Iecep Math

MATH IECEP COMPILATION 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 at

What is the Laplace transform of e ? 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)

AxB = i – 10j + 4k Given a scalar function f (x, y , z), find the gradient of f.

∂f ∂f ∂f i+ j+ k ∂x ∂ y ∂z

a.

grad f =

b.

grad f =

∂f ∂f ∂f + + ∂x ∂y ∂z

c.

grad f =

∂ 2 f ∂2 f ∂2 f + + ∂ x2 ∂ y 2 ∂ z2

d.

grad f =

∂2 f ∂2 f ∂2 f i+ j+ k ∂ x 2 ∂ y 2 ∂ z2

Given a vector A = axi + ayj + azk, find the divergence of A. a.

¿f =

∂f ∂f ∂f i+ j+ k ∂x ∂ y ∂z

b.

¿f =

∂f ∂f ∂f + + ∂x ∂ y ∂z

c.

¿f =

∂2 f ∂ 2 f ∂2 f + + ∂ x2 ∂ y2 ∂ z2

d.

¿f =

∂ f ∂ f ∂ f i+ 2 j+ 2 k 2 ∂x ∂y ∂z

2

2

2

What is the Laplace transform of cosh (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2) d. s / (s2 - w2)

Given a scalar function f (x, y , z), find the Laplacian of f. a.

¿( grad f )=

∂f ∂f ∂f i+ j+ k ∂x ∂ y ∂z

What is the Laplace transform of sinh (wt)? a. 1 / (s2 + w2) b. 1 / (s2 - w2) c. s / (s2 + w2) d. s / (s2 - w2)

b.

¿( grad f )=

∂f ∂f ∂f + + ∂x ∂y ∂z

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°

c.

d.

∂2 f ∂2 f ∂2 f ¿( grad f )= 2 + 2 + 2 ∂ x ∂ y ∂z ¿(grad f )=

∂2 f ∂2 f ∂2 f i+ j+ k ∂ x 2 ∂ y 2 ∂ z2

Given a scalar function f (x, y , z), find the curl of the gradient of f a. 1 b.

∂2 ax ∂2 ay ∂2 az curl( grad f )= i+ j+ 2 k 2 2 ∂x ∂y ∂z

c. inf d. 0

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°

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

Given vectors A = 4i + k and B = -2i + j + 3k, find AxB a. 0 b. -8i + 3k c. -12 d. i – 10j + 4k

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)

AxB =

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

|

|

i j k 4 0 −1 −2 1 3

MATH IECEP COMPILATION 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

2 x =1−√ 2−x

a.1 b.-1 c.1 / 4 d. – 1 / 4 Find the y-intercept of the graph

y=x 2−2 a. 0 b. 1.7 c.-2 d. No answer because only lineaf function have y-intercept Given the equation

2

2

x + y =4

, 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 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

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

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.

80 √97

c.

−97 √ 80

b.

d.

−80 √ 97 97 √80

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

MATH IECEP COMPILATION y (t )=cos t b. c.

y (t )=sinht

d.

y (t )=cosh t

Taking the Laplace transform of the DE

s 2 Y ( s )−sy ( 0 ) − y ' ( 0 ) +Y ( s )=0 Y ( s )= 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

1 s +1 2

y (t )=sint Find the first derivative of uv.

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.

v uv−1 du+u v lnudv

b.

v uv−1 dv +uv lnvdu

c.

v uv−1 du+ vu lnvdv

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

d.

v uv−1 dv +uv lnudu lny=vlnu

d (lny )=d ( vlnu )

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

y' 1 =v du+lnudv y u

()

uv v y= du+u v lnudv u '

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

d ( uv ) =v uv−1 du+uv lnudv 3

If 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

a. b. c. d.

L−1 { f ( s ) }=x L−1 {f (s−a)}

the inverse Laplace transform of a function f(s). Find x a. e-at b. eat c. a-et d. aet

3 2

where L-1 is

1

Evaluate:

∫ x 2011 ( 1−x )2 0

a.

1 4078507092

b.

1 8144863716

Solve the initial value problem

y ( t ) + y ( t )=0 ; y ( 0 )=0, y ' ( 0 )=1 y (t )=sint

2

.

1 ( infinite geometric series ) 1−x

dx =−ln |1−x||32=−ln |−2|+ln |−1|=ln 2−1 =ln 0 .5 ∫ 1−x

''

a.

, determine

∫ f (x) dx

ln 0.5 ln 2 ln 3 ln 1.5

f ( x )=1+ x + x 2+ x 3 +…=

The logarithm of the reciprocal of a number is called _____. a. Inverse Logarithm b. Cologarithm c. Index d. Briggsian Logarithm

Given the equation:

f ( x )=1+ x + x 2+ x 3 +…

c.

MATH IECEP COMPILATION

1 2011

a. b. c. d.

d.

Simplify: a. b. c. d.

Find the mean deviation for the following set of data: {35,40, 45} a. 10/3 b. 5 c. 25 d. 5/3

1 −1 Ta n +Ta n−1 2 3

( )

−1

Oscillating series Geometric series Bilateral Series Di-valued Series

¿ X −40∨¿ a 1 M . D .= ∑ ¿ 3 i=1

Tan-1 1/7 Tan-1 1/6 -Tan-1 1/7 -Tan-1 1/6

1 M . D .= (|−5|+|0|+|5|) 3

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

6! =15 4!2!

M . D .=10/3

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

ave= 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.

45 91

b.

2 25

c.

4 15

d.

20 91

σ=

√

1 ( 54 +57+59+…+ 90 )=68 41 41

1 ∑ ( Xi−68)2 41 i=1

( 54−68 )2 + ( 57−68 )2+ …+(90−68)2 ¿ ¿ ¿ σ=√ ¿ σ =7.1 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

This is a case of hypergeometric probability distribution.

Meiko’s average speed is the Harmonic mean of 30 mph and 55 mph

5 C 1 ⋅10 C 2 45 = 15 C 3 91

w h ere nCr=

H=

n! r ! ( n−r ) !

1

1 1 + 30 55

=38.

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

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 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.

Q=

√

1.32+1.5 2+1.7 2+ 1.02+ 1.12 =1.34 5

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

For the probability, we have: 2

1 r π 2 area of success 1 P= = = 2 area of possible 4 πr

( )

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?

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 5

C3 p3q2 + 5C4 p4q + p5 = 10(1/6)3(5/6)2 + 5(1/6)4(5/6) + (1/6)5 = 23/648

Find the probability of throwing at least 2 aces in 10 throws of a die

MATH IECEP COMPILATION 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)

¿

C (13,2) C (52,2)

=

13 ! 11 ! 2 ! 78 1 = = 52 ! 1326 17 50 ! 2!

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

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)

¿

C ( 6,3 ) . C (7,2) C ( 6,4 ) .C (7,1) C ( 6,5 ) .C (7,0) + + C(13,5) C(13,5) C (13,5) = 140/429 + 35/429 + 2/429 = 177/429 = 59/143

P (2 defective )=

C (3,2) 3 1 = = C (15,2) 105 35

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

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

10.9 .8 .7 .6 1 5 1 5 ( ) ( ) =63/256 5.4 .3.2 .1 2 2

Odds = P(s)/(P(f) = (1/36) / (35/36) = 1/35

C(10,5) T F =

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

Find the median of the following set of data: {4,10,1,6} a. 4 b. 10 c. 7 d. 5.25

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

5 5

The median is the mean of the two middle values. {1, 4, 10, 61} Thus,

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

Md=

4 +10 =7 2

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) =

P( A∧B) 3 /36 1 = = P(B) 6 /36 2