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18MA41A
Mathematics
Sikkim Manipal University
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18MA41A
USN
RV COLLEGE OF ENGINEERING
® (An autonomous Institution affiliated to VTU) IV Semester B. E. Grade Improvement Examinations October 2021
Common to CSE/ISE
GRAPH THEORY, STATISTICS AND PROBABILITY THEORY
Time: 03 Hours Maximum Marks: 100 Instructions to candidates: Answer any FIVE full questions out of TEN. Each carries 20 marks.
1 1 A regular graph with vertices has _________ edges. 01 1 All positive integer values of such that the complete graph isEulerian is _________. 01 1 Find the incidence matrix to represent the graph shown in Fig 1.
Fig 1 01 1 How many different spanning trees does , has? 01 1 Let be a graph with 1000 vertices and 3000 edges. Check if the graph is planar or not. 01 1 Given the regression lines 19. 13 0. 87 ; 11. 64 0. 5 , compute the coefficient of correlation. 01 1 Write the normal equations to fit a curve of the form . 02 1 The first four central moments of a distribution are given as 0 , 7. 058 , 36. 151 & 408. 735 respectively. Find , and hence comment on the skewness and kurtosis of the distribution. 02 1 Give reasons why the graphs shown in Fig 1 are not isomorphic.
Fig 1 02 1 A tree has five vertices of degree 2 , three vertices of degree 3 and four vertices of degree 4. Compute the number of vertices of degree 1 , given max degree is 4. 02 1 Construct ring sum of the graphs shown in Fig 1.
Fig 1 02
1 A random variable has ! 2 "#, 1 , 2 , 3 ... Show that ! is a probability mass function. 02 1 If the probability of a bad reaction from a certain injection is 0. 001 , what is the chance that out of 2000 individuals, more than two will get a bad reaction? 02
2 a Prove that if a graph contains a % & walk of length ', then contains a % & path of length atmost '. 07 b Verify whether the graphs shown in Fig 2b are isomorphic, with proper reasons.
Fig 2b 07 c Show that the graph shown in Fig 2c has no Eulerian circuit but has an Eulerian trail. Give reasons to justify your answer.
Fig 2c 06
3 a Prove that every graph has an even number of odd degree vertices. 06 b Suppose the graphs and ( are respectively ) and . Construct * ( and + * (,. 06 c Check the graphs shown in Fig 3c for isomorphism, by mapping the vertices. Also check for the isomorphism of their complements.
Fig 3c 08
4 a Apply Kruskal’s algorithms to find a minimal spanning tree for the weighted graph shown in Fig 4a.
Fig 4a Mention the steps followed to obtain the spanning tree, the weight of the spanning tree. Draw the minimum spanning tree as well. 08
7 a If ∆ ! is the maximum degree of the vertices of a connected graph , then prove that 4 ! 5 1 ∆ !. 10 b The diagram in Fig 7b shows the traffic lanes at the intersection of two streets. A traffic light is located at this intersection. During a certain phase of traffic light, those vehicles in the lanes for which the light is green may proceed safely through the intersection. What is the minimum number of phases needed for the traffic light so that all vehicles may proceed through the intersection?
Fig 7b Draw the graph which represents the situation and hence obtain the solution. 10
8 a Calculate skewness and kurtosis coefficients from the following grouped data. 6'77 2 4 4 6 6 8 8 10 89:;%:< 3 4 2 1 b Compute the coefficient of correlation and equation of lines of regression for the data shown below. 1 2 3 4 5 6 7 9 8 10 12 11 13 14
9 a In a distribution of the daily consumption of electric power, the following information is available: mean 1 , variance 16 , 1 and = 0. 63. Determine the first four moments about the origin. Comment on the corresponding skewness and kurtosis coefficients. 10 b The average normal daily temperature (in degree Fahrenheit) and the corresponding average monthly precipitation (in inches) for a particular month are shown here for seven randomly selected cities. Fit the linear relationship of the form for precipitation () or temperature ( ) and compute the corresponding correlation coefficient. Also determine the precipitation at a temperature of 70° F. 0&:92: ?@' A:BC:9A%9: 86 81 83 89 80 74 64 0&:92: C9:<@C@AA@D 3. 4 1. 8 3. 5 3. 6 3. 7 1. 5 0. 2 10
10 a A continuous random variable has the density function
E! F
1
4
1 !- 1 5 5
0 DAG:9H@7:
I
Find the cumulative distribution function. 06 b In 800 families with 5 children each, how many families would be expected to have i) 3 boys ii) 5 girls iii) Either 2 or 3 boys iv) At most 2 girls Assume probabilities for boys and girls to be equal. 08 c An electrical firm manufactures light bulbs that have a life before burning out, that is normally distributed with mean equal to 800 GD% and a standard deviation of 40 GD%97. Find the probability that a bulb burns i) Between 778 and 834 GD% ii) More than 834 GD%97, if it has burnt for more than 778 GD%97. Also represent the corresponding normal probability curve. 06
18MA41A
Course: Mathematics
University: Sikkim Manipal University
- More from:MathematicsSikkim Manipal University666 Documents
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