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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the adjacency matrix of the given graph or digraph. (1) еб е1 e A ег (3) (2) ез
State the most important facts about distributions of two random variables and their marginal distributions.
Graph a sample space for the experiments:Drawing gaskets from a lot of 10, containing one defective D, unitil D is drawn, one at a time and assuming sampling without replacement, that is, gaskets
Let X [millimeters] be the thickness of washers. Assume that X has the density f(x) = kx if 0.9 < x < 1.1 and 0 otherwise. Find k. What is the probability that a washer will have thickness
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Efficiency [%] of seven Voith Francis turbines of runner diameter 2.3 m under a head range of 185 m91.8 89.1
In 1910, E. Rutherford and H. Geiger showed experimentally that the number of alpha particles emitted per second in a radioactive process is a random variable X having a Poisson distribution. If X
If the diameter X [cm] of certain bolts has the density f(x) = k(x - 0.9)(1.1 - x) for 0.9 < x < 1.1 and 0 for other x, what are k, μ, and σ2? Sketch f(x).
In how many different ways can 6 people be seated at a round table?
If we inspect photocopy paper by randomly drawing 5 sheets without replacement from every pack of 500, what is the probability of getting 5 clean sheets although 0.4% of the sheets contain spots?
A manufacturer knows from experience that the resistance of resistors he produces is normal with mean μ = 150 Ω and standard deviation σ = 5 Ω. What percentage of the resistors will have
Graph a sample space for the experiments:Recording the daily maximum temperature X and the daily maximum air pressure Y at Times Square in New York
Let X be the number of cars per minute passing a certain point of some road between 8 A.M. and 10 A.M. on a Sunday. Assume that X has a Poisson distribution with mean 5. Find the probability of
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Release time [sec] of a relay1.3 1.2 1.4 1.5 1.3 1.3
When is the Poisson distribution a good approximation of the binomial distribution? The normal distribution?
What are the mean thickness and the standard deviation of transformer cores each consisting of 50 layers of sheet metal and 49 insulating paper layers if the metal sheets have mean thickness 0.5 mm
Of a lot of 10 items, 2 are defective.(a) Find the number of different samples of 4. Find the number of samples of 4 containing(b) No defectives(c) 1 defective(d) 2 defectives.
What is sampling with and without replacement? What distributions are involved?
If the lifetime X of a certain kind of automobile battery is normally distributed with a mean of 5 years and a standard deviation of 1 year, and the manufacturer wishes to guarantee the battery for 4
Five fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of heads and compute the probabilities of obtaining no heads, precisely 1 head, at least 1
Graph f and F when f(-2) = f(2) = 1/8, f(-1) = f(1) = 3/8. Can f have further positive values?
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Weight of filled bags [g] in an automatic filling203 199 198 201 200
State the definition of probability from memory. Give simple examples.
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = 4e4x (x ≥ 0)
Find the density of the marginal distribution of Y in Fig. 524. az 0 01 B,
In how many different ways can we select a committee consisting of 3 engineers, 2 physicists, and 2 computer scientists from 10 engineers, 5 physicists, and 6 computer scientists? First guess.
If a box contains 10 left-handed and 20 right-handed screws, what is the probability of obtaining at least one right-handed screw in drawing 2 screws with replacement?
What is a random variable? Its distribution function? Its probability function or density?
Let X be normal with mean 50 and variance 9. Determine c such that P(X < c) = 5%, P(X > c) = 1%, P(50 - c X < 50 + c) = 50%
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Systolic blood pressure of 15 female patients of ages 20–22156 158 154 133
Find the mean and variance of the random variable X with probability function or density f(x).Uniform distribution on [0, 2π]
What do we mean by an experiment? An outcome? An event? Give examples.
If a box contains 4 rubber gaskets and 2 plastic gaskets, what is the probability of drawing(a) First the plastic and then the rubber gaskets(b) First the rubber and then the plastic ones? Do this by
Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of the event that all 3 screws drawn are nondefective, assuming that we draw(a) With
What properties of data are measured by the mean? The median? The standard deviation? The variance?
In rolling 2 fair dice, what is the probability of a sum greater than 3 but not exceeding 6?
Let X be normal with mean 10 and variance 4. Find P(X > 12), P(X < 10), P(X < 11), P(9 < X < 13).
Graph a sample space for the experiments:Drawing 3 screws from a lot of right-handed and left handed screws
Graph the probability function f(x) = kx2 = (x = 1, 2, 3, 4, 5; k suitable) and the distribution function.
Mark the positions of ? in Fig. 517. Comment. 0.5 p = 0.1 p = 0.2 p = 0.5 GL 5 p = 0.8 0 p = 0.9
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:Length of nails [mm]19 21 19 20 19 20 21
What are stem-and-leaf plots? Boxplots? Histograms? Compare their advantages.
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = kx (0 ≤ x ≤ 2, k suitable)
Let f(x, y) = k when 8 ≤ x ≤ 12 and 0 ≤ y ≤ 2 and zero elsewhere. Find k. Find P(X ≤ 11, 1 ≤ Y ≤ 1.5) and P(9 ≤ X ≤ 13, Y ≤ 1).
In how many ways can a company assign 10 rivers to n buses, one driver to each bus and conversely?
In rolling 3 fair dice, what is the probability of obtaining a sum not greater than 16?
Find a graph, as simple as possible, that cannot be vertex colored with three colors. Why is this of interest in connection with Prob. 24?Data from Prob. 24The famous four-color theorem states that
Using augmenting paths, find a maximum cardinality matching. (1) (3) 5 7 (2 (4) 6 (8)
What would be the answer to Prob. 22 if only the five ships S1, · · ·, S5 had to be accommodated?Data from Prob. 22How many piers does a harbor master need for accommodating six cruise ships with
How many piers does a harbor master need for accommodating six cruise ships S1, · · ·, S6 with expected dates of arrival A and departure D in July, (A, D) = (10, 13), (13, 15), (16, 18), (14, 17),
Company A has offices in Chicago, Los Angeles, and New York; Company B in Boston and New York; Company C in Chicago, Dallas, and Los Angeles. Represent this by a bipartite graph.
How many colors do you need for vertex coloring any tree?
Find a shortest spanning tree. (5) 5 7 (3) 1 4 4 2 2 3 8 2)
Three teachers x1, x2, x3 teach four classes y1, y2, y3, y4 for these numbers of periods: Show that this arrangement can be represented by a bipartite graph G and that a teaching schedule for one
If a network has several sources s1, · · ·, sk, show that it can be reduced to the case of a single-source network by introducing a new vertex s and connecting s to s1, · · ·, sk, by k edges of
The definition is B? = [bjk], where Find the incidence matrix of the digraph in Prob. 11. Data from Prob. 11 Find the adjacency matrix of the given graph or digraph. bjk (-1 1 0 otherwise. if edge
Find the maximum flow by inspection: 1 5, 3 10, 7 3 (2) 6, 2 3, 1 8,4 8,5 5) t 4 7,4
If two vertices in a tree are joined by a new edge, a cycle is formed.
Find shortest paths by Dijkstra?s algorithm. (3) 6 4) 5 3 8 4 (1) 2 2
What is the smallest number of exam periods for six subjects α, b, c, d, e, f if some of the students simultaneously take α, b, f, some c, d, e, some α, c, e, and some c, e? Solve this as follows.
The definition is B = [bjk], where Find the incidence matrix of the graph in Prob. 8. Data from Prob. 8 Find the adjacency matrix of the given graph or digraph. bjk 1 if vertex j is an endpoint of
Find a shortest path and its length by Moore?s BFS algorithm, assuming that all the edges have length 1. S t
In Prob. 15, the cut set contains precisely all forward edges used to capacity by the maximum flow (Fig. 501). Is this just by chance? Data from Prob. 15 Find a minimum cut set in Fig. 500 and its
In what case are all the off-diagonal entries of the adjacency matrix of a graph G equal to one?
Find the maximum flow by inspection: s(1) 5, 2 (2) 8,5 6,3 4,2 3 4 11,7 2, 2 5, 2 4,1 5 13,9 6)t
A tree with exactly two vertices of degree 1 must be a path.
Make it for the graph in Prob. 15. Data from Prob. 15 Sketch the graph whose adjacency matrix is: 0 1 0 1 1 0 0 0 0 0 11 1 1 0
A planar graph is a graph that can be drawn on a sheet of paper so that no two edges cross. Show that the complete graph K4 with four vertices is planar. The complete graph K5 with five vertices is
Is the graph in Fig. 484 an Euler graph. Give reason. 2 s (1 (2) 1 2 (3) 4 (4) 3 5 5 4 (6)
Fi Sketch the graph whose adjacency matrix is: 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0
Find a minimum cut set in Fig. 500 and its capacity. (1) 20, 5 10, 4 2 4 11,8 4,3 7,4 (3) 5, 2 5 13,6 3,3 6t
Sketch the graph for the given adjacency matrix. 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0
An Euler graph G is a graph that has a closed Euler trail. An Euler trail is a trail that contains every edge of G exactly once. Which subgraph with four edges of the graph in Example 1,is an Euler
Using augmenting paths, find a maximum cardinality matching:In Prob. 12Data from Prob. 12Find an augmenting path: (1) (3) (5) (7) (2) (4) (6) (8)
Find flow augmenting paths: 2) Co 4, 2 10,3 3, 1 (2) 5,3 6,0 (3) 8,5 1, 1 5
Prove the following.If in a graph any two vertices are connected by a unique path, the graph is a tree.Data from Prob. 14Prove the following.The path connecting any two vertices u and v in a tree is
Sketch the graph for the given adjacency matrix. 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0
Sketch the graph whose adjacency matrix is: 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0
Sketch the graph whose adjacency matrix is: 0 1 0 1 1 0 0 0 0 0 11 1 1 0
Find a shortest spanning tree by Prim?s algorithm. Write a program and apply it to Probs. 6. Data from Prob. 6 Find a shortest spanning tree by Prim?s algorithm. (3) 3 10 2 14 4) 6 1 9 2 5 15
Are the consecutive flow augmenting paths produced by Ford–Fulkerson unique?
Find the adjacency matrix of: (1) (4) (2) (3)
Find flow augmenting paths: g (1) 8,3 6, 2 2 3 10, 2 4,1 14, 1 12, 3 4, 2 (5) t
Using augmenting paths, find a maximum cardinality matching:Data from Prob. 11Find an augmenting path: (1) im 3 (5) (2) 4 (6) (7)
The postman problem is the problem of finding a closed walk W: s?s (s the post office) in a graph G with edges (i, j) of length lij?> 0 such that every edge of G is traversed at least once and the
How can you see that Ford–Fulkerson follows a BFS technique?
Find the adjacency matrix of the given graph or digraph. (1) (4) (2) (5) (3)
A (not necessarily connected) graph without cycles is called a forest. Give typical examples of applications in which graphs occur that are forests or trees.
Write a program and apply it to Probs. 6?9. Data from Prob. 6 Find the maximum flow by Ford-Fulkerson: (1) SO (1 2, 1. S 2,1 2 2, 1 (3) 1,0 4, 2 1,0 4 8, 1 7,1 (6) 8, 1 LO 5
Find a shortest spanning tree by Prim?s algorithm. For the graph in Prob. 4 Data from Prob. 4 Find a shortest spanning tree by Kruskal?s algorithm. Sketch it. 2 5 20 4 (2) 6 8 Pol 2 7 (3) 3 4 5
Find the adjacency matrix of the given graph or digraph. (1) e₁ N (2) 23 e4 e2 (4) (3)
Which edges could be omitted from the network in Fig. 499 without decreasing the maximum flow? 8 (1) 8,4 7,5 4 2 6, 1 6, 5 8,5 4, 2 5 3 2, 1 4, 2 10, 8 6 6, 1 (7) t
Apply the algorithm in Prob. 10 to the graph in Example 1. Compare with the result in Example 1.Data from Prob. 10Design an algorithm for obtaining longest spanning trees.
Find the adjacency matrix of: (3) 4) (2) (1)
Find an augmenting path: (1) im 3 (5) (2) 4 (6) (7)
Find and sketch a Hamiltonian cycle in Prob. 1.Data from Prob. 1Find a shortest path P: s?t and its length by Moore?s algorithm. Sketch the graph with the labels and indicate P by heavier lines as in
Find the adjacency matrix of the given graph or digraph. (1) €3 (3) es es 5 €6 €2 (2) e. 4
Sketch the network in Fig. 499, and on each edge (i, j) write cij - fij and fij. Do you recognize that from this ?incremental network? one can more easily see flow augmenting paths? 1 8,4 7,5 4 2
What is BFS? DFS? In what connection did these concepts occur?
Find and sketch a Hamiltonian cycle in the graph of a dodecahedron, which has 12 pentagonal faces and 20 vertices (Fig. 483). This is a problem Hamilton himself considered.
Find the maximum flow by Ford-Fulkerson: 4, 2 (1) S 3,2 (2) 2, 1 (3) 5, 3 3, 2 6,3 (4) 10, 4 3, 1 5) 1,0 (6)

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