Mathematics

(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, (ex^1/x+1), if it exists.

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

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?

Determine the median wage graphically from the following data:

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

Is f a well defined function?

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

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

In a group of 1000 wage earners the monthly wages of 4% are below Rs.60 and those of 15% are under Rs.62.50, 15% earned Rs.95 and over, and 5% get Rs.100 and over. Find the median wage.

 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:

(b) Show that the graphs are isomorphic.

(a) Show that 

(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 =

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.

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

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

Show that it is a lattice

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.

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

When is mode preferred over other form of average?

For the following data of the frequency of visit of customers in a shop in the month of April , year 2017 starting from 1st to 30th (row wise) 

(1) Draw a frequency table and find the dates on which customer are more frequent.

(2) Find the average number of visit in this month.

(3)Which of the week showing more favorable visit for the customers? 

For what value of x, the matrix

 is singular.

Show that function is maximum when x = 

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

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

 Compute quartile deviation from the following data:

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

(a) Let f:  be defined by 

(b) Show that  logically implies 

(a) Express  in factorial form and hence evaluate 

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?

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?

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

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

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.

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

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

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

Give the table of values

Evaluate  using Lagrange's interpolation and estimate the error.

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

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

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

 Are (i) and (ii) tautologies?

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

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 

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.

(a) Maximize subject to

Using the Simplex method

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.

The marks obtained by 9 students in a test are 25,20,15,45,18,7,10,38 and 12. Find the value of Q1, Q3, I.R and Q.D.

A consulting firm is preparing a study on consumer behavior. The company the following data in thousand rupees to determine whether there is a relationship between consumer income and consumption levels:

Calculate correlation coefficient for the above data. Write your comments about the correlation coefficient' value

A company has 4 machines to be assigned to 4 of the 5 workers available for the purpose. The expected production from each machine operated by each worker is given below: 

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

Using properties without expanding prove that:

Show that  is continuous at x = 2.

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

Verify Cayley Hamilton theorem for the matrix  Also,  find 

Find eigen values and eigen vectors of the matrix 

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). 

Evaluate lim 

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

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

 Discuss the nature of discontinuity at x = 0 of     

Find all the asymptotes of y⁴ - 2xy³ +2x³y -x⁴ -3x³ + 3x²y  + 3xy² - 3y³ - 2x² + 2y² - 1 = 0

If  and 

If y =  x then show that

If y =  x then show that

Examine the function for maxima/minima 

Evaluate

i) 

Evaluate

ii) 

Show that

Evaluate 

Obtain the reduction formula for  .Also  evaluate 

Define

(i) ogives

Define

(ii) histogram

Define

(iii) relation between mean, median and mode.

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 .

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? 

Obtain the dual of the following Linear Programming Problem:

Maximize z = 7x₁ + 5x₂

Subject to constraints:

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.

 Calculate the correlation coefficient from the following data:

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.

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

Two random variables have the regression equations

3x+2y-26 = 0 L

6x+y-31 = O

Calculate

(i) Mean value of X and Y.

Two random variables have the regression equations

3x+2y-26 = 0 L

6x+y-31 = O

Calculate

(ii) The coefficient of Correlation 

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.

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

Min z = 8x₁ + 9x₂ 

Sub to Constraints:

Solve the following linear programming problem by Graphical Method:

Max 5x₁ - 4x₂ 

Subject to constraints:

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 

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.

 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

 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

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

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

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 ..

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

Define isomorphic and Hamilton Graphs with example.

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.

Define Tautology and contradictions,

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

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

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.

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

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

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}.

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

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

In a lattice, prove that

In a lattice, prove that 

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

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

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.

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

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

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

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

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

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

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

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

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

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

(b) to show that 

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

(d) Define Tautology and Contradiction.

(e) By means of truth table, prove that 

(f) If    , then find for all 

(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). 

(h) Define Distributed & Complemented Lattice.

(f) Define POSET with example.

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

(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) 

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

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

(i) 

(ii) 

(iii) 

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

(a) Find the complements of each element in D₄₂

(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). 

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

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

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

(a) Prove that    is a tautology. 

(a) Prove that 

(b) Prove that 

(b) Estimate the missing term in the following table

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

find the value of k. 

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

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

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

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

Also interpolate at x=1.5 

(h) Events A and B are such that

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

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

(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. 

(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. 

(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.

(b) For what value of n is 

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

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

(a) Find   and variance for the following distribution.

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

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

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

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

(a) Solve by Jacobi's Method

4x + y + 32 = 17

x + 5y + z = 14

2x-y + 8z = 12 

(b) Given that

Find  and  at  and 

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

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

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

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

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

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

Prove that for any two events A and B

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

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?

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.)?

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.

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?

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

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.

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

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

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.

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?

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

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

Consider the system of equations:

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

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.

Solve the system of equations:

Using LU decomposition method.

Solve the system of equations:

Using LU decomposition method.

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

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

(b) Find the rank of the matrix A = 

(a) Discuss the continuity of the function

f(x) = 

(b) Solve 

(a) Discuss the continuity of the function

(b) Evaluate 

(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). 

a if y =

and calculate yn(O).

blind all the asymptotes of the curve

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

(a) Prove that(m,n) = 

(b) (1)Evaluate                     (ii) Evaluare 

(a} If I_n =  xdx ,show that 

(b) Evaluate 

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

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

Let   

Find

Show that the graph G₁ =( V_1'E₁ ) and G₂=(V_2,E₂) , are isomorphic 

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 

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.

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

b) Prove the following

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.

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.

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.

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

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

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

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

Define symmetric and skew-symmetric matrix with an example. Express

as a sum of symmetric and skew-symmetric matrices. 

Find the value of 'a' if

If

find 

. find the derivative 

Prove that  where denotes Gamma function

Solve the following system of equations using Cramer's Rule ;

5x-7y+z = 11

6x-8y-z = 15

3x+2y-6z= 7

Find the characteristic roots and characteristic vectors of the matrix.

Test the consistency of the given system of linear equations using rank of matrices 

x + y + z = 6

x + 2y + 3z = 14

x + 4y + 72 = 30

Find the characteristic equation of the matrix :

Evaluate :

Discuss the nature of discontinuity of :

Show that the function

is discontinuous at x -0.

Evaluate

(i)

Evaluate

(ii)

Find the maximum value of f(x) =

Find all the asymptotes of the curve

Verify Rolle's theorem for f(x) =

Evaluate :

(i) 

(ii) 

Integrate

(i)

Integrate

(ii)

Express  dx as beta function.

Obtain the reduction formula for . Hence evaluate

Evaluate  x dx as the limit of sum. 

How would you account for the predominant choice of A.M. as a measure of central tendency? Under what circumstances would it be How would you account for the predominant choice of A.M. as a measure of central tendency? Under what circumstances would it be appropriate to use mode or median?

Define (i) ogives, (ii) histogram (iii) Lorenz curve.

Define (i) Transportation problem (i) LPP (iii) Stepping-Stone method

Coefficient of variation of two series is 60% and 80%. Their standard deviations are 24 and 20. What are their arithmetic means?

(b) Prove that the Standard deviation is independent of any change of origin but is dependent on the change of scale.

Find the standard deviation and coefficient of variation from the following data:

Given standard deviation with respect to data X is 3 and regression equation are

4X-5Y+33=0,

20X-9Y-107=0,

Find

(a) A.M. of data X

(b) A.M. of data Y

(c) Standard deviation of data Y

(d) Correlation coefficient 

From the following table, find correlation coefficient between age and playing habit of students:

Define Regression. Why are there two regression lines? Under what conditions can there be only one regression line? 

The manager of an oil refinery must decide on the optimal mix of two possible blending processes of which the inputs and outputs per production run are as follows:

The maximum amounts available of crude A and B are 200 units and gasoline X and 80 units of gasoline Y must be produced. The product per production run from process 1 and 2 is Rs.300 and Rs.400 respectively. Solve the above Linear Programming Problem by GRAPHICAL method. 

Minimize z=2x1+3x2

Subject to: X1-2x2<=0

-2x1+3x2=>-6

X1, X2 unrestricted.

 07. Maximize Z = 4x1 + x2 + 3x3 + 5x4 

Given below is a transportation table taken from the solution process for a transportation problem:

Answer the following questions, giving brief answers:

(1) is this solution feasible?

(2) Is this solution degenerate?

(3) Is this solution optimal? If not, find the optimal solution.

(4) Does the problem have alternative optimal solution? If yes, give another optimal solution.