3𝑥(𝑥𝑦 − 2)𝑑𝑥 + (𝑥 3 + 2𝑦)𝑑𝑦 = 0 * B. (𝑥𝑦 + 1)𝑑𝑥 + (𝑥𝑦 − 1)𝑑𝑦 = 0 D. sin 𝑡 cos 𝑥 𝑑𝑥 − sin 𝑥 cos 𝑡 𝑑𝑥 = 0 8. Find the solution of the exact differential equation from the previous problem. A. 𝑥 3 𝑦 − 3𝑥 2 + 𝑦 2 = 𝐶 * C. 𝑥 3 𝑦 + 3𝑥 2 − 𝑦 2 = 𝐶 3 2 2 B. 𝑥𝑦 + 3𝑦 − 𝑥 = 𝐶 D. 𝑥𝑦 3 − 3𝑦 2 + 𝑥 2 = 𝐶 9. Which among the following is a linear differential equation? A. 𝑦 ′ = 𝑥 sin 𝑦 + 𝑒 𝑥 C. 3𝑥 2 𝑦𝑑𝑥 + (𝑦 + 𝑥 3 )𝑑𝑦 = 0 2 )𝑑𝑥 B. 2(𝑦 − 4𝑥 + 𝑥𝑑𝑦 = 0 * D. 𝑥𝑦𝑑𝑥 + 𝑦 2 𝑑𝑦 = 0

10. Find the integrating factor of the linear differential equation from the previous problem. A. 𝑥 C. 𝑥 2 * B. ln 𝑥 D. 𝑒 𝑥 11. Which of the following is the solution of the linear differential from Question No. 9? A. 𝑥𝑦 2 = 2𝑥 4 + 𝐶 C. 𝑥𝑦 2 = 2𝑦 4 + 𝐶 2 4 B. 𝑥 𝑦 = 2𝑦 + 𝐶 D. 𝑥 2 𝑦 = 2𝑥 4 + 𝐶 * 12. Which among the following is a Bernoulli differential equation? 2 2 2 A. (𝑥𝑦 + 1)𝑑𝑥 − 𝑑𝑦 = 0 C. (2𝑥𝑦𝑒 −𝑥 + 𝑥𝑒 −𝑥 )𝑑𝑥 − 𝑒 −𝑥 𝑑𝑦 = 0 B. 𝑥𝑦𝑑𝑥 − 𝑑𝑦 = 0

D.

𝑑𝑦 𝑑𝑥

+

𝑦 𝑥

=

𝑦2 𝑥2

*

13. Which of the following gives the solution of the Bernoulli differential equation from the previous problem? 2𝑥 2𝑥 A. 𝑦 = C. 𝑦 = 2 * B. 𝑦 =

1+𝐶𝑥 2𝑥 2 1+𝐶𝑥

D. 𝑦 =

1+𝐶𝑥 2

1+𝐶𝑥 2

14. Some Non-exact differential equations can be made exact by multiplying them by a certain factor. Which of the following factors will make the equation (4𝑥𝑦 + 3𝑦 2 − 𝑥)𝑑𝑥 + 𝑥(𝑥 + 2𝑦)𝑑𝑦 = 0 exact? A. 𝑥 C. 𝑥 2 * B. ln 𝑥 D. 𝑒 𝑥 15. Which among the following is the solution of the given differential equation on question No. 14? A. 𝑥 3 (4𝑥𝑦 + 4𝑦 2 − 𝑥) = 𝐶 * C. 𝑥 3 (4𝑥 2 𝑦 + 4𝑦 2 + 𝑥) = 𝐶 3 (5𝑥𝑦 2 B. 𝑥 − 4𝑦 − 𝑥) = 𝐶 D. 4𝑥𝑦 + 4𝑦 2 − 𝑥 = 𝐶 16. Find the solution of 𝑦(2𝑥𝑦 + 1)𝑑𝑥 − 𝑥𝑑𝑦 = 0. A. 𝑥 2 𝑦 + 2𝑥 + 𝑦 = 𝐶 C. 𝑥 2 𝑦 + 𝑥 2 = 𝐶 2 B. 𝑥 𝑦 + 𝑥 = 𝐶𝑦 * D. 𝑥𝑦 2 + 𝑥 = 𝐶𝑦 17. Find the orthogonal trajectories of the family of curves 𝑦 = 𝑎𝑥 5 . A. 𝑥 2 + 5𝑦 2 = 𝐶 * C. 5𝑥 2 − 𝑦 2 = 𝐶 2 2 B. 𝑥 − 5𝑦 = 𝐶 D. 5𝑥 2 + 𝑦 2 = 𝐶 18. Radium decomposes at a rate proportional to the amount present. If the half-life is 1600 years, that is, if half of any given amount is decomposed in 1600 years, find the percentage remaining at the end of 200 years. A. 61. 7% C. 81.7% B. 71.7% D. 91.7% * 19. A turkey is removed from an oven when it has reached an internal temperature of 165 °F. After 20 minutes, the turkey probe reads 150 °F. The temperature in the room is 65 °F. Assuming Newton's law of cooling, after how many minutes will the turkey read 120 °F? A. 60 C. 45 B. 74 * D. 84 20. A tank contains 1000 liters of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 liters/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?

A. 12.28 kg * B. 14. 24 kg

C. 10.52 kg D. 12.54 kg

Find the general solution of each of the following homogenous differential equations. 21. 𝑦 ′′ − 4𝑦 ′ + 4𝑦 = 0 A. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑒 2𝑥 B. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 2𝑥 * 22.

𝑑3𝑦 𝑑𝑥 3

+5

𝑑2𝑦 𝑑𝑥 2

−2

𝑑𝑦 𝑑𝑥

C. 𝑦 = 𝐶1 𝑥 2 + 𝐶2 𝑒 2𝑥 D. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 𝑥

− 24𝑦 = 0

A. 𝑦 = 𝐶1 𝑒 −3𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 −4𝑥 * B. 𝑦 = 𝐶1 𝑒 3𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 −4𝑥

C. 𝑦 = 𝐶1 𝑒 −3𝑥 + 𝐶2 𝑥𝑒 2𝑥 + 𝐶3 𝑥 2 𝑒 −4𝑥 D. 𝑦 = 𝐶1 𝑒 3𝑥 + 𝐶2 𝑒 −2𝑥 + 𝐶3 𝑒 −4𝑥

23. 𝑦 ′′ − 4𝑦 ′ + 13𝑦 = 0 A. 𝑦 = 𝑒 2𝑥 (𝐶1 cos 3𝑥 ± 𝐶2 sin 3𝑥) B. 𝑦 = 𝑒 3𝑥 (𝐶1 cos 2𝑥 + 𝐶2 sin 2𝑥)

C. 𝑦 = 𝑒 2𝑥 (𝐶1 cos 3𝑥 − 𝐶2 sin 3𝑥) D. 𝑦 = 𝑒 2𝑥 (𝐶1 cos 3𝑥 + 𝐶2 sin 3𝑥) *

24. A differential equation has a characteristic equation whose roots are -5, -5, 8, 7 – 2i and 7 + 2i. What is the general solution of the differential equation? A. 𝑦 = 𝐶1 𝑒 −5𝑥 + 𝐶2 𝑒 −5𝑥 + 𝐶3 𝑒 8𝑥 + 𝑒 7𝑥 (𝐶4 cos 2𝑥 + 𝐶5 sin 2𝑥) B. 𝑦 = 𝐶1 𝑒 −5𝑥 + 𝐶2 𝑥𝑒 −5𝑥 + 𝐶3 𝑒 8𝑥 + 𝑒 2𝑥 (𝐶4 cos 7𝑥 + 𝐶5 sin 7𝑥) C. 𝑦 = 𝐶1 𝑒 −5𝑥 + 𝐶2 𝑥𝑒 −5𝑥 + 𝐶3 𝑒 8𝑥 + 𝑒 7𝑥 (𝐶4 cos 2𝑥 + 𝐶5 sin 2𝑥) * D. 𝑦 = 𝐶1 𝑒 −5𝑥 + 𝐶2 𝑥𝑒 −5𝑥 + 𝐶3 𝑥 2 𝑒 8𝑥 + 𝑒 7𝑥 (𝐶4 cos 2𝑥 + 𝐶5 sin 2𝑥) 25. Solve

𝑑2𝑦 𝑑𝑥 2

−5

𝑑𝑦 𝑑𝑥

+ 6𝑦 = 𝑒 5𝑥 . 5

A. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑒 3𝑥 + 𝑒 5𝑥 6 1

B. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑒 3𝑥 + 𝑒 5𝑥 * C. 𝑦 = 𝐶1 𝑒

2𝑥

+ 𝐶2 𝑒

3𝑥

6 2

+ 𝑒 5𝑥 5 1

D. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑒 3𝑥 + 𝑒 5𝑥 2

Given a differential equation 𝑑2𝑦 𝑑𝑦 +5 + 2𝑦 = 𝑓(𝑥) 2 𝑑𝑥 𝑑𝑥 solve the particular solution for each of the following cases: 26. if 𝑓(𝑥) = 3 A. 2 B. 3/2 *

C. 2/3 D. 5/2

27. if 𝑓(𝑥) = 7𝑒 2𝑥 A. 7𝑒 2𝑥

C.

B.

7 11

𝑒 2𝑥 *

28. if 𝑓(𝑥) = 3 cos 2𝑥

11 7

𝑒 2𝑥

D. 11𝑒 2𝑥

A. B.

3 25 3 25

(15 sin 2𝑥 + 3 cos 2𝑥)

C.

(15 sin 2𝑥 − 3 cos 2𝑥)

D.

29. if 𝑓(𝑥) = 12𝑥 2 A.6𝑥 2 + 30𝑥 + 69 B. 6𝑥 2 − 30𝑥 − 69

1 52 1 52

(15 sin 2𝑥 + 3 cos 2𝑥) (15 sin 2𝑥 − 3 cos 2𝑥) *

C. 6𝑥 2 − 30𝑥 + 69 * D. −6𝑥 2 − 30𝑥 + 69

30. Determine the particular solution of the following differential equation: 𝑑2𝑦 𝑑𝑦 –4 + 4𝑦 = 𝑒 2𝑥 𝑑𝑥 2 𝑑𝑥 A. B.

𝑒 2𝑥 3 𝑥𝑒 2𝑥 2

C. D.

𝑥 2 𝑒 2𝑥 2 𝑥 2 𝑒 2𝑥

*

3

31. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? A. 4.75 * C. 1.58 B. 3.17 D. 2.60 32. A puppy weighs 2.0 lbs at birth and 3.5 lbs two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 3 months old? A. 4.2 lbs C. 4.8 lbs B. 4.6 lbs* D. 5.6 lbs 33. How can the following differential equation best be described? A

d2 x 2

dt

+ B(t)

dx + C = D(t) dt

A. linear, homogenous and first order B. homogenous and first order C. linear, second order and nonhomogenous * D. linear, homogenous and second order 34. The appropriate form of a particular solution 𝑦𝑝 of the equation 𝑦" + 2𝑦′ + 𝑦 = 𝑒 −𝑡 is A. 𝐴𝑒 −𝑡 C. 𝐴𝑡 2 𝑒 −𝑡 * −𝑡 B. 𝐴𝑡𝑒 D. (𝐴+𝐵𝑡)𝑒 −𝑡 𝑑𝑦

35. Find the integrating factor of the differential equation cos 𝑥 + 𝑦 𝑠𝑖𝑛 𝑥 = 1. 𝑑𝑥 A. 𝑐𝑜𝑠 𝑥 C. 𝑡𝑎𝑛 𝑥 B. 𝑠𝑒𝑐 𝑥 * D. 𝑠𝑖𝑛 𝑥

d2y dy +4 +3y=e2x 2 dx dx e2x A. y=Ae-x +Be-3x + * 15 36. Solve:

C. y=Ae-x +Be-3x +

e-2x 12

B. y=Ae-x +Be-3x +

ex 12

D. y=Ae-x +Be-3x +

e-2x 16

37. What is the order and degree of the differential equation y’’’ + xy’’ + 2y(y’) 2 +xy = 0. A. first order, second degree B. second order, third degree C. third order, first degree * D. third order, second degree 38. Solve the differential equation y’ = 1 + y2 A. y = tan -1 (x) + c C. y = tan (x + c) * B. y = tan (x) + c D. y = tan (x) 39. Solve the initial value problem y ‘ = -y / x, where y(1) = 1 A. y = c / x C. y = x B. y = x / c D. y = 1 / x * 40. Under certain conditions, cane sugar in water is converted into dextrose at a rate proportional to the amount that is unconverted at any time. If, of 75 kg at time t = 0, 8kg are converted during the first 30 minutes, find the amount converted in 2 hours. A. 72.73 kg C. 27.23 kg * B. 23.27 kg D. 32.72 kg 41. A tank contains 400 liters of brine holding 100 kg of salt in solution. Water containing 125 g of salt per liter flows into the tank at the rate of 12 liters per minute, and the mixture, kept uniform by stirring, flows out at the same rate. Find the amount of salt at the end of 90 minutes. A. 53.36 kg * C. 53.63 kg B. 56.33 kg D. 65.33 kg 42. A thermometer reading 18oC is brought into a room where the temperature is 70oC; 1 minute later the thermometer reading is 31oC. Determine the thermometer reading 5 minutes after it is brought into the room. A. 62.33oC C. 56.55oC B. 58.99oC D. 57.66oC * 43. Given that ẍ + 3x = 0, and x(0) = 1, ẋ (0) = 0, what is x(1)? A. –0.99 C. 0.16 B. –0.16 * D. 0.99 44. Which is not true for the differential equation y’ + x–2y = x–2 A. it is linear in y B. it is separable C. it is homogeneous * D. it has the integrating factor e–1/x 45. Obtain the differential equation of the family of straight lines with slope and y-intercept equal. A. 𝑦𝑑𝑦 + (𝑥 + 1)𝑑𝑥 = 0 C. 𝑦𝑑𝑥 + (𝑥 + 1)𝑑𝑦 = 0 B. 𝑦𝑑𝑦 − (𝑥 + 1)𝑑𝑥 = 0 D. 𝑦𝑑𝑥 − (𝑥 + 1)𝑑𝑦 = 0 * 46. Obtain the general solution of the differential equation 𝑥𝑦𝑑𝑥 − (𝑥 + 2)𝑑𝑦 = 0. A. 𝑒 𝑥 = 𝑦(𝑥 + 2)2 C. 𝑒 𝑥 = 𝑐𝑦(𝑥 + 2) 𝑥 2 B. 𝑒 = 𝑐𝑦(𝑥 + 2) *y D. 𝑒 𝑥 = 2𝑐𝑦(𝑥 + 2)

47. The differential equation

𝑑𝑦 𝑑𝑥

=

3𝑥 2 + 𝑦 2

can be made exact by using the integrating factor

2𝑥𝑦

2

C. 𝑦 2 D. 1/𝑦 2

A. 𝑥 B. 1/𝑥 2 * 48. Find the particular solution of

𝑑2𝑦 𝑑𝑦 –3 + 2𝑦 = 10𝑒 3𝑥 𝑑𝑥 2 𝑑𝑥 A. 7𝑒 3𝑥 B. 5𝑒 3𝑥 *

C. 2𝑒 3𝑥 D. 11𝑒 3𝑥

49. Determine the particular solution of (𝐷 2 + 9) 𝑦 = 𝑐𝑜𝑠 4𝑥. 1

1

A. − cos 4𝑥 * 1

C. − sin 4𝑥

7

7

1

B. cos 4𝑥

D. 7 sin 4𝑥

7

50. Solve (𝐷3 – 𝐷 2 – 4𝐷 + 4) 𝑦 = 0

A. 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2𝑒 3𝑥 + 𝐶3𝑒 −2𝑥 B. 𝑦 = 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 −3𝑥

C. 𝑦 = 𝐶1𝑒 𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 −2𝑥 * D. 𝑦 = 𝐶1𝑒 3𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 −2𝑥

Advance Mathematics/Discrete Mathematics 1. The modulus of the complex number (

3 + 4i 1 –2i

) is

C. 1/√5 D. 1/5

A. 5 B. √5 *

2. Find the argument of (−2 + 2i)/3i. A. π C. π/3 B. π/2 D. π/4 * 3. If 𝑧 = 3 − 𝑖, find the value of cos 𝑧. A. −1.528 + 0.166𝑖 * B. 1.422 + 0.158𝑖

C. 0.923 + 1.648𝑖 D. 1.723 − 0.342𝑖

4. The conjugate of a complex number is A. B.

1

C.

𝑖−1 −1

D.

𝑖−1

5. Find the value of (2 + 3𝑖)4+3𝑖 A. 0.957 − 1.342𝑖 B. 0.143 + 0.562𝑖 2 3 6. If A = [5 –3 9 2 A. 21 B. –96

1 𝑖−1

4 8 ] , then trace of A is: 16 C. 10 D. 15 *

. Then that complex number is 1

𝑖+1 −1 𝑖+1

*

C. 0.544 − 7.435𝑖 D. 0.667 + 8.835𝑖 *

7. Which of the following matrices is a singular matrix? 2 –3 –3 21 A. ( ) C. ( )* 4 –28 21 4 B. (

5 –1 ) 6 8

D. (

5 –12

4 ) –8

1 2  3  5 then its adjoint is     5  2   1  2 A.  C.    * 5  3 1  3  5  2   5  2 B.  D.     3 1   3  1 8. If A =

9. Identify the Hermitian matrix.

æ3

æ3

7-2iö ÷ø

7+2iö

C. ç è 7-2i -2 ÷ø

A. ç è 7i -2

æ2

3e-2i ö ÷ è 3e2i 1 ø

B. ç

10. For the matrix [ A. 3 and –3

D. Both B & C* 4 1 ] the eigenvalues are: 1 4 C. 3 and 5 *

B. –3 and –5

D. 5 and 0

11. What is the Laplace transform of teat? A. 1 / (s – a)2 * B. 1 / (s – a )

C. (s + a)2 D. s – a

12. If the unilateral Laplace transform of f(t) is A. B. -

s (s2 +s+1) 2s+1

2

C.

(s2 +s+1)

2

D.

s

A. B.

√𝑠 √𝜋 √𝑠

2

(s2 +s+1)

C. *

. What is the unilateral Laplace transform of t f(t)?

2

(s2 +s+1) 2s+1

*

13. Find the Laplace transform of the function 𝜋

1 s2 +s+1

D.

𝜋

1 √𝑡

.

𝑠 √𝜋 𝑠2

14. Find the Laplace transform of ei5t where i = √–1. s – 5i s + 5i A. 2 C. 2 B.

s – 25 s + 5i s2 + 25

*

D.

s – 25 s – 5i

s2 + 25

15. If 𝑓"(𝑥) − 𝑓′(𝑥) − 2𝑓(𝑥) = 0,𝑓′(0) = −2, and 𝑓(0) = 2, then 𝑓(1) =? A. 1 C. 𝑒 2 B. 0 D. 2𝑒 −1 * 16. What is the approximate value of cos 2 obtained by using sixth degree Taylor Polynomial about 𝑥 = 0 for cos 𝑥? A. - 0.4222 * C. - 0.4161 B. 0.4325 D. 0.4153 17. The scalar or dot product of two vector quantities is defined as the product of their magnitudes ⃑⃑ and 𝑩 ⃑⃑ when multiplied by the cosine of the angle between them. Find the scalar product of 𝑨 ⃑𝑨 ⃑ = i + 2j - k and ⃑𝑩 ⃑ = 2i + 3j + k. A. 5 C. 7 * B. 6 D. 8 18. Given: A=2i+aj+k & B=4i-2j-2k. Compute the value of a so that A and B are perpendicular. A. 1 C. 2 B. 3* D. 4 19. Determine the divergence of the vector function F(x,y,z)=xzi+e xyj+7x3yk. A. z+ex* C. z+yex+21x2y B. x+y D. x+yex 20. Determine the curl of the vector function F(x,y,z)=3x 2i+7exyj. A. 7exy C. 7exyi x B. 7e yj D. 7exyk* 21. Determine the Laplacian of the scalar function (1/3)x 3-9y+5 at the point (3,2,7). A. 0 C. 1 B. 6* D. 18 22. If A is a subset of universal set U, then which of the following is incorrect? A. 𝐴 ∪ ∅ = 𝐴 C. 𝐴 ∩ 𝑈 = 𝐴 B. 𝐴 ∩ ∅ = ∅ D. 𝐴 ∪ 𝑈 = 𝐴* 23. The set A consists of elements {1, 3, 6}, and the set B consists of elements {1, 2, 6, 7}. Both sets come from the universe of {1, 2, 3, 4, 5, 6, 7, 8}. What is the intersection, 𝐴̅ ∩ 𝐵? A. {2, 7}* C. {2, 4, 5, 7,8} B. {2, 3, 7} D. {4, 5, 8} 24. Of the 80 students in class, 25 are studying German, 15 French and 13 Spanish. 3 are studying German and French; 4 are studying French and Spanish; 2 are studying German and Spanish; and none is studying all 3 languages at the same time. How many students are not studying any of the three languages? A. 18 C. 62 B. 53 D. 36* 25. What is the cardinality of the set of odd positive integers less than 10? A. 10 C. 5 * B. 3 * D. 20 26. Which of the following is the Laplace transform of √t?

A. B.

√π

C.

√s √π

D.

2√s

√π 2s3/2 √π

*

2s

27. The curl of vector V(x,y,z) = 2x2 i +3z2 j + y3 k at x = y = z = 1 is: A. – 3i * C. 3i – 4j B. 3i D. 3i – 6k 28. Find the orthogonal trajectories of the family of hyperbolas x2 – y2 = ay. A.𝑥 2 + 3𝑥𝑦 = 𝐶 C. 𝑥 3 + 2𝑥𝑦 2 = 𝐶 3 2 B. 𝑥 + 3𝑥𝑦 = 𝐶 * D. 𝑥 3 + 2𝑥 2 𝑦 = 𝐶 29. The inverse of the matrix [ −5 A. [ 2 −1 B. [ −2

−3 ] −1 −3 ]* −5

5 −2

−3 ] is: 1 −1 3 C. [ ] 2 −5 1 3 D. [ ] 2 5

30. (3 + 4i)4i A. 0.2124 B. 0.1224 *

C. 0.1422 D. 0.1442

31. Determine the determinant of the matrix 1 [ 2 −1

3 6 0

0 4] 2 C. –12 * D. –24

A. 12 B. 24 32. Evaluate ln(6 – 3i)

C. 1.9e−j0.464 D. 1.96e−j0.239 *

A. 3.49ej1.032 B. 1.79ej3 8 33. The eigenvalues of 𝐴 = [−6 2 A. 0, 3, -15 B. 0, -3, -15

−6 7 −4

2 −4] are 3 C. 0, 3, 15 * D. 0, -3, 15

34. If A = x2 z i – 2y3 z2 j + xy2 z k, find the divergence of A at the point (1, –1, 1) A. –2 C. 1 B. 0 D. –3 * 35. The inverse Laplace transform of 1/(𝑠 2 + 𝑠) is A. 1 + 𝑒 𝑡 C. 1 − 𝑒 −𝑡 * 𝑡 B. 1 − 𝑒 D. 1 + 𝑒 −𝑡 36. Given vectors A = i + j + k and B = 2i – 3j + 5k, find A∙B. A. 2i -3j + 5k C. 0 B. 2i + 3j + 5k D. 4 *

37. Given vectors A = i + 2j and B = 3i – 2j + k, find the angle between them. A. 0° C. 96.865° * B. 36.575° D. 127.352° 38. Adjoint of a matrix is the transpose of matrix of A. Cofactors * B. Origin values 0 39. If 𝐴 = [−1 2 A. 0 B. -2 *

C. Origin vectors D. Unit element 1 0 −2

−2 3 ] is a singular matrix, then 𝜆 is 𝜆 C. 2 D. -1 1

40. The inverse Laplace transform of the function 𝐹(𝑠) = A. 𝑓(𝑡) = sin 𝑡 B. 𝑓(𝑡) = 𝑒 −𝑡 sin 𝑡

𝑠(𝑠+1)

is given by

−𝑡

C. 𝑓(𝑡) = 𝑒 D. 𝑓(𝑡) = 1 − 𝑒 −𝑡 *

41. The product of two complex numbers 1 + I and 2 – 5i is A. 7 – 3i * C. -3 – 4i B. 3 – 4i D. 7 + 3i 42. The Laplace transform of a function 𝑓(𝑡) is −𝑡

A. 𝑡 − 1 + 𝑒 * B. 𝑡 + 1 + 𝑒 −𝑡

1

. The function 𝑓(𝑡) is

𝑠 2 (𝑠+1) −𝑡

C. −1 + 𝑒 D. 2𝑡 + 𝑒 𝑡

43. A series expansion for the function sin 𝜃 is A. 1 − B. 𝜃 −

𝜃2 2! 𝜃3 3!

+ +

𝜃4 4! 𝜃5 5!

−⋯ −⋯ *

C. 1 + 𝜃 + D. 𝜃 +

𝜃3 3!

𝜃2

+

+

2! 𝜃5 5!

𝜃3 3!

+⋯

+⋯

44. The divergence of the vector field 3𝑥𝑧𝒊 + 2𝑥𝑦𝒋 − 𝑦𝑧 2 𝒌 at a point (1, 1, 1) is equal to A. 7 C. 3 * B. 4 D. 0 45. Given vectors A = 4i – k and B = -2i + j + 3k, find AxB. A. 0 C. –12 B. –8i + 3k D. i – 10j + 4k * 46. In the Taylor expansion of 𝑒 𝑥 about 𝑥 = 2, the coefficient of (𝑥 − 2)4 is A. 1/4! C. e2/4! * B. 24/4! D. e4/4! 47. The divergence of the vector field (𝑥 − 𝑦)𝒊 + (𝑦 − 𝑥)𝒋 + (𝑥 + 𝑦 + 𝑧)𝒌 is A. 0 C. 2 B. 1 D. 3 * 48. Determine the product of the conjugates of 4 + 3i and 2 + 5i A. −7 – 26i * C. 7 + 26i

B. −7 + 26i

D. 7 – 26i

̅ = AT . 49. An n x n complex matrix A is _____ if and only if A A. Unitary C. Singular B. Hermitian * D. Skew–Hermitian 50. What is the modulus of 9 - 3i? A. 5√2 B. 6√2

Prepared by: Angelo T. Lopez, ECE

C. 2√3 D. 3√10 *