EcE Math Board Exam

Name: Year: Block:

_____ 5. What is the laplace transform of tan(x)? A. sec2(s) B. tan(s)sec(s)

C. ln[tan(s)+sec(s)] D. exp[tan(s)+sec(s)]

Score: _____ 6. What is a quantity representing the power to which a fixed number must be raised to produce a given number?

EcE Quizzer 2016-2017 MATH INSTRUCTION: Please do not write anything on this questionnaire except writing your answers on the left part of the question. Make sure that the answers except in identification will be capitalized. If the answer is not on the choices, answer it with letter E, otherwise if all the choices are the answer, answer it with letter F. Put all the solutions in the scratch paper properly. No solution, no points. No erasures.

A. Exponent B. Characteristic

_____ 7. Determine the middle term in the expansion of (a-2b)10. A. 8064a5b5 B. -8064a5b5

_____ 1. These equations are equalities that involve trigonometric functions and are true for every single value of the occurring variables where both sides of the equality are defined. A. Trigonometric Functions B. Pythagorean Theorem C. Identity Matrix D. Trigonometric Identities _____ 2. What do you call the method in integrating a complex equation wherein you will substitute tangent half angle? A. Integration by Parts B. Miscellaneous Substitution C. Algebraic Substitution D. Trigonometric Substitution _____ 3. In solving higher order linear differential equations, this method will determine if R(x) is linearly dependent or independent to the complementary solution of the differential equation. What is that method? A. Wronskian Determinant B. Fermat Last Theorem

C. Chios Method D. Gauss-Jordan Method

_____ 4. Kobe Bryant will be a three-pointer in the next game. In his last five games, his mean three-point is 60%. What is the probability that he will gain 16 threepoints out of 25 attempts? A. 16 % B. 15 %

C. 12% D. 19%

C. Logarithm D. Mantissa

C. -8032a5b5 D. 8032a5b5

_____ 8. These theorem states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between then – that is, a point where the first derivative is zero. A. Mean-value theorem B. Rolle’s theorem

C. Intermediate value D. Bifurcation theory

_____ 9. It is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. A. Mean-value theorem B. Rolle’s theorem

C. Intermediate value D. Bifurcation theory

_____ 10. Determine the integral of ∫(x2+2x+1)(x-1)/ (x+1)dx. A. ½ x2 + x + C B. ½ x2 - x + C

C. ⅓ x3 + x + C D. ⅓ x3 - x + C

_____ 11. Given a function, y=f(x) whose f’(x)=-f(x), determine the function. A. Logarithmic B. Exponential

C. Transcendental D. Sinusoidal

_____ 12. Find C so that the line y=4x+3 is tangent to the curve y=x2+C. A. 4

B. 5

C. 6

D. 7

_____ 13. At point of inflection, A. y = 0

B. y’ = 0

C. y” = 0

D. y’ = y”

_____ 14. A 5-foot boy is walking toward a 20-foot lamp post at the rate of 6 feet per second. How fast is the length of the girl’s shadow changing? A. 2 fps, dec B. 3 fps, inc

C. 2 fps, inc D. 3 fps, dec

_____ 15. A frustum of a pyramid has an upper base 100m by 10m and a lower base of 80m by 8m. If the altitude of the frustum is 5m, find its volume. A. 4567.67 cu. m. B. 3873.33 cu. m.

C. 4066.67 cu. m. D. 3345.98 cu. m.

_____ 16. Find the polar coordinates for the point whose rectangular coordinate of (-6, -8). A. (10, -233.13°) B. (10, 126.87°)

C. (10, 233.13°) D. (10, -126.87°)

_____ 17. If today is Sunday, what day of the week will be 2016 days after today? A. Sunday B. Tuesday

C. Monday D. Thursday

_____ 18. A die is rolled and a coin is tossed, find the probability that the die shows an odd number and the coin shows a head. A. 0.75 B. 0.125

C. 0.50 D. 0.25

_____ 19. A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white? A. 1/2 B. 1/3

C. 1/4 D. 1/5

_____ 20. You purchase a certain product. The manual states that the lifetime T of the product, defined as the amount of time (in years) the product works properly until it breaks down, satisfies P(T≥t)=exp(−t/5), for all t≥0. I purchase the product and use it for two years without any problems. What is the probability that it breaks down in the third year? A. 0.1713 B. 0.1712

C. 0.1813 D. 0.1812

_____ 21. Two skaters, Micah and Angelo, are at points A and B, respectively, on a flat frozen lake. The distance between A and B is 100 feet. Micah leaves A and skates at a speed of 2.44 m/s on a straight line that makes a 60° angle with AB. At the same time Micah leaves A, Angelo leaves B at a speed of 7 fps and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Micah skate before meeting Angelo? A. 36.58 m B. 42.67 m

C. 48.77 m D. 54.86 m

_____ 22. In a circle, parallel chords of length 2, 3, and 4 determine central angles of A, B, and A+B radians, respectively, where A+B < π. Determine the exact value of cos A. A. 15/47 B. 17/32

C. 32/59 D. 23/74

_____ 23. An inscribe angle in a semicircle in a right angle. A. Exterior Angle B. Thale’s Theorem

C. Morley’s Theorem D. Angle Bisector

_____ 24. An angle 2π = 360° corresponding to the central angle of an entire circle. A. straight angle B. perigon

C. myriagon D. polygon

_____ 25. A cat is tethered by a 6 mile rope to the outside corner of a shed measuring 4 mi by 5 mi in a sand field. What area of sand can the cat graze? A. 113π/4

B. 112π/5

C. 131π/4

D. 121π/5

_____ 26. Find the limit of (2-x)tan πx/2 as x approaches 1. A. e2π

B. e-2/π

C. 0

D. ∞

_____ 27. Find the rectangle of largest area that can be inscribed in an equilateral triangle of side 20. A. 50sqrt(3) B. 25sqrt(3)

C. 50sqrt(2) D. 25sqrt(2)

_____ 28. An open box is formed by cutting squares if equal size from the corners of a 24 by 15-inch piece of sheet metal and folding up the sides. Determine the maximum volume of the box. A. 400 B. 486

C. 386 D. 300

_____ 33. A balloon leaving the ground 18 m from the observer rises 3 m/s. How fast is the angle of elevation of the line of sight increasing after 8 seconds? A. 0.03 rad/s B. 0.04 rad/s

C. 0.06 rad/s D. 0.02 rad/s

_____ 34. Determine y’ implicitly: xy+y2=1 A. y/(x+2y) B. y/(x-2y)

C. –x/(x+2y) D. –y/(x+2y)

_____ 35. In trapezoid ABCD, AD is parallel to BC, Angle A and Angle D is equal to 45°. If AB=6, and the area of the trapezoid is 30, find BC, A. 3sqrt(2)

B. 2sqrt(2)

C. 3sqrt(3)

D. 2sqrt(3)

_____ 29. Consider the function defined by y = x 3 + kx2 + 3x – 4. For what value(s) of the constant k will the graph have exactly one horizontal tangent?

_____ 36. Two circles have radius of 4 cm, and 12 cm, respectively. If the distance between their centers is 30 cm, compute the length of the common internal tangent to the two circles in cm.

A. ±4 B. ±3

A. 27.32

C. ±2 D. ±1

_____ 30. Two posts, one 8 m and the other 12 m high are 15 m apart. If the posts are supported by a cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance to the lower post to the stake to use minimum amount of wire. A. 6 m B. 8 m

C. 9 m D. 4 m

_____ 31. The charge in coulombs that passes through a wire after t seconds is given by the function: Q(t) = t 3 – 2t2 + 5t + 2. Determine the average current during the first two seconds. A. 5 B. 7

C. 9 D. 11

_____ 32. Determine the second derivative of the function y=(x+1)(x-3)3. A. 12(x-3)(x-1) B. 5(x-4)(x-1)2

C. 3(x-3)2(x+1)+(x-3)3 D. 3x(x-3)2

B. 57.70

C. 25.37

D. 24.06

_____ 37. The length of a chord of a circle is 14 m. A point is selected on the chord so that its distance from one end of the chord is 8 m, while its distance from the center of the circle is 4 m. What is the length of the chord that is perpendicular to the radius that passes through this point? A. 15.37

B. 13.86

C. 10.31

D. 19.64

_____ 38. The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice? A. 32

B. 31

C. 33

D. 38

_____ 39. A man walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend walks up to the top of the escalator and counts 75 steps. If the man’s speed of walking (in steps per unit time) is three times his friend’s walking speed, how many steps are visible on the escalator at a given time? A. 150

B. 75

C. 120

D. 80

_____ 40. Find the value of x if |4x + 4|=12. A. 2, -4

B. -2,4

C. 3,-4

D. 1,-4

_____ 41. The ____ is a statement of general application which will aid us in factoring polynomials when the factors are not immediately evident. A. Cramer’s Rule B. Remainder Theorem

C. Binomial Theorem D. Factor Theorem

_____ 42. The portion of a sphere that is bound by the two intersections of two parallel planes with the sphere. A. zone B. spherical sector

C. lune D. spherical wedge

_____ 43. A solid had a circular base of radius 20 cm. Find the volume of the solid if every plane section perpendicular to a certain diameter is an equilateral triangle. A. 18,475.21 cm3 B. 20,356.36 cm3

C. 19,343.71 cm3 D. 17,253.86 cm3

_____ 44. What is the specific name given to the set of five regular polyhedrons? A. Platonic solids B. regular polyhedrons

C. regular solids D. Euclidean solid

_____ 45. A circle is circumscribing a triangle formed by the lines y=0, y=x, and 2x+3y=10. Find the equation of the circle. A. x2+y2-5x+y=0 B. x2+y2-2x+y=0

C. x2+y2+5x+y=0 D. x2+y2+2x+y=0

_____ 46. A line segment has its ends on the coordinate axes and forms with them a triangle of area equal to 36 sq. units. The segment passes through the point (5,2). Compute for the length of the line segment intercepted by the coordinate axes. A. 14.52

B. 13.42

C. 12.12

D. 11.02

_____ 47. The major axis of the elliptical path in which the earth moves around the sun is approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. A. 93,000,000 mi B. 94,335,100 mi

C. 91,450,000 mi D. 94,550,000 mi

_____ 48. The polar equation of a curve is equal to r2(4sin2θ + 9cos2θ)=36. Determine the eccentricity of the given curve. A. 0.845 B. 0.334

C. 0.745 D. 0.232

_____ 49. The axis of the hyperbola which passes through the center, the foci and vertices. A. transverse axis B. asymptotic axis

C. coordinate D. ordinate

_____ 50. The surface area of a solid of revolution is equal to the length of the generating arc times the circumference of the circle described by the centroid of the arc, provided that the axis of revolution does not cross the generating arc. A. First proposition of Pappus Theorem B. Second proposition of Pappus Theorem C. Third proposition of Pappus Theorem D. Fourth proposition of Pappus Theorem _____ 51. Water is flowing in a conical vessel 15 cm deep and having a radius of 3.75 cm across the top. If the rate at which the water rises is 2 cm/s, how fast is the water flowing into the conical vessel when the water is 4 cm deep? A. 6.28 cc/s B. 2.37 cc/s

C. 4.57 cc/s D. 5.73 cc/s

_____ 52. A hole of radius 2 is drilled through the axis of a sphere of radius 3. Compute the volume of the remaining solid. A. 46.832

B. 56.235

C. 26.315

D. 18.327

_____ 53. Determine the length of the curve y=x 3 from x=0 to x=1. A. 1.45

B. 1.55

C. 1.65

D. 1.75

_____ 61. Evaluate: (1+j)1+j A. 0.27 + j0.58 B. 0.58 – j0.27

C. 0.27 – j0.58 D. 0.58 + j0.27

_____ 54. How many distinct permutations do the word INDEPENDENCY has?

_____ 62. Given a scalar field g = 3x 2y – y3z2, determine the gradient of g at point (1, -2, -1).

A. 6653800

A. -12i -9j -16k B. 37

B. 6652900

C. 6642800

D. 6652800

C. 12i + 9j + 16k D. -37

_____ 55. Suppose there is an average of 2 suicides per year per 50,000 populations. In a city of 100,000, find the probability that in a given year there are 0 suicides.

_____ 63. Given a discrete function x[n] = (7/8) n u(n), determine the z-transform.

A. 0.0183

A. 1/[1-(7/8)z] B. z/[1-(7/8)z]

B. 0.0412

C. 0.0353

D. 0.0512

_____ 56. In a class of 28 students, the teacher selects four people at random to participate in a geography contest. What is the probability that this group of four students includes at least two of the top three geography students in the class? A. 37/819

B. 23/476

C. 34/911

C. 1/[1-(7/8)z-1] D. z-1/[1-(7/8)z-1]

_____ 64. Determine b2 in the Fourier series expansion of the function f(x) = x2 in the interval of –π < x < π and has a period of 2π. A. 4/9

B. -4/9

C. 0

D. -2/9

D. 10/259 _____ 65. Determine the value of tanh(j2).

_____ 57. In a certain college, 4% of the men and 1% of the women are taller than 6 feet. Furthermore, 60% of the students are women. Now, if a student is selected at random and is taller than 6 feet, what is the probability that the student is a woman? A. 1/3

B. 3/11

C. 1/4

D. 4/11

_____ 58. It states that, under mild conditions, the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution. A. Normal Distribution B. Gaussian Theorem

C. Normal Curve Symmetry D. Central Limit Theorem

_____ 59. The mean of a binomial distribution is given by: A. np(1-p)

B. np

C. n2

D. p2

_____ 60. What is the variance of a std. normal curve? A. 1

B. 0

C. 1/2

D. -1/2

A. 2.19e-j1.57

B. 1.12e-j1.57

C. 1.57e-j2.19

D. 1e-j1.57

_____ 66. The captain of the ship views the top of a lighthouse at an angle of 60° with the horizontal at an elevation of 6 meters above sea level. Five minutes later, the same captain of the ship views the top of the same lighthouse at an angle of 30° with the horizontal. Determine the speed of the ship in m/sec if the lighthouse is known to be 50 meters above sea level. A. 0.265

B. 0.155

C. 0.169

D. 0.210

_____ 67. In a tank are 100 liters of brine containing 50 kg total of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 L/min. Brine runs out of the tank at the rate of 2 L/min. The instantaneous concentration in the tank is kept uniform by stirring. How much salt is in the tank at the end of one hour? A. 15.45 kg

B. 19.53 kg

C. 12.62 kg

D. 20.62 kg

_____ 68. Solve: [y – sqrt(x2 + y2)]dx – xdy = 0 A. sqrt(x2+y2)+y=C B. sqrt(x2+y2+y)=C

C. sqrt(x+y)+y=C D. sqrt(x2-y)+y=C

_____ 76. The area (in sq. m.) of spherical triangle ABC whose parts are A=93°40’, B=64°12’, C=116°51’, and the radius of the sphere is 100 m is A. 16513

_____ 69. Which of the following equations is an exact Differential Equation? A. (x2 + 1)dx – xydy = 0 B. x dy + (3x – 2y) dx = 0 C. 2xy dx + (2 + x2) dy = 0 D. x2y dy – y dx = 0

_____ 71. Find the area (in sq. units) bounded by the parabolas x2 – 2y = 0 and x2 + 2y – 8 = 0. B. 4.7

C. 9.7

D. 10.7

_____ 72. The area in the second quadrant of the circle x2 + y2 = 36 is revolved around the line y + 10 = 0. What is the volume generated? A. 2218.33

B. 2228.83

C. 2233.43

D. 2208.53

_____ 73. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0. A. 5

B. 4

B. 3/4

C. 2/3

D. 1/2

B. degree

C. radian

D. grad

C. 8000

D. 80000

_____ 79. Express 45° in mils A. 80

A. 11.7

D. 16530

_____ 78. An angular unit equivalent to 1/400 of the circumference of a circle is called A. mil

A. (1/2)exp[sin(2x)] + C B. -(1/2)exp[sin(2x)] + C C. -exp[sin(2x)] + C D. exp[sin(2x)] + C

C. 16545

_____ 77. Determine the sum of the infinite series: S=1/3 + 1/9 + 1/27 + … + (1/3)n A. 4/5

_____ 70. What is the integral of cos(2x)exp[sin(2x)]dx?

B. 16531

C. 3

B. 800

_____ 80. Convert the polar equation to its Cartesian equivalent: r = 6cosθ. A. x2 – 6x + y2 = 0 B. 3x2 – 6x + 2y2 = 0

C. x2 – 3x + y2 = 0 D. 2x2 – 3x + 3y2 = 0

81. Solve for the inverse laplace transform of the following:

(

2

1 s 1 s −4 s+ 20 F ( s )= Co t −1 + ln 2 2 2 4 s −4 s+ 8

()

)

D. 2 Answer: ______________________________________

_____ 74. The distance (in units) from a point (1,3) to the line 4x + 3y + 12 = 0 is: A. 4

B. 5

C. 6

D. 7

82. Solve for the particular equation of the following using inverse differential operator:

y -3y'+2y=4t+6 {e} ^ {-t} ;y(0)=1, y'(0)=2

_____ 75. If Greenwich mean time (GMT) is 6 AM, what is the time at a place located 30° East longitude? A. 7 AM

B. 8 AM

C. 9 AM

D. 4 AM

Answer: ______________________________________ _____ 83. Find the area of the region bounded by the parabola x = y2 + 2 and the line y = x – 8. A. 125/6

B. 125/3

C. 121/5

D. 121/2

_____ 84. A plane 3000 ft. from the earth is flying east at the rate of 120 mph. It passes directly over a car also going east at 60 mph. How fast are they separating when the distance between them is 5000 ft? A. 74.4 fps

B. 84.4 fps

C. 70.4 fps

D. 63.7 fps

_____ 85. A ladder 20 feet long is placed against a wall. The foot of the ladder begins to slide away from the wall at the rate of 1 ft/sec. How fast is the top of the ladder sliding when the foot of the ladder is 12 feet from the wall? A. 3/4 fps

B. 3/5 fps

C. -3/5 fps

D. -3/4 fps

86. An equilateral hyperbola has an equation of x 2 – y2 = 9. Compute the location of the vertices. Answer: ______________________________________ 87. Triangle ABC has right angle at B, and contains a point P for which PA=10, PB=6, and