What is the Laplace Transform of a unit step function? a. 1 b. s c. 1 / s d. u(t) What is the Laplace transform of t? a. 1 / s b.1 / s2 c 1. d. s What is the Laplace transform of eat? a. 1 / (s – a) b. 1 / (s + a) c. s + a d. s – a What is the Laplace transform of teat? a. 1 / (s – a)2 b. 1 / (s – a ) c. (s + a)2 d. s – a What is the Laplace transform of sin (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2) d. s / (s2 - w2) What is the Laplace transform of cos (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2)
d. s / (s2 - w2) What is the Laplace transform of cosh (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2) d. s / (s2 - w2) What is the Laplace transform of sinh (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2) d. s / (s2 - w2) Given vectors A = i + j + k and B = 2i – 3j + 5k, find A∙B. a. 2i -3j + 5k b. 2i + 3j + 5k c. 0 d. 4
Given vectors A = i + 2j and B = 3i – 2j + k, find the angle between them. a. 0° b. 36.575° c. 96,865° d. 127.352° Φ = cos -1 ( A∙B) / (|A| |B|) A∙ B = (1)(3) + (2)(-2) + (0)(1) = -1 |A| = sqrt (12+22) = sqrt(5) |B| = sqrt (32 + (-2)2 + 12) = sqrt (14) Φ = cos -1 ( -1) / (sqrt (5) x sqrt (14) ) Φ = 98.865°
Given vectors A = 4i + k and B = -2i + j + 3k, find AxB a. 0 b. -8i + 3k c. -12 d. i – 10j + 4k
c. d. Given a scalar function f (x, y , z), find the curl of the gradient of f a. 1 b.
AxB = |
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AxB = i – 10j + 4k Given a scalar function f (x, y , z), find the gradient of f. a. b. c. d. Given a vector A = axi + ayj + azk, find the divergence of A. a.
c. inf d. 0 It is an equation that contains one or several derivatives of an unknown function called y(x) and which we want to determine from the equation. a. homogeneous differential equation b. ordinary differential equation c. partial differential equation d. linear constant coefficient differential equation Solve the differential equation y’ = 1 + y2 a. y = tan -1 (x) + c b. y = tan (x) + c c. y = tan (x + c) d. y = tan (x)
b. c. d. Given a scalar function f (x, y , z), find the Laplacian of f. a. b.
Solve the initial value problem y ‘ = -y / x, where y(1) = 1 a. y = c / x b. y = x / c c. y = x d. y = 1 / x A ______ is a collection of objects, and these objects are called the elements. a. set
b.subset c.venn diagram d.union Solve the equation √ a.1 b.-1 c.1 / 4 d. – 1 / 4 Find the y-intercept of the graph
minute. How fast is the third side of the triangle growing when the angle is 60deg? a. 5π/6 in/m b. 5π/36 in/m c. 6π/25 in/m d. 6π/5 in/m
a. 0 b. 1.7 c.-2 d. No answer because only lineaf function have y-intercept Given the equation , find its symmetry. a. symmetric with respect to x-axis b. symmetric with respect to y-axis c. symmetric with respect to the origin d.all of the above The charge in coulombs that passes through a wire after t seconds is given by the function Q(t) = t3 − 2t2 + 5t + 2. Determine the average current during the first two seconds. a. 2 amperes b. 3 amperes c. 4 amperes d. 5 amperes
Two sides of a triangle are 5 and 10 inches, respectively. The angle between them is increasing at the rate of 5◦ per
Two cars begin a trip from the same point P. If car A travels north at the rate of 30 mi/h and car B travels west at the rate of 40 mi/h, how fast is the distance between them changing 2 hours later? a. 20 mi/h b. 30 mi/h c. 40 mi/h d. 50 mi/h
Postal regulations require that the sum of the length and girth of a rectangular package may not exceed 108 inches (the girth is the perimeter of an end of the box). What is the maximum volume of a package with square ends that meets this criteria? a. 11,646 in3 b. 11,466 in3 c. 11,464 in3 d. 11,664 in3 A baseball diamond is a square whose sides are 90 ft long. If a batter hits a ball and runs to first base at the rate of 20 ft/sec, how fast is his distance from second base changing when he has run 50 ft? a. b. √
c.
√
√
d.
√
Any differential equation of the form y= px + f(p) where f(p) contains neither x nor y explicitly is called a/n _______. a. Bernoulli’s Equation b. Clairaut’s Equation c. Homogenous Equation d. Laguerre Polynomials
The graphs of the equations of the forms r = asinnϴ and r = acosnϴ where n is a positive integer, greater than 1, are called _____. a. Lemniscates b. Rose Curves c. Cardioids d. Limacons The graph of an equation of the form r = b + asinϴ or r = b + acosϴ is called a ________. a. Lemniscates b. Rose Curves c. Cardioids d. Limacons A/n ______ is the set of all points P in a plane such that the sum of the distances of P from two fixed points F and G of the plane is constant. a. Ellipse b. Circle c. Conic d. Parabola
These variables are dimensionless combinations of the physical variable and parameters of the original. a. Canonical Variables b. Dependent Variables c. X and Y Variables d. Controlled Variables This states that every integral rational equation has at least one root. a. Fundamental Theorem of Arithmetic b. Fundamental Theorem of Counting c. Fundamental Theorem of Algebra d. Fundamental Theorem of Equations
The logarithm of the reciprocal of a number is called _____. a. Inverse Logarithm b. Cologarithm c. Index d. Briggsian Logarithm { } Given the equation: { } where L-1 is the inverse Laplace transform of a function f(s). Find x
a. b. c. d.
e-at eat a-et aet
If a. b. c. d.
, determine ∫
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ln 0.5 ln 2 ln 3 ln 1.5
Solve the initial value problem
a. b. c. d.
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Taking the Laplace transform of the DE Evaluate: ∫ a. b. v
Find the first derivative of u . a. b. c. d.
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c. d. Simplify: a. b. c. d.
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-1
Tan 1/7 Tan-1 1/6 -Tan-1 1/7 -Tan-1 1/6
How many different signals, each consisting of 6 flags hung in a vertical line, can be formed from 4 identical red flags and 2 identical blue flags? a. 15
b. 672 c. 720 d. 34560 This is a case of permutations of indistinguishable objects
c. 1/3 d. 1 For the circle given, draw a concentric circle with a radius half of the radius of the given circle. A point that lies on the inner circle is closer to the center of the original circle than to its circumference. For the probability, we have:
Three light bulbs are chosen at random from 15 bulbs from which 5 are defective. Find the probability that one light bulb drawn is defective. a. b. c. d. This is a case of hypergeometric probability distribution.
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Consider the series Sn =1 -1 +1 -1 +1 + -… If n is even, the sum is zero and if n is odd, the sum is 1. What do you call this kind of infinite series? a. Oscillating series b. Geometric series c. Bilateral Series d. Di-valued Series Find the mean deviation for the following set of data: {35,40, 45} a. 10/3 b. 5 c. 25 d. 5/3 ∑|
A point is selected at random inside a circle. Find the probability that the point is closer to the center than to its circumference. a. ¼ b. ½
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Compute the standard deviation for {54, 57, 59, 59, 60, 61, 61, 62, 62, 62, 63, 64, 65, 65, 66, 66, 66, 66, 67, 67, 68, 68, 68, 68, 68, 69, 69, 69, 70, 71, 71, 72, 72, 73, 75, 75, 77, 79, 81, 83, 90} a. 50.41 b. 7.1 c. 68 d. 8.6
Find the probability of obtaining an ace on both the first and second draws from a deck of cards when the first is not replaced before the second is drawn. a. 1/256 b. 1/17 c. 1/21 d. 1/221
P1P2 = (5/42)(3/51) = 1/221
√
∑ The probability of throwing at least 3 aces in 5 throws of a die. a. 8/243 b. 23/648 c. 125/3888 d. 126/3888
√
Meiko King travels 100 miles at the rate of 30 mph and then on a free way travels the next 100 miles at the rate of 55 mph. What is her average speed? a. 38.8 mph b. 42.5 mph c. 45.2 mph d. 48.8 mph Meiko’s average speed is the Harmonic mean of 30 mph and 55 mph
Find the quadratic mean of {1.3, 1.5, 1.7, 1.0, 1.1} a. 9.04 b. 1.1 c. 1.8 d. 1.34 √
p3q2 + 5C4 p4q + p5 = 10(1/6)3(5/6)2 + 5(1/6)4(5/6) + (1/6)5 = 23/648 5C3
Find the probability of throwing at least 2 aces in 10 throws of a die a. 0.484 b. 0.333 c. 0.515 d. 0.238 The probability of 0 or 1 aces is (5/6)10 + 10(5/6)9(1/6) = 9762625/20155392 The probability of throwing at least 2 aces is 1 - 9762625/20155392 = 10389767/20155392 = 0.5154832513 Two cards are drawn at random from a standard deck of 52 cards. What is the probability that both are hearts? a. 13/52 b. 1/17 c. 7/13 d. 7/26
P(two hearts)
=
A collection of 15 transistors contains 3 that are defective. If 2 transistors are selected at random, what is the probability that at least 1 of them is good? a. 1/35 b. 1/5 c. 34/35 d. 4/5
Thus, the probability of selecting at least one good transistor is 1 -1/35 = 34/35 What are the odds of getting 2 ones in a single throw of a pair of dice? a. 25 to 36 b. 35 to 36 c. 1 to 36 d. 1 to 35 There are 6x6 or 36 possible outcomes when throwing two dice P(s) = 1/36 P(f) = 1 – 1/36 = 35/36 Odds = P(s)/(P(f) = (1/36) / (35/36) = 1/35
Find the probability of getting a sum of 7 on the first of two dice and a sum of 4 on the second throw. a. 1/72 b. 1/6 c. 11/36 d. 6/36 Let A be a sum of 7 on the first throw. Let B be a sum of 4 on the second throw. P(A) = 6/36 P(B) = 3/36 P(A and B) = P(A).P(B) = (6/36)(3/36) = 1/72
A new phone is being installed at the Steiner residence. Find the probability that the final three digits in the telephone number will be even. a. 1/8 b. 1/4 c. 1/2 d. 3/8 P(any digit being even) = 5/10 or ½ P(final three being even) = (1/2)(1/2)(1/2) = 1/8 There are 5 red, 3 blue, and 7 black marbles in a bag. Three marbles are chosen without replacement. Find the probability of selecting a red one, then a blue one, and then a red one. a. 2/91 b. 1/5 c. 2/225 d. 1/26 P(red, blue, and red) = (5/15)(3/14)(4/13) = 2/91 Find the probability of a sum of 6 or a sum of 9 on a single throw of two dice. a. 1/4 b. 5/324 c. 5/9 d. 15/36 P(sum of 6) = 5/36 P(sum of 9) = 4/36 P(sum of 6 or sum of 9) = 5/36 + 4/36 = 9/36 = 1/4 What is the probability of drawing a king or a black card? a. 15/25 b. 7/13 c. 1/2 d. 6/13 King black black king 4/52 + 26/52 2/52
black or king = 28/52 or 7/14
A committee of 5 people is to be selected from a group of 6 men and 7 women. What is the probability that the committee will have at least 3 men? a. 59/143 b. 140/429 c. 84/145 d. 37/429
P(at least 3 men) = P(3 men) + P(4 men) + (5 men) = 140/429 + 35/429 + 2/429 = 177/429 = 59/143 Suppose that three dice are thrown at the same time. Find the probability that at least one 4 will show. a. 1/216 b. 91/216 c. 25/36 d. 1/12 P(at least one 4) = p43 + 3p42q4 + 3p4q42 + q43 = (1/6)3 + 3(1/6)2(5/6) + 3(1/6)(5/6)2 =91/216 Peggy guesses on all 10 questions on a true-false quiz. What is the probability that exactly half of the answers are correct? a. 1/2 b. 1/32 c. 1/8 d. 63/256 C(10,5) T5F5 = Find the median of the following set of data: {4,10,1,6} a. 4 b. 10 c. 7 d. 5.25 The median is the mean of the two middle values. {1, 4, 10, 61} Thus, A pair of dice is thrown. Find the probability that their sum is greater than 7 given that the numbers are match. a. 6/36 b. 3/36 c. 1/2 d. 1/11 P(B) = 6/36 P(A and B) = 3/36
P(A/B) =
