Math

Excel Review Center Find the convolution  g  f  t  of f  t   tu  t  and g  t   sin t  u  t 

Answer:  g  f  t   t  sin t Find the convolution of the Laplace transform of f  t   tu  t  and g  t   sin t  u  t  ? Answer: 1/s2 – 1/(s2 + 1) Find the convolution of e2t u  t  and e  t u  t  . Answer: e t  e2t Find the convolution of e t u  t  and tu  t  .

ECE Coaching Course that there will be a seat available for every person who shows up for the flight? Answer: 0 Workers in a factory incur accidents at the rate of two accidents per week. Calculate the probability that there will be at most two accidents, (a) during 1 week, (b) during 2 weeks, (c) in each of 2 weeks. Answer: (a) 0.677, (b) 0.238, (c) 0.458 How many children should a family have so that with probability 0.95 it has at least a boy and at least a girl? Answer: 6

Answer:  t  1  e t

Suppose that the duration in minutes of long distance telephone conversation follows an

Determine the poles and zeros of the system, whose transfer function is given by 30  s  6  . H s  s  s 2  4s  13

1 x exponential density function f  x   e 5 for x > 5 0. Find the probability that the duration of a conversation; a. Will exceed 5 minutes; b. Will be between 5 and 6 minutes; c. Will be less than 3 minutes; d. Will be less than 6 minutes given that it was greater than 3 minutes. 92 Answer: a. 0.368, b. 0.248, c. 0.451, d. 0.451

Answer: Zero: s = 6, Poles: s = 0, –2 + 3i, –2–3i A die is thrown as long as necessary for an ace or a 6 to turn up. Given that no ace turned up at the first two throws, what is the probability that at least three throws will be necessary? Answer: 16/25 Two athletic teams A and B play a series of independent games until one of them wins 4 games. The probability of each team winning in each game equals to 1/2. Find the probability that the series will end, (a) in at most 6 games, (b) in 6 games given that team A won the first two games. Answer: (a) 11/16, (b) 1/4 At the college entrance examination each candidate is admitted or rejected according to whether he has passed or failed the test. Of the candidates who are really capable, 80% pass the test; and of the incapable, 25% pass the test. Given that 40% of the candidates are really capable, find the proportion of capable college students. Answer: 68% Three players P1, P2 and P3 throw a die in that order and, by the rules of the game, the first one to obtain an ace will be the winner. Find their probabilities of winning. Answer: 36/91, 30/91, 25/91 The Pap test makes a correct diagnosis with probability of 95%. Given that the test is positive for a lady, what is the probability that she really has the disease? Assume that one in every 2,000 women, on average, has the disease. Answer: 19/2018 What is the conditional probability that a hand at poker consists of spades, given that it consists of black cards? Answer: 9/460 A machine normally makes items of which 4% are defective. Every hour the producer draws a sample of size 10 for inspection. If the sample contains no defective items he does not stop the machine. What is the probability that the machine will not be stopped when it has started producing items of which 10% are defective? Answer: 0.349 One per thousand of a population is subject to certain kinds of accident each year. Given that an insurance company has insured 5,000 persons from the population, find the probability that at most 2 persons will incur this accident. Answer: 0.104 A certain airline company, having observed that 5% of the persons making reservations on a flight do not show up for the flight, sells 100 seats on a plane that has 95 seats. What is the probability

In a lottery that sells 3,000 tickets the first lot wins $1,000, the second $500 and five other lots that come next wins $100 each. What is the expected gain of a man who pays 1 dollar to buy a ticket? Answer: –1/3 A die is thrown until the result “ace or even number” appears three times. Find the expected number of throws (a) in one performance of throws, (b) in ten repetitions. Answer: (a) 9/2, (b) 45 The height of men is normally distributed with mean   167 cm and standard deviation   3 cm. What is the percentage of the population of men that have height (a) greater than 167 cm, (b) greater than 170 cm, (c) between 161 cm and 173 cm? In a random sample of four men what is the probability that (d) all will have height greater than 170 cm, (e) two will have height smaller than the mean (and two bigger than the mean)? Answer: (a) 50%, (b) 16%, (c) 95.45%, (d) 0.07%, (e) 37.5% A machine produces bolts the length of which (in centimeters) obeys a normal probability law with mean 5 and standard deviation 0.2. A bolt is called defective if its length falls outside the interval (4.8, 5.2). a. What is the proportion of defective bolts that this machine produces? b. What is the probability that among ten bolts none will be defective? Answer: a. 0.32, b. 0.0211 Wronskian of f  x   x 2 sin x and g  x   x 2 cos x Answer: –x4

Math Take Home 1 Find the value of c such that the circles x2 + y2 + 2x + 2y + 1 = 0 and x2 + y2 + 2x + 2y + c = 0 touch each other. Answer: 1 The integrating factor of the differential equation x 1  y 2  dy  y 1  x 2  dx  0 . Answer: 1/xy In solving any problem, odds against A are 4 to 3 and odds in favor of B in solving the same problem are 7 to 5. The probability that the problem will be solved is Answer: 16/21 Determine the area enclosed by the curve x2 – 10x + 4y + y2 = 196. Answer: 225 Calculate the area bounded by x = –y2 + 9 and 1 x  y 2  6y  9 . 2 Answer: 128 The three vectors (1, 1, –1, 1), (1, –1, 2, –1) and (3, 1, 0, 1) are Answer: Linearly dependent Find the second order Taylor expansion of   f  x, y   sin  x 2  1 y  about the point  0,  .  2 Answer: f(x,y) = 1 – (y –  /2)2 Find the eigenvectors of the matrix  8 6 2  A   6 7 4  using   15  2 4 3  2 Answer:  2  1 

A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 11 steps, he is just one step away from the starting point. Answer: 0.3679 9 3 3 0 Find the rank of the matrix  1 1  0 6 Answer: 4

1 0 1 6  1 1  1 9

sin 3 x dx  x

Evaluate the integral of 



Answer: 3  /4 Given the ellipse 9x2 + 4y2 – 54x – 56y + 241 = 0. Find its vertices. Answer: (3, 10) and (3, 4) sin mx  sin nx , where m  n is equal to x Answer: m – n lim x 0

The angle between 2i + j – 3k and 3i – 2j – k. Answer: 60o Find the equation of the straight line which passes through the intersection of the straight line 2x – 3y +4 = 0 and 3x + 4y + 5 = 0 and is perpendicular to the straight line 6x – 7y + 8 = 0. Answer: 119x + 102y + 205 = 0

 3x 2  2x  : whenx  0  If f (x)    , what must be x   k : whenx  0   the value of k in order for f(x) to be a continuous function? Answer: 2

Give it your best shot!

Given A = {1, {4}, {2}, 3, 4, 5}, B = {{{1, 4, 5, 3, 1}}}, C = {1, {3}, 2, 1}, D = {1, 1, 3}, E = {1, 4, {5}, {3}}, F = {1, 8, {1, 2, 3, 4}}. Calculate the set (a) B  F , (b) C   D  F  Answer: (a)  , (b) 1, 2, 3 Find the equation of tangent to 16x2 + 9y2 = 144 at (x1, y1) where x1 = 2 and y1 > 0. 2x 5 Answer:  y 1 9 12 Find the equation of the circle whose center lies on the line x – 4y = 1 and which passes through the points (3,7) and (5,5). Answer: x2 + y2 + 6x + 2y – 90 = 0

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ECE Coaching Course

Find the term independent of x in the expansion 12

1  of  x   . x  Answer: 924 If 5 times the 5th term of an A.P. is equal to the 10 times the 10th term, find the 15th term of the A.P. Answer: 0 Find an estimate of the standard deviation. Income (1000’s Frequency Pounds) 9 10  i  15 16 15  i  20 22 20  i  25 8 25  i  30 5 30  i  35 Answer: 5617 Find the square root of 12 – 6i. Answer: –3.856 + 0.842i Find a unit vector normal to the plane of vectors A = 3i – 2j + 4k and B = i + j – 2k. 2j  k Answer:  5 Find the Fourier series of the function defined by  x  , 0  x   f x    x  ,    x  0 Answer:    2  2 n n    1  1 cos nx  1   1  sin nx  2 n 1  n 2   n 

Find the arc length of f  x   x 3/2  2 on [1, 4]. Answer: 7.63 In drawing two balls, from urn containing 10 balls of each of the colors, red, white and blue, find the probability of getting two different colors. Answer: 20/29 1 The value of the determinant 3 5

3 1 4

5 4 , 1

where  is an imaginary cube root of unity is Answer: 3 Find the general solution u xx  3 if u  x, y  is a function of x and y. Answer: u  1.5x 2  xf  y   g  y  Suppose the diameter of a certain car component follows the normal distribution with X  N 10,3 . Find the proportion of these components that have diameter larger than 13.4 mm. Or, if we randomly select one of these components, find the probability that its diameter will be larger than 13.4 mm. Answer: 0.1292 1

1 dx using Simpson’s 2 0 1 x

For what real values of  is y  cos t a solution

The first Newton approximation x1 for a zero f(x) = x3 – 2x with initial approximation x0 = 2. Answer: 8/5

If the mean and variance of binomial variate are 12 and 4, then the probabilities of the distribution are given by the terms in the expansion of Answer: (1/3 + 2/3)18

The remainder of 534 when divided by 17 is Answer: 8 Find the remainder on dividing 320 by 7 Answer: 2 Refer to the data set below, the number of patient visits per week at a chiropractor’s office over a ten-week period. Number of patients per week (75 86 87 90 94 102 105 109 110 120) Calculate the (a) first quartile, (b) third quartile, (c) interquartile range (IQR) of the data. Answer: (a) 87, (b) 109, (c) 22 What is the area in sq. m of the zone of a spherical segment having a volume of 1470.265 cu. m if the diameter of the sphere is 30 m? Answer: 565.5 m2 For which values of parameter a, the vectors (1, a, 2) and (a, 4, 4) are (a) parallel, (b) orthogonal? Answer: (a) 2, (b) –8/5 8 2 5  What is the nullity of 16 6 29  ?  4 0 7  Answer: 1

Define sets A = {1, …., 10}, B = {3, 7, 11, 12}, C = {0, 1, …, 20}. Which of the following are propositions? I.  A  B   C 3

II. 8  22  /102 III.  B  C   9 IV. 7  A Answer: I and IV only Today is Monday, 1 July 2002. What day of the week will be 29833 days from now? Answer: Friday

If coversine θ is 0.134, find the value of θ. Answer: 60o Find the area of a regular pentagon whose side is 25 m and apothem is 17.2 m. Answer: 1075 m2 A trough of water is 8 meters deep and its ends are in the shape of isosceles triangles whose width is 5 meters and height is 2 meters. If water is being pumped in at a constant rate of 6 m3/sec. At what rate is the height of the water changing when the water has a height of 120 cm? Answer: 0.25 m/s For the plane curve y = 4x3/2, find the curvature function. 3 Answer: 3/2 x 1  36x  Sketch the solid obtained by rotating the region  3 bounded by y = 0 and y = cos(x) for  x  2 2 about the y-axis and find its volume. Answer: 4 2 Let A be a 3 x 3 matrix with A  5. Find (a) A T , (b) A  I , (c) 2A Answer: (a) 5, (b) not enough info, (c) 40 The functions x, x2, x3 defined on an interval I, are always Answer: Linearly independent The complementary function for the solution of the differential equation 2x 2 y '' 3xy ' 3y  x 3 is obtained as Answer: Ax  Bx



3 2

Let V1 = (1, -1, 0), V2 = (0, 1, -1), V3 = (0, 0, 1) be elements of R3. The set of vectors {V1, V2, V3} is Answer: Linearly independent

The gross domestic product (GDP) of a certain country is N(t) = t2 + 3t + 80 in billions of dollars when t is measured in years. This growth was valid from 1980 to 1990. Find the percentage rate of change in 1986. Answer: 11% per year

If X has a Poisson distribution such that P(X = 2) = 9P(X = 4) + 90P(X = 6) then the variance of the distribution is Answer: 1

A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of AB is 6 cm, and that of BC is 2 cm. The angle ABC is a right angle. Find the square of the distance (in centimeters) from B to the center of the circle.

rule with a partition having four intervals. Answer: 0.785392

A certain auditorium has 30 rows of seats. Row 1 has 11 seats, while Row 2 has 12 seats, Row 3 has 13 seats, and so on to the back of the auditorium where Row 30 has 40 seats. A door prize is to be given away by randomly selecting a row (with equal probability of selecting any of the 30 rows) and then randomly selecting a seat within that row (with each seat in the row equally

to the equation y '' 9y  0 ? Answer: 3

Given  1.5   0.8862 , (a)   3.5  , (b)   0.5  Answer: (a) 3.32325, (b) –3.5448

Estimate the integral 

The average value of the function f(x) = sin(x) – x on the interval  0,  . Answer: –0.93418

Math Take Home 1

likely to be selected). Find the probability that Seat 15 was selected given that Row 20 was selected. Answer: 1/30

Answer: 26 Find the volume obtained if the region bounded by y = x2 and y = 2x is rotated about the x-axis? Answer: 13.4 Find the inflection points of f(x) = (x – 2)ex. Answer: x = 0

Give it your best shot!

The value of integral



xy  x  y  dxdy over

the region bounded by the line y = x and the curve y = x2 is Answer: 3/56 The family of orthogonal trajectories to the family y = (x – k)2, where k is an arbitrary constant, is 3 3 Answer: y 2   c  x  4 Find the general solution of the differential equation x 3 y ''' x 2 y '' 2xy ' 2y  x 3 . 1 Answer: y  c1x  c 2 x ln x  c3 x 2  x 3 4 As the tide changes, the water level in a bay varies sinusoidally. At high tide today at 8 AM, the water level was 15 feet; at low tide, 6 hours later at 2 PM, it was 3 feet. How fast, in feet per hour, was the water level dropping at noon today?

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ECE Coaching Course Find the real values of x and y if 3  ix 2 y and

 3 Answer: 2

x 2  y  4i are complex conjugate to each other.

The odds that a Ph.D. thesis will be favorably reviewed by three independent examiners are 5 to 2, 4 to 3 and 3 to 4. What is the probability that a majority approve the thesis? Answer: 209/343 A force field is said to be conservative if Answer: curl = 0 The value of the integral  ydS where C is the

Answer:  1, 4 

1

    If f  x        Answer: 0

2 1 2 1

 sin x 1      cos x x  , then f   is 4  1 x2   

C

curve y  2 x from x = 3 to x = 24 is Answer: 156

If Z2  iZ , then Answer: Im(Z) = 0

If X is a binomial variate with p = 1/5, for the experiment of 50 trials, then the standard deviation is equal to Answer: 2 2 If   3x 2 y  y3z 2 ,grad at (1,- 2,- 1) is equal to Answer: – (12i + 9j + 16k)

Forces F1, F2, F3 of magnitudes 5, 3, 1 units respectively, act in the directions 6i + 2j + 3k, 3i – 2j + 6k, 2i – 3j – 6k respectively on a particle. If the particle is displaced from the point (2,-1,-3) to the point (5,-1,1) find the work done by the resultant force. Answer: 231

Find the difference of the area of the square inscribed in a semi-circle having a radius of 15 m. The base of the square lies on the diameter of the semi-circle. Answer: 173.5 cm2

The constant forces 2i – 5j + 6k, –i + 2j – k and 2i + 7j act on a particle which is displaced from position 4i – 3j – 2k to position 6i + j – 3k. Find the total work done. Answer: 17 N

Find the number of hours for 25 people to sweep and polish the floors of a building if the supervisor notes that it takes 4 hours for 20 people to do the job. Answer: 3.2 hours

It is given that the event A and B are such that 1 1 2 P  A   , P  A B  , P  B A   . P  B = 4 2 3 Answer: 1/3

2

The number of integers that satisfy the inequality x2 + 48 < 16x is Answer: 7 A parabola with a vertical axis has its vertex at the origin and passes through point (7,7). The parabola intersects line y = 6 at two points. The length of the segment joining these points is Answer: 13 |2x – 1| = 4x + 5 has how many numbers in its solution set? Answer: 1 A circular table is tangent to two adjacent walls of a room as shown in the figure. Point N is 10 inches from one wall and 5 inches from the other wall. What is the area of the circular table? Answer: 625 A sphere with a 10-cm diameter sits in a cone so that the point of tangency is 12 cm up the cone’s edge from the vertex. How much liquid can be under the sphere if the liquid and sphere just touch one another? Answer: 93.08 cm3 Write in rectangular coordinates form r  6 tan  sec  Answer: x2 = 6y

What is the angle in degrees between an asymptote of the hyperbolax2 – 4y2 – 2x – 63 = 0 and the x-axis? Answer: 26.6o

Take a semicircle with a rectangle on its diameter. If the perimeter of the figure is 20 feet, find the radius of the semicircle in order that its area may be maximum. Answer: 2.8

x2 x2 y ' 2 y  xe x if y = x is one of x x its solutions. Answer: y  C1x  C 2 xe x  xe x  x 2 e x

Solve y ''

Compute the Taylor expansion about 0 as far as degree 4 for the solution of y’’ – e7xy’ + xy = 0, which satisfies y(0) = 2 and y’(0) = 1. 1 2 1 Answer: y  2  x  x 2  x 3  x 4  ... 2 3 2 Solve for the particular solution to the Ricatti 4 1 equation y '   2  y  y 2 t t 2 Answer: y  t 1 Solve the IVP y ' y   y 2 , y(1) = 1, t > 0. t Answer: y = 2t/(t2 + 1)

A restricted access lake is stocked with 400 fish. It is estimated that the lake will be able to hold 10,000 fish. The number of fish tripled in the first year. Assuming that the fish population follows a logistic model and that 10,000 is the limiting population, find the length of time needed for the fish population to reach 5,000. Answer: 2.68 years Find the Laplace transform of g  t   e3t  cos  6t   e3t cos  6t  Answer:

1 s s3   s  3 s 2  36  s  32  36

If today is Monday, and 1234 days passed, what day will it be? Answer: Wednesday

Find the Laplace transform of g  t   t 2

3

If a and b are two unit vectors inclined at an angle  and are such that a + b is a unit vector, then  is equal to Answer: 2  /3

Answer:

3  5

4s 2 3

Find the Laplace transform of f  t   10t  2 3

Answer: 10 2 If v, w, x, y, and z are consecutive integers whose zv sum is 0, then what is the value of ? yv Answer: 0 Find the differential equation of x 2  y3  Cx 4  0

dy x 2  y3 4 0 Answer: 2x  3y dx x 2

11

Find the area, if it exists, of the region above the 4 x-axis, between the curve y  and its 1 x2 asymptotes. Answer: 4

4

Find the Laplace transform of f  t   t cosh  3t  Answer: (s2 + 9)/(s2 – 9)2

Answer: y   2ln cos x  C1

Answer: 1 – 11x3 + (99/2)x6 + …

5

The divergence of the vector field (x – y)i + (y – x)j + (x + y + z)k is Answer: 3

Find the orthogonal trajectories of y  Csin  x  .

First three terms in expansion of 1  2x 3  2 are

Math Take Home 1     Solve y  4y  5y''' 6y'' 36y' 40y  0 Answer: y  c1e2x  c2 xe2x  c3e2x  c4ex cos  2x   c5 ex sin  2x 

The half–life of radium is 1600 years. If a sample initially contain 50 g, how long will it be until it contains 45 g? Answer: 243.2 years A tank contains a salt water solution consisting initially of 20 kg of salt dissolved into 10 L of water. Fresh water is being poured into the tank at a rate of 3 L/min and the solution, kept uniform by stirring, is flowing out at 2 L/min. Find the amount of salt in the tank after 5 minutes. Answer: 8.89 kg Solve y’’ + 5y’ + 6y = 0. Answer: y  c1e3x  c2e2x

Give it your best shot!

3  5

4s 2 Find the Laplace transform of f  t   tg '  t  Answer: G  s   sG '  s  Interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated. Answer: Interval estimate of a parameter A specific numerical value estimate of a parameter. Answer: Point estimate. The best point estimate of the population mean is the sample mean. The probability that the interval estimate will contain the parameter. Answer: Confidence level of an estimate of a parameter A parameter of a specific interval estimate determined by using data obtained from a sample and by using the specific confidence level of the estimate. Answer: Confidence interval Three common confidence level used. Answer: 90%, 95%, 99%

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Math Take Home 1

Critical value of (a) 90%, (b) 95% and (c) 99% confidence level. Answer: (a) 2.58, (b) 1.96, (c) 1.645

RMSE Answer: Root mean square error. The square root of MSE

The margin of error for a sample of size n is 1 Answer: n

Inverse Laplace transform of 1/(s + 5) at t = 0.5 Answer: 0.082

The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 50 students is selected, and the mean is found to be 23.2 years. Find the 95% confidence interval of the population mean. Answer: between 22.6 and 23.8 years old A survey of 30 adults found that the mean age of a person’s primary vehicle is 5.6 years. Assuming the standard deviation of the population is 0.8 year, find the 99% confidence interval of the population mean. Answer: between 5.2 and 6.0 years The college president asks the statistics teacher to estimate the average age of the students at their college. How large a sample is necessary? The statistics teacher would like to be 99% confident that the estimate should be accurate within 1 year. From a previous study, the standard deviation of the ages is known to be 3 years. Answer: 60 What are the three properties of a good estimator? Answer: unbiased, consistent and relatively efficient What is the maximum error of estimate? Answer: Margin of error Confidence interval can be calculated using the formula  Answer: x   z critical value  n Minimum sample size needed for an interval estimate of the population mean can be calculated using the formula 2

  z critical value    Answer:   where E is the E   maximum error of estimate

An estimate that is based on the variances within the samples. An estimate of the variance whether or not the null hypothesis is true. Answer: Mean Square Error (MSE) The estimate is based on the variance of the sample means. An estimate of the variance if the null hypothesis is true. Answer: Mean Square Between (MSB)

Inverse Laplace transform of 1/(s + 2)2 at t = 0.2 Answer: 0.134 Inverse Laplace transform

1 2

 s  2   s  1

at t = 0.2

Answer: 0.0055395

Mean square error can be calculated using the formula  si 2 Answer: MSE  a Given the following data below, compute the MSE and MSB. Aspirin Tylenol Placebo 3 2 2 5 2 1 3 4 3 5 4 2 Answer: MSE = 1.111, MSB = 4

The length of diameter AB is a two digit integer. Reversing the digits gives the length of a perpendicular chord CD. The distance from their intersection point H to the center O is a positive rational number. Determine the length of AB.

Find the Laplace transform of the second derivative of f  t   sin 2 t . Answer: 2s/(s2 + 4) Find the Laplace transform of the second derivative of f  t   t sin  at  Answer:

2as3

s

2

 a2 

2

t

Find the Laplace transform of  32d 0

Answer: 3!/s4

Answer: 65 In the adjoining figure, two circles with radii 6 and 8 are drawn with their centers 12 units apart. At P, one of the points of intersection, a line is drawn in such a way that the chords QP and PR have equal length. P is the midpoint of QR. Find the square of the length of QP.

t

Find the Laplace transform of  2 cos  2  d 0

Answer: 2/(s2 + 4) z 1 , residue at (a) z = 0, (b) z = 4z3  z 1/2, (c) z = –1/2 is Answer: (a) –1, (b) 3/4, (c) 1/4

Given F  z  

2    d   z  i   lim    z i  dz 2 2    1  z   Answer: i/4

 d   z  i 2 z    lim   2  z i dz    z  i    Answer: 1 z 1 i  z2 dz where z = e where     0 Answer: 2  i dz where (a) z = ei where     0 , (b) z = z2 t + 0i where 1  t  2 . Answer: (a) –2, (b) 1/2





Answer: 130 A point P is chosen in the interior of triangle ABC such that when lines are drawn through P parallel to the sides of triangle ABC, the resulting smaller triangles t1, t2 and t3 in the figure, have area 4, 9 and 49 respectively. Find the area of triangle ABC.

Answer: 144 When a triangle is rotated about one leg, the volume of the cone produced is 800 cm3. When the triangle is rotated about the other leg, the volume of the cone produced is 1920 cm3. What is the length of the hypotenuse of the triangle? Answer: 26 cm An ellipse has foci at (9, 20) and (49, 55) in the xy – plane and is tangent to the x – axis. What is the length of its major axis? Answer: 85

log  z dz where z = ei and 0    

Answer: 2i If the null hypothesis is (a) true, (b) false then Answer: (a) MSE and MSB should be about the same value, (b) MSB is larger than MSE

Answer: 288

Determine the radius of convergence of 

 2 

n n

 x  3 . n 1 Answer: 1/2

 n 0

Three 12 cm x 12 cm squares are each cut into two pieces A and B, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are than attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume of this polyhedron?

Find the radius of convergence of the power  xn n series   1 n n 0  2  1 n 2  1 Answer: 2 The solid has a square base of side length s. The upper edge is parallel to the base and has length 2s. All other edges have length s. Given that s = 6 2 , what is the volume of the solid?

Mean square between can be calculated using the formula Answer: MSB = ns M 2

Answer: 864 cm3 In triangle ABC, AB = 425, BC = 450 and AC = 510. An integer point P is then drawn and segments are drawn through P parallel to the sides of the triangle. If these three segments are of an equal length d, find d. Answer: 306 Let triangle ABC be a right triangle in the xy – plane with a right angle at C. Given that the

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length of the hypotenuse AB is 60, and that the medians through A and B lie along the lines y = x + 3 and y = 2x + 4 respectively, find the area of triangle ABC. Answer: 400 Find the area of the region enclosed by the graph x x  60  y  4 Answer: 480 Triangle ABC has a right angle at B, and contains a point P for which PA = 10, PB = 6 and angle APB = angle BPC = angle CPA. Find PC.

Answer: 33 The value of a for which the sum of the squares of the roots of the equation x 2   a  2  x  a  1  0 assume the value is Answer: 1 If the cube roots of unity are 1,  , 2 then the 3

roots of the equation  x  1  8  0 are

A matrix of non-negative real numbers, such that the entries in each row sum to 1. Answer: Markov matrix

The eigenvalues of a unitary matrix or orthogonal matrix have Answer: Absolute value 1

A matrix whose off-diagonal entries are nonnegative. Answer: Metzler matrix

The matrix containing minors of a given square matrix. Answer: Adjugate matrix

A square matrix with exactly one non-zero entry in each row and column. Answer: Monomial matrix

A matrix in which successive columns have a particular function applied to their entries. Answer: Alternant matrix

A square matrix satisfying Aq = 0 for some positive integer q. Answer: Nilpotent matrix

Synonym for Skew–Hermitian matrix. Answer: Anti–Hermitian Matrix

If matrix A has an inverse. Answer: Non–singular matrix

Synonym for skew–symmetric matrix. Answer: Anti–symmetric matrix

A real unitary matrix. Answer: Orthogonal matrix

A matrix whose rows are concatenations of the rows of two smaller matrices. Answer: Augmented matrix

A symmetric matrix. Answer: Real Hermitian matrix

A matrix with nonzero entries on the main diagonal and on sloping parallel to it. Answer: Band matrix

Answer: 1, 1  2, 1  22 2

If the roots of the equation x  bx  c be two consecutive integers, then b2 – 4c equals Answer: 1 If both the roots of the quadratic equation x 2  2kx  k 2  k  5  0 are less than 5, then k lies in the interval Answer:  , 4  All the values of m for which both roots of the equation x 2  2mx  m2  1  0 are greater than –2 but less than 4, lie in the interval Answer: –1 < m < 3 If the difference between the roots of the equation x 2  ax  1  0 is less than 5 , then the set of possible values of a is Answer:  3,3 The quadratic equations x2 – 6x + a = 0 and x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is Answer: 2 If p and q are the roots of the equation x2 + px + q = 0, then Answer: p = 1, q = –2 If (1 – p) is a root of quadratic equation x2 + px + (1 – p) = 0 then its roots are Answer: 0, –1 Fourier transform of Dirac delta or impulse function   t  . Answer: 1 Fourier transform of unit step or Heaviside function u  t  . 1 Answer:     i

Fourier transform of negative time unit step function u  t  Answer:    

1 i

The eigenvalues of a Hermitian matrix or a symmetric matrix are Answer: Real

Math Take Home 1

The eigenvalues of a skew-Hermitian matrix or a skew-symmetric matrix are Answer: Pure imaginary or zero

A matrix with all elements either 0 or 1. Answer: Binary, Boolean, Logical or (0,1) matrix A square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal. Answer: Bisymmetric matrix A matrix with all rows and columns mutually orthogonal, whose entries are unimodular. Answer: Complex Hadamard matrix Two matrices A and B are congruent if there exists an invertible matrix P such that PT A P = B. Answer: Congruent matrix Square matrix that can have nonzero entries only on the main diagonal. Any entry above or below the main diagonal must be zero. Answer: Diagonal matrix A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal. Answer: Frobenius matrix An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Answer: Hessenberg matrix A matrix which is equal to its conjugate

A skew – symmetric matrix. Answer: Real Skew–Hermitian matrix If all the diagonal entries of a diagonal matrix are equal. Answer: Scalar matrix A matrix whose entries are either +1, 0, or −1. Answer: Sign matrix A diagonal matrix where the diagonal elements are either +1 or −1. Answer: Signature matrix If matrix A has no inverse. Answer: Singular matrix A matrix where its negative equivalent is equal to T

its conjugate transpose A  A or a ij  a ij Answer: Skew–Hermitian matrix A square matrix whose transpose equals minus the matrix. Answer: Skew–symmetric matrix A matrix with relatively few non-zero elements. Answer: Sparse matrix A square matrix with all entries nonnegative and all column sums equal to 1. Answer: Stochastic matrix A square matrix whose entries come from coefficients of two polynomials. Answer: Sylvester matrix

T

transpose. A  A or a ij  a ij Answer: Hermitian matrix A square matrix of second partial derivatives of a scalar-valued function. Answer: Hessian matrix A square matrix whose main diagonal comprises only zero elements. Answer: Hollow matrix A matrix that has the property A² = AA = A. Answer: Idempotent or Projection matrix A square matrix which is its own inverse, AA = I. Answer: Involutory matrix A matrix of first-order partial derivatives of a vector-valued function. Answer: Jacobian matrix A square matrix that can have nonzero entries only on and below the main diagonal, whereas any entry above the diagonal must be zero. Answer: Lower triangular matrix

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A matrix is symmetric if it equals its transpose. Answer: Symmetric matrix A matrix with constant diagonals. Answer: Toeplitz matrix A matrix with nonzero entries on the main diagonal and on the two sloping parallels immediately above or below the diagonal. Answer: Tridiagonal matrix An invertible matrix with entries in the integers (integer matrix). Necessarily the determinant is +1 or –1. Answer: Unimodular matrix A square matrix with all eigenvalues equal to 1. Answer: Unipotent matrix A scalar matrix whose entries on the main diagonal are all 1. Answer: Unit or Identity matrix A matrix where is inverse is equal to its conjugate T

transpose. A  A 1

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ECE Coaching Course

Answer: Unitary matrix

Answer: Tricomi equation

A square matrix that can have nonzero entries only on and above the main diagonal, whereas any entry below the diagonal must be zero. Answer: Upper triangular matrix A row consists of 1, a, a², a³, etc., and each row uses a different variable. Answer: Vandermonde matrix

G '' yG  0 or y '' k 2 xy  0 Answer: Airy equation

A square matrix, with dimensions a power of 2, the entries of which are +1 or -1. Answer: Walsh matrix

y ' p  x  y  q  x  y n Answer: Bernoulli equation

y'  Ay  By2 (a special type of Bernoulli equation) Answer: Logistic or Verhulst equation x 2 y '' xy '  x 2  v2  y  0 (the parameter v is a

A square matrix the entries of which are in {0, 1, −1}, such that AAT = wI for positive integer w. Answer: Weighing matrix

given number, real and nonnegative) Answer: Bessel’s equation

A matrix with all off-diagonal entries less than 0. Answer: Z – matrix

1  x  y'' 2xy ' n  n  1 y  0

The differential equation in the form of y '  P  x  y 2  Q  x  y  R  x  is called a/an Answer: Ricatti equation The second order differential equation x 2 y" Axy ' By  0 (with A and B are constants) is called Answer: Euler – Cauchy equation

u tt  c 2u xx where t and x stands for time and spatial coordinated respectively, c is wave speed and u represents the amplitude. Answer: Wave equation u tt  c 2 u xx  f  x, t  Answer: Non homogeneous wave equation u tt  c 2 u xx  du Answer: Klein Gordon equation u tt  c 2 u xx  du  f  x, t  Answer: Non homogeneous Klein Gordon equation u tt  au t  bu  c 2 u xx Answer: Telegraph equation u t   2 u xx

where  2 is called the thermal diffusivity of the rod and u represents the temperature if the equation represents heat conduction through a rod. Answer: Heat equation u t   2 u xx  f  x, t  Answer: Non homogeneous heat equation u t   2 u xx  bu x  cu  f  x, t  Answer: Convective heat equation

parameter  ) Answer: Sturm – Liouville equation

u xx  u yy  f  x, y   0

Answer: Helmholtz equation

x 2 y ''  x 2  n  n  1  y  0 Answer: Ricatti equation

y''  1  y2  y ' y  0

Partial DE

Au xx  2Bu xy  Cu yy  F  x, y, u, u x , u y  is hyperbolic if Answer: AC  B2  0 or  2B  4AC  0 (example is Wave equation) Partial DE

Au xx  2Bu xy  Cu yy  F  x, y, u, u x , u y  is

u u T  c2 2 , c  t 2 x  Answer: One dimensional wave equation

parabolic if

u 2u  c2 2 t x Answer: One dimensional heat or diffusion equation

Partial DE

2

2u 2u  0 x 2 y 2 Answer: Two dimensional Laplace equation 2 u 

 2u  2 u  2 u  2  2  f  x, y  x y Answer: Two dimensional Poisson equation

2

Answer: AC  B2  0 or  2B  4AC  0 (example is Heat equation)

Au xx  2Bu xy  Cu yy  F  x, y, u, u x , u y  elliptic if 2

Answer: AC  B2  0 or  2B  4AC  0 (example is Laplace equation) versine(x) or vers(x) is equivalent to Answer: 1 – cos(x) coversine(x) or covers(x) is equivalent to Answer: 1 – sin(x) exsecant(x) or exsec(x) is equivalent to Answer: sec(x) – 1

  2 u  2u  2u  c2  2  2  2 t y   x

haversine(x) or hav(x) is equivalent to Answer: vers(x)/2

Answer: Two dimensional wave equation

The complement of a set A is

 2u  2 u  2u   0 x 2 y 2 z 2 Answer: Three dimensional Laplace equation

Answer: A  U  A

A  U  A, A    A Answer: Identity Laws

x 2 y '' axy ' by  0 Answer: Euler Cauchy equation

A  U  U, A     Answer: Domination Laws

x 1  x  y'' c   a  b  1 x  y' aby  0 Answer: Gauss’s hypergeometric ODE

A  A  A, A  A  A Answer: Idempotent Laws

xy '' 1  x  y ' ny  0 Answer: Laguerre’s equation

A  A

 2 u 1 u 1  2 u   r 2 r r r 2 2 Answer: Laplacian in polar coordinates 2 u 

Answer: Poisson equation

u xx  u yy  au  f  x, y   0

r  r  1  bo r  co  0 (quadratic equation) Answer: Indicial equation of the ODE

2

f  z   u  x, y   iv  x, y  should satisfy what type of equation to be analytic. Answer: Cauchy – Riemann equations u x  v y and u y   v x

Elliptic equation u xx  u yy  0 Answer: Laplace equation

 2u  2u   v 2u  0 x 2 y 2 Answer: Helmholtz equation

Answer: Van der Pol equation

 p  x  y' '  q  x   r  x   y  0 (involving a

2

The equation y" p  x  y  q  x  y  0 is Answer: Second order linear homogeneous

y ' p  x  y  g  x  y 2  h  x  Answer: Riccati equation y  xy ' g  y '  Answer: Clairaut equation

2

Answer: Legendre’s equation A matrix with all entries zero. Answer: Zero matrix

Math Take Home 1  1  y ''  1  y '2  y' y  0,   0  3  Answer: Rayleigh equation

 2 u 1 u 1  2 u  2 u    r 2 r r r 2 2 z 2 Answer: Laplacian in cylindrical coordinates 2u 

y '' o 2 y  y3  0 Answer: Duffing equation

Answer: Complementation Laws

A  B  B  A, A  B  B  A Answer: Commutative Laws

A   B  C    A  B  C A   B  C    A  B  C Answer: Associative Laws

A   B  C    A  B   A  C  A   B  C    A  B   A  C  Answer: Distributive Laws

yu xx  u yy  0

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Excel Review Center A  B  A  B, A  B  A  B Answer: De Morgan’s Laws

A   A  B  A A   A  B  A Answer: Absorption Laws A  A  U, A  A   Answer: Complement Laws xx Answer: Law of double complement x  x  x, x  x  x Answer: Idempotent Laws

x  0  x, x 1  x Answer: Identity Laws x  1  1, x  0  0 Answer: Domination Laws x + y = y + x, xy = yx Answer: Commutative Laws x + (y + z) = (x + y) + z, x(yz) = (xy)z Answer: Associative Laws x + yz = (x + y)(x + z), x(y + z) = xy + xz Answer: Distributive Laws

 xy   x  y,  x  y   xy Answer: De Morgan’s Laws x + xy = x, x(x + y) = x Answer: Absorption Laws

x  x 1 Answer: Unit property

xx  0 Answer: Zero property p  T  p, p  F  p (T denotes the compound proposition that is always true and F denotes the compound proposition that is always false.) Answer: Identity Laws p  T  T, p  F  F (T denotes the compound proposition that is always true and F denotes the compound proposition that is always false.) Answer: Domination Laws

p  p  p, p  p  p Answer: Idempotent Laws   p   p Answer: Double negation law

p  q  q  p, p  q  q  p Answer: Commutative Laws

p  q  r  p  q  r , p  r   r  p  q  r  Answer: Associative Laws

ECE Coaching Course Answer: Negation Laws For all real numbers x and y, x + y is a real. Answer: Closure law for addition For all real numbers x and y, x  y is a real number. Answer: Closure law for multiplication For all real numbers x, y and z, (x + y) + z = x + (y + z). Answer: Associative law for addition For all real numbers x, y and z,  x  y  z  x   y  z Answer: Associative law for multiplication For all real numbers x and y, x + y = y + x. Answer: Commutative law for addition For all real numbers x and y, x  y  y  x Answer: Commutative law for multiplication For every real number x, x + 0 = 0 + x = x. Answer: Additive identity law For every real number x, x 1  1 x  x Answer: Multiplicative identity law The additive identity 0 and the multiplicative identity 1 are distinct, that is 0  1. Answer: Identity elements axiom For every real number x, there exists a real number –x called the additive inverse of x, such that x + (–x) = (–x) + x = 0. Answer: Inverse law for addition For every nonzero real number x, there exists a real number 1/x called the multiplicative inverse of x, such that x  1 / x   1/ x   x  1 Answer: Inverse law for multiplication For all real numbers x, y and z, x  y  z   xy  xz and  x  y  z  xz  yz Answer: Distributive laws For all real numbers x and y, exactly one of x = y, x > y or y > x is true. Answer: Trichotomy law For all real numbers x, y and z, if x >y and y > z, then x > z. Answer: Transitivity law For all real numbers x, y and z, if x > y, then x + z > y + z. Answer: Additive compatibility law For all real numbers x, y and z, if x > y and z > 0, then xz  yz Answer: Multiplicative compatibility law Every nonempty set of real numbers that is bounded has a least upper bound. Answer: Completeness property Every nonempty subset of the set of positive integers has a least element. Answer: The Well-ordering property

p  q  r   p  q   p  r ,

If S is a set of positive integers such that 1  S and

p  q  r   p  q   p  r

for all positive integers n if n  S , then n  1  S , then S is the set of positive integers. Answer: Mathematical induction axiom

Answer: Distributive Laws   p  q   p  q,   p  q   p  q Answer: De Morgan’s Laws p   p  q   p, p   p  q   p Answer: Absorption Laws

Modus ponens  p   p  q    q Answer: Law of detachment (Modus ponens is Latin for mode for affirms) This is law states that is  0 , as n becomes arbitrarily large the probability approaches 1 that

p  p  T, p  p  F

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Math Take Home 1 the fraction of times an event E occurs during n trials is within  of p(E). Answer: Law of large numbers For every real number x there exists an integer n such that n > x. Answer: Archimedean Property This law states that if the sample space S is the disjoint union of the events S1 , S2 ,...,Sn and X is a n



  

random variable, then E  X    E X S j P S j j1

Answer: Law of total expectation A statement that is true or false. Answer: Proposition A variable that represents a proposition. Answer: Propositional variable True or false Answer: True value Proposition with truth value opposite to the truth value of p. Answer: Negation Operators used to combine propositions. Answer: Logical operators A proposition constructed by combining propositions using logical operators. Answer: Compound proposition A table displaying all possible truth values of propositions. Answer: Truth table The proposition “p or q”, which is true if and only if at least one of p and q is true. Answer: Disjunction The proposition “p and q”, which is true if and only if both p and q are true. Answer: Conjunction Proposition “p XOR q”, which is true when exactly one of p and q is true. Answer: Exclusive OR The proposition “if p, then q”, which is false if and only if p is true and q is false. Answer: Implication The proposition “p if and only if q”, which is true if and only if p and q have the same truth value. Answer: Biconditional Either a 0 or 1. Answer: Bit A variable that has a value 0 or 1. Answer: Boolean variable A list of bits. Answer: Bit string A compound proposition that is always true. Answer: Tautology A compound proposition that is always false. Answer: Contradiction A compound proposition that is sometimes true and sometimes false. Answer: Contingency Compound propositions for which there is an assignment of truth values to the variables that makes all these propositions true. Answer: Consistent compound propositions A compound proposition for which there is an assignment of truth values to its variables that makes it true. Answer: Satisfiable compound proposition

Excel Review Center Compound proposition that always have the same truth values. Answer: Logically equivalent compound propositions A sequence of statements. Answer: Argument An invalid argument form often used incorrectly as a rule of inference (or sometimes, more generally, an incorrect argument) Answer: Fallacy Mathematical assertion can be shown to be true. Answer: Theorem A mathematical assertion proposed to be true, but that has not been proved. Answer: Conjecture

ECE Coaching Course The secant method of finding roots of nonlinear equations falls under the category of ____ methods. Answer: open Highest order of polynomial integrand for which Simpson’s 1/3 rule of integration is exact is Answer: Third

Exclusive disjunction Answer: Symbol: ,  , Should be read as: xor

Newton-Raphson method of solution of numerical equation is not preferred when Answer: The graph of f(x) is nearly horizontal – where it crosses the x–axis.

Contradiction Answer: Symbol: , F, 0 , Should be read as: bottom, falsum

Newton-Raphson method is applicable to the solution of Answer: Both algebraic and transcendental equations

A demonstration that a theorem is true. Answer: Proof

The order of errors the Simpson’s rule for numerical integration with a step size h is Answer: h^2

A statement that is assumed to be true and that can be used as a basis for proving theorems. Answer: Axioms

In which method proper choice of initial value is very important? Answer: Newton-Raphson

A theorem used to prove other theorems. Answer: Lemma

Errors may occur in performing numerical computation on the computer due to Answer: Rounding errors

A proposition that can be proved as a consequence of a theorem that has just been proved. Answer: Corollary A statement containing one or more variables that becomes a proposition when each of its variables is assigned a value or is bound by a quantifier. Answer: Propositional function In which of the following method, we approximate the curve of solution by the tangent in each interval. Answer: Euler’s method Jacobi’s method is also known as Answer: Simultaneous displacement method The convergence of which of the following method is sensitive to starting value? Answer: Newton-Raphson method To perform a Chi-square test Answer: Data conform to a normal distribution. Data be measured on a nominal scale. Each cell has equal number of frequencies. In the Gauss elimination method for solving a system of linear algebraic equations, triangularization leads to Answer: upper triangular matrix Newton-Raphson Answer: Root finding Runge-kutta Answer: Ordinary differential equations Gauss-seidel Answer: Solution of system of linear equations Simpson’s rule Answer: Integration The expected value of the random variable Answer: Is another term for the mean value. Solving an engineering problem requires four steps. In order of sequence, the four steps are Answer: Formulate, solve, interpret, implement True error is defined as Answer: True value – approximate value The Newton-Raphson method of finding roots of nonlinear equations falls under the category of ____ methods. Answer: open

Math Take Home 1

Tautology Answer: Symbol: , 1 Should be read as: top, verum

Universal quantification Answer: Symbol: , () Should be read as: for all; for any; for each Existential quantification Answer: Symbol:  Should be read as: there exists Uniqueness quantification Answer: Symbol: ! Should be read as: there exists exactly one Definiton Answer: Symbol: :,  , :  Should be read as: is defined as

_______ distinct _______ points form a plane. Answer: Three; non collinear

Precedence grouping Answer: Symbol: ( ) Should be read as: parentheses, brackets

Two parallel lines intersected by a transverse line, the alternating interior angles are ________. Answer: Congruent

Turnstile Answer: Symbol:

, Should be read as: provable

Double turnstile Answer: Symbol:

, Should be read as: entails

Two parallel lines intersected by a transverse line, the alternating exterior angles are ________. Answer: Congruent Two parallel lines intersected by a transverse line, the same side interior angles are ________. Answer: Supplementary Two parallel lines intersected by a transverse line, the same side exterior angles are ________. Answer: Supplementary Solid generated by a line revolved and intersected by another is a/an Answer: Cone Generated by a parabola on a plane with a perpendicular line is a/an Answer: Parabolic cylinder Generated by an ellipse on a plane with a perpendicular line is a/an Answer: Elliptical cylinder The mean of the sides of a triangle meet at what point? Answer: Centroid The altitudes of the sides of a triangle meet at what point? Answer: Orthocenter Material implication Answer: Symbol: ,  ,  Should be read as: implies; if… then Material equivalence Answer: Symbol: ,  ,  , Should be read as: if and only if; means the same as Negation Answer: Symbol: , ! , ~, Should be read as: not Logical conjunction Answer: Symbol: , , & ,Should be read as: and Logical disjunction Answer: Symbol: , +,

, Should be read as: or

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Complement of set A Answer: The set of all elements in U that are not in set A. Denoted by A, A' or AC Intersection of set A and B Answer: The set of elements belonging to both A and B. Denoted by A  B Union of set A and B Answer: The set of elements belonging to either A or B. Denoted by A  B Difference (or Relative Complement) of set A and B Answer: The set of elements of A that do not belong to B. Denoted by A – B or A\B Symmetric difference of set A and B Answer: The set of elements that belong to A or B but not both. Denoted by A  B or AB Property of a relation R on a set A where  x, y   R and  y, x   R implies x = y. Answer: Anti–symmetric or Asymmetric The relation R  1,1 , 1, 2  ,  3, 2  ,  3,3 is Answer: Anti–symmetric, Transitive Property of a relation R on a set A where x  A ,  x, x   R or xRx Answer: Reflexive Property of a relation R on a set A where x, y  A

 x, y   R implies  y, x   R or xRy implies yRx. Answer: Symmetric Property of a relation R on a set A where x, y, z  A ,  x, y   R and  y, z   R implies

 x, z   R . Equivalently, for all x, y, z  A , xRy and yRz implies xRz. Answer: Transitive

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A relation between elements of a set A that is reflexive, symmetric and transitive. Answer: Equivalence relation A relation that is symmetric and transitive. Answer: Partial Equivalence Relation “Don’t quit on yourself.”

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Math Take Home 1