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10

Let A={1,2,3} and R={(1,1), (1,2), (2,1), (2,2), (3,3)} Show that R is equivalence relation.

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7

From a pack of 52 cards, two are drawn at random. What is the probability that one is a king and the other a queen?

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3

a) Is H a subgraph of I shown in the following Figure? Explain it.


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2

(b) Show that the graphs are isomorphic.

              

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2

Find all paths of length 2 in the following graph G.


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2

Check the validity of the argument:

Dr. Das buys a car, then he can go home in time. If he goes in time, then his family will be happy.

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2

b) Define bipartite graphs. Verify whether following graph is bipartite or not.

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1

For what value of x, the matrix

 is singular.

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1

Show that function is maximum when x = 

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1

(a) Show that the relation  is partial order relation on the set of natural numbers. (Where  means less than or equal to). 

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1

(a) Let f:  be defined by 

   

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1

(b) Show that  logically implies 

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1

(a) Show that 

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1

(a) Express  in factorial form and hence evaluate 

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1

A grade school boy has 5 blue and 4 white marbles in his left pocket and 4 blue and 5 white marbles in his right pocket. If he transfer one marble from his left pocket to right pocket, what is the probability of bis then drawing a blue marble from his right pocket? From a pack of 52 cards, two are drawn at random. What is the probability that one is a king and the other a queen?

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1

A can solve 75% of the problems and 3 can solve 70 of the problems. What is the probability that either A or B can solve problem chosen at random?

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1

A fire insurance company has 3,840 policyholders. If the probability is 1/1200 that any one of the policyholders will file atleast one claim in any given year. Find the mean of the distribution.

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1

(a) Evaluate the determinant of the matrix

(b) Use Cramer's rule to sole the system of equations

x+y+z+ 1 = 0; ax + by + cz + d = 0; a2x+ b2y + c2 z+ d2 = 0

(c) Find the maximum value of y=(1/x)2

(d) Evaluate 

(e) Evaluate lim, (ex1/x+1), if it exists.

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1

 For all n>=1, let a recursive function f(n) be defined as

Is f a well defined function?

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1

If p(x) and Q(x) are propositions, then prove that


 Are (i) and (ii) tautologies?

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1

Consider a poset {(1,2,4,5,10,201.)} as described by Hasse diagram in the following 


Show that it is a lattice

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1

B) Let R be the set of real niamber and let  is invertible ?

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1

For the data X and Y given below:


(i)Find Spearman's rank correlation coefficient.

(ii)Find the regression line

(iii)Find the coefficient of determination for the regression line fit.


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1
Write short notes on PERT VS. CPM This question has 0 answers so far.
0
A bag contains four white and two black balls and a second bag contains three of each colour A bag is selected at random and a ball is then drawn at random from the bag choses. What is the probability that the ball drawn is white? This question has 0 answers so far.
0
Define normal distribution and hypergoometric distribution. This question has 0 answers so far.
0

 Prove that every square matrix is uniquely expressible as the sum of asymmetric matrix and a skew symmetric matrix. 

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0

Using properties without expanding prove that:


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0

Show that  is continuous at x = 2.

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0

 If the matrix is orthogonal, then find the values of a, b and c where matrix is 

        

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0

Verify Cayley Hamilton theorem for the matrix  Also,  find 

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0

Find eigen values and eigen vectors of the matrix 


 

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0

Examine the following system of vectors for linearly dependence. If dependent, find the relation between them

X1 = (1,-1, 1); X2 = (2, 1, 1); X3 = (3,0,2). 

 

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0

Evaluate lim 

 

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0

For what choice of 'a' and 'b' is the function continuous 

 

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0

For what value of  does the lim f(x) exists, where f(x) exists, where f is defined by the rule 

 

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0

 Discuss the nature of discontinuity at x = 0 of     

 

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0

Find all the asymptotes of y4 - 2xy3 +2x3y -x4 -3x+ 3x2y  + 3xy2 - 3y3 - 2x2 + 2y2 - 1 = 0

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0

If  and 

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0

If y =  x then show that




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If y =  x then show that




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Examine the function for maxima/minima 


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0

Evaluate

i) 

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Evaluate

ii) 

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Show that


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Evaluate 

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0

Obtain the reduction formula for  .Also  evaluate 

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0

Define

(i) ogives

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Define

(ii) histogram

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Define

(iii) relation between mean, median and mode.

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Karl Pearson's coefficient of skewness of a distribution is 0.32, its standard deviation is 6.5 and mean is 29 6, find the mode of the distribution .

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0

A contractor employs three types of workers - male, female and children. To a male worker he pays Rs. 200 per day to a female worker Rs. 150 per day and to a child worker Rs. 85 per day. What is the average wage per day paid by the contractor? 

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0

Find the two lines of regression and coefficient of correlation for the data given below: 

  


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0

Obtain the dual of the following Linear Programming Problem:

Maximize z = 7x+ 5x2

Subject to constraints:

                                  

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0

Find the missing frequencies for the class intervals using the following data of 150 students given that the mean marks is 65.

              

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 Compute quartile deviation from the following data:

         

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0

 Suppose that samples of Polythene bags from two Manufacture, A and B are tested by a Prospective buyer for bursting pressure, with the following results:

           

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0

The mean of 5 observations is 4,4 and variance is 8.24. If three of the five observation are 1, 2 and 6. Find the other two.

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0

 Calculate the correlation coefficient from the following data:

     


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0

Ten Participants in a Context are ranked by three judges as follows:

  

Calculate the rank correlation coefficient and also discuss which pair of judges has similar taste.

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0

Calculate (i) two regression lines,(ii) two regression coefficient and(iii) coefficient of correlation from the following data:-

   

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0

Two random variables have the regression equations

3x+2y-26 = 0 L

6x+y-31 = O

Calculate

(i) Mean value of X and Y.

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0

Two random variables have the regression equations

3x+2y-26 = 0 L

6x+y-31 = O

Calculate

(ii) The coefficient of Correlation 

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0

Two random variables have the regression equations

3x+2y-26 = 0 L

6x+y-31 = O

Calculate

(iii)If var(x) = 25, find standard deviation of x from the data given above.

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0

Write the dual of the following primal problems and also find its optimal solution using simplex method

Min z = 8x1 + 9x

Sub to Constraints:

                      

                               

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0

Solve the following linear programming problem by Graphical Method:

Max 5x1 - 4x2 

Subject to constraints:

                                 

   

                    

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0

A branch of PNB has only one typist. Since the typing work varies in length, the mean service rate 8 letters/hr. The letters arrive at a rate of 5/hr during the entire 8 hr work day. If the type write is valued 1.50/- Rs. per hour, determine: 

(i) The equipment utilization 

                    

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0

The average system time A branch of PNB has only one typist. Since the typing work varies in length, the mean service rate 8 letters/hr. The letters arrive at a rate of 5/hr during the entire 8 hr work day. If the type write is valued 1.50/- Rs. per hour, determine:

ii) The Percent time that an arriving letter has to wait.


                    

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0

 The The average system time A branch of PNB has only one typist. Since the typing work varies in length, the mean service rate 8 letters/hr. The letters arrive at a rate of 5/hr during the entire 8 hr work day. If the type write is valued 1.50/- Rs. per hour, determine:

(iii) The average system time


                    

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0

 The The average system time A branch of PNB has only one typist. Since the typing work varies in length, the mean service rate 8 letters/hr. The letters arrive at a rate of 5/hr during the entire 8 hr work day. If the type write is valued 1.50/- Rs. per hour, determine:

(iv) The average cost due to waiting on the part of the type writer

                    

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0

Given the following data, determine the least cost allocation of the available machines to five jobs

 

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Solve the following transportation problem and test for optimality to find optimal solution by MODI method

    

 

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0

Let A = {1, 2, 3) and B = {a, b, c). Let R = {(1.b)(1.c)(3.b)}. Find the domain and range of the relation. Determine ..

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Let D denote the set of all positive divisors of the positive integer n. Determine  and represent it by Hasse Diagram.

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Define isomorphic and Hamilton Graphs with example.

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Let f, g, be functions from N to N (set of natural numbers) for such that f(n) = n + 1, g(n) = 2n. Find fog and gof.

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Define Tautology and contradictions,

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Show that the relation of parallel lines in the set of lines on a plane is an equivalence relation.

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Choose any two statements p and q as you like. Draw the truth table for and

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Consider the graph G (V, E) where v consists of Four vertices A, B, C,D and E of five edges where represent this undirected graph diagrammatically. Determine the degree of each vertex.

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Let f be a mapping from R to R such that f(x) = 2x + 3. Show that f is invertible and find its inverse.

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.If n (A) = 40, n(B) = 30, = 20. Then find

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0

Let A = {1,2,5,6), B = {2,5,7), C = (1,3,5,7,9).
Verify (A x B)  {A X C} = A x B x C}.

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0

Let N = {1,2,3,........), denote the set of all positive integers and A = {, 3 < x < 12), B = {, x is even, x < 15). Find

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If R is an equivalence relation in a set A. Then prove that is also equivalence relation.

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In a lattice, prove that

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In a lattice, prove that 

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Define complemented lattice, also find the complement (if exists) of all elements of

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Let A = {1,2,3,4,6,8,9, 12, 18, 24} be equipped with relation x divides y.Draw the Hasse diagram.

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Let G be an undirected graph with m vertices, sayDefine the adjacent matrix A of G. Consider the undirected graph G with 5 vertices shown in the following diagram.

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Draw the directed graph for the following incident matrix. Also find the degree of all vertex. 

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Show that the two graphs shown in the figure are Isomorphic.

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Show that the two graphs shown in the figure are Isomorphic.

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Prove that the union of two graphs  will be a graph such that.

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Prove that the union of two graphs  will be a graph such that.

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By means of truth tables, justify that the conditional statement "If p then q" is logically equivalent to the statement "Not p or q".

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Define a proposition. Let p and q be propositions and denote compound proposition, "if p then q" Draw the truth table for the compound proposition
Let p: you try, and q: you will succeed. Justify the truth table for

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Verify De-morgan's laws for propositions. And also prove that.

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Consider the following:
P: Today is Tuesday, Q: It is raining, R: It is cold.
Write in simple sentence the meaning of

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Consider the following:
P: Today is Tuesday, Q: It is raining, R: It is cold.
Write in simple sentence the meaning of .

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0

(b) to show that 

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(c) Find the domain and range of the function f(x) = 

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(d) Define Tautology and Contradiction.

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(e) By means of truth table, prove that 

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(f) If    , then find for all 

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(g) Verify De-morgan's laws for universal set U = {1, 2, 3, 4, 5, 6, 7),A = {4, 1, 2, 5) and B = {1, 2, 4, 6). 

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(h) Define Distributed & Complemented Lattice.

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(f) Define POSET with example.

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(a) R is an equivalence relation in a set A, then prove that  is also and quivalence relation. 

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(b)  Let U = {a,b,c,d,e), A = {a,b,d} and B = {b,d,e). Find (i)  (ii)  (iii) B-A  (iv)  (v) 

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(b) Define the relation congruence modulo m." Show that it is and quivalence relation in set of integers. 

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(a) Let   be a lattice, then prove that for every element a,b 

(i) 

(ii) 

(iii) 

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(b) Prove that every finite lattice L is bounded.

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(a) Find the complements of each element in D42

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(b) Let R be the relation of divisibility on set A. Draw the Hasse diagram of poset relation R where A = 2, 4, 8, 16, 32). 

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(a) Find the chromatic number of the graph given below using the Welch-Powell algorithm. 

                             

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(a) Draw a 3-regular graph with 6 vertices.

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(b) Prove that the degree of any vertex in a simple graph of n vertices cannot exceed n - 1. 

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(a) Prove that    is a tautology. 

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(a) Prove that 

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(b) Prove that 

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(b) Estimate the missing term in the following table

 

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(c) A random variable x has the following probability function 


find the value of k. 


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(d) If a random variable has a Poisson distribution such that P(1) - P(2),find the mean of the distribution. 

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(e) For a normally distributed variable with mean 1 and S.D 3, find the probability that 

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(f) If X and Y are independent binomial variables such that X-b(5, 1/2),  (7, 1/2). Find P [X+Y=3). 

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(g) Find the Lagrange interpolating polynomial that fit the following data values.

  

Also interpolate at x=1.5 

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(h) Events A and B are such that


    

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(i) The expected value of a random variable x is 2 and its variance is 1.Find the variance of 3x+4. 

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0

(i) if two regression coefficients are 0.8 and 0.2. What would be the value of coefficient of correlation? 

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0

(a) A bag containts 40 tickets numbered 1,2,3,.,40 of which four are drawn at random and arranged in ascending order  Find the probability of t3 being 25. 

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0

(b) The probability that Nirmal will solve a problem is 2/3 and the probability that Satyajit will solve it is  What is the probability that (i) the problem will be solved (b) neither can solve it. 

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0

(a) The content of three urns are: 1 white, 2 red, 3 green balls; 2 white, 1 red, 1 green balls and 4 white, 5 red, 3 green balls. Two balls are drawn from an urn chosen at random. These are found to be one white and one green. Find the probability that the balls so drawn come from the third urn.

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(b) For what value of n is 

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(a) of the moment generating function of normal distribution is 

       

         

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(b) Calculate the coefficient of rank correlation from the following. 

          

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0

(a) Find   and variance for the following distribution.

  

        

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(b) From the given data obtain the two regression equations using the method of least square.

              

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(b) Using Newton-Raphson method evaluate to two decimal figures, the root of the equation ex = 3x, lying between 0 and 1. 

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0

(a) Use Newton's interpolating formulae to find y when x =1.85 and x=2.4 from the data. 

     

       

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(b) Find a root of x3-5x+3 = 0, by Newton-Raphson method.

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0

(a) Solve by Jacobi's Method

4x + y + 32 = 17

x + 5y + z = 14

2x-y + 8z = 12 

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(b) Given that

  

Find  and  at  and 

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0

(a) Apply LU Method to solve the equations 3x+2y+7z =4,  2x+3y+z=5,  3x+4y+z=7 

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(b) Calculate the value of  by simpson' s 1/3 rule using 1 1 ordinals. 

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A coin is tossed three times. What is the probability of occurrence of head and tail alternately?

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Find the value of so that the following function is a probability density function 

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Find the value of so that the following function is a probability density function 

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For any positive integers r and n>r, prove that 

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Using Lagrange's interpolation find the interpolation polynomial for the function y(x) for the given values below.

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Find two successive approximation of a real roots of the equation by using Bisection method.

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Prove that for any two events A and B

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A coin is tossed until a a head appears. What is the expectation of the number of tosses?

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A coin is tossed until a a head appears. What is the expectation of the number of tosses?

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A perfect cube die is a large number of times in a set of 8. The occurrence of 5 or 6 is called a success. What is probability of exactly 3 success?

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Cut of (2n+1) tickets consecutively numbered, three are draws at random. What is the probability that the numbers on them are in arithmetic progression (A. P.)?

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Find the probability that at most 5 defective fuses will be found in a box of 200 fuses if experience show that 2% of such fuses are defective.

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The four attendants A, B, C and D of a company are supposed to wash the windshield of each customer's car. A, who service 20% of all cars, fails to wash the windshield one time in 20; B, who services 60% of all cars, fails to wash the windshield one time in 10: C, who services 15% of all cars, fails to wash the windshield one time in 10, and D, who services 5% of all cars, fails to wash the windshield one time in 20. If a customer complains later that his windshield was not washed, what is the probability that his car was serviced by A?

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Out of 3n consecutive number 3 numbers are selected at random. What is the probability that their sum is divisible by 3?

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Prove that the Binomial distribution reduces to Poisson distribution under the conditions that the number of trial n is very large and the probability of success p is small such that , a constant.

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If X is normally distributed with mean 2 and variance. Find 

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Find the moment generating function of Binomial distribution of random variable X. Hence find its mean and variance.

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It has been claimed that in 60% of all solar heat installations utility bill reduced by atleast one-third. Accordingly, what are the probability that the utility bill will be reduced by at least one-third in four of five installation.

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It has been claimed that in 60% of all solar heat installations utility bill reduced by atleast one-third. Accordingly, what are the probability that the utility bill will be reduced by at least four of five installation?

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Find a real root of the equation correct to 3 decimal places using method of False position.

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0

From the following table of value of x and f(x).

Determine the value of (0.23) and (0.29).

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Find a real root of the equation using Newton-Raphson method with an accuracy of 10^-3.

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Give the table of values

Evaluate  using Lagrange's interpolation and estimate the error.

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Consider the system of equations:

Find the inverse of the coefficient matrix of the above system of equations by Gauss Jordan method.

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A rocket is launched from the ground. Its acceleration a(t) measured in every 5 seconds is tabulated below.

Find the velocity of the rocket at t = 30 second using Simpson's 1/3 rule.

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Solve the system of equations:

Using LU decomposition method.

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Solve the system of equations:

Using LU decomposition method.

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From the following table find the value of dy/dx and d^2y/dx^2 at the point x=1.5.

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From the following table find the value of dy/dx and d^2y/dx^2 at the point x=1.5.

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0

(a) Show that the vectors

x1 =(1, 2, 4),     x2 = (2,-1,3),      x3 = (0,1,2) and      x4= (-3,7,2) dependent and find the relation between them.

(b) Find the eigen values and eigen vectors of A =

 

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(a) Given A = find adj (A) by using Cayley-Hamilton  theorem.

(b) Find the rank of the matrix A = 

 

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(a) Discuss the continuity of the function

f(x) = 

(b) Solve 

 

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(a) Discuss the continuity of the function

(b) Evaluate 

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(a) Verify Lagrange's Mean value Theorem for


(b) Expand logx in powers of (x-1) by Taylor's theorem and hence find the value of loge (1.1). 

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0

a if y =

and calculate yn(O).

blind all the asymptotes of the curve

y3 + 4xy2 + 4x2 y + 5y2 + 15xy + 10x2 - 2x + 1 = 0 

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(a) Prove that(m,n) = 

(b) (1)Evaluate                     (ii) Evaluare 

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(a} If I=  xdx ,show that 

(b) Evaluate 

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Let A= (1,2,3.9.181}. Consider the partial order of divisibility on A. Draw the Hasse diagram of the poset (A, <=)

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Construct the truth table to determine whether each of the following is a tautology, or an absurdity?



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Show that Pvq = qvp, - (Pvq) = ~p^ ~q


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Let   

Find


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Show that the graph G=( V1'E1 ) and G2=(V2,E2) , are isomorphic 


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b) Let   be a lattice shown in the following figure where L - (a,b,c,d,e,f,g).  Define an algebraic system on A 


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a) IN Q 6a, consider the following subset of Lattice

i) S-2.4.10.20)

ii) S-14,5,10,20,)

iii) S 11.2.5,10)

Determine whether the above subset of the lattice is a sublattice.

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b) Define distributed and complemented lattices  In Q 6a, find the component the element 4, Is the lattice given in Q6a complemented lattice.

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b) Prove the following


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a) Let A - B-R the set of real numbers. : Let  be given by the formula f(x)-4x5-1 and let g - be given by  Show that f is bijection and g is also bejection.

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The following data which is the number of tonnes shipped weekly across the Pacific by a shipping company.

398, 412, 560, 476, 544, 690, 587, 600, 613, 457, 504, 477, 530, 641, 359, 566, 452, 633, 474, 499, 580, 606, 344, 455, 505, 396, 347, 441, 390, 632, 400, 582.

Assume these data represent an entire population. Find the population mean and the population standard deviation.

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The following data are numbers of colour television sets manufactured per day at a given plant: 15, 16, 18, 19, 14, 1-2, 22, 23, 25, 20, 32, 17, 34, 25, 40, 41. Draw a frequency polygon and an ogive for these data.

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Given the set of data 4, 8, 9, 8, 6, 5, 7, 5, 8, find each of the following sample statistics:

(i) Mean

(ii) Median

(iii)Mode

(iv) Midrange 

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The following table shows the average weights for given heights in a population of men. 


(a) The relationship between the variables is modelled by the regression equation y-ax+b. Write down the value of a and of b.

(b) Use this relationship to estimate the weight of a man whose height is 177 cm.

(c) Find the correlation coefficient.

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Correlation and Causality are one and the same. Critically comment.

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(a) Maximize subject to


Using the Simplex method

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Explain the concept of duality in context to linear programming problems.

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Define and explain (1) pessimistic time (2) Optimistic time

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The project activities, precedence relationships and durations are described in the table. The critical path of the project is


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A trucking company has a contract to move 115 truckloads of sand per week between three sand-washing plants W,X and Y, and three destinations, A,B and C.

Cost and volume information is given below. Compute the optimal transportation cost and the transportation plan



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A head of department has four lecturers to assign to pure maths (1),mechanics (2), statistics (3) and Quantitative techniques (4). All of the teachers have taught the courses in the past and have been evaluated with a score from 0 to 100. The scores are shown in the table below. 


The head of department wishes to know the optimal assignment of teachers to courses that will maximize the overall total score. Give the optimal assignment of teachers and courses / papers as well as the optimal score.

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