Suppose we wish to estimate the total number of Cows in 2026 in certain state. The total number of cows for 2024 was X = 5000. The Sampling unit was the farm, and it is assumed that there has been no change in the number of farms which we shall assume to be N=500. A sample of n=20 farms is selected, and the data is as follows:

Estimate the average number of cows for 2026 by Ratio Method of estimation and obtain the estimate of the MSE of Ratio estimator of Population Mean

State whether the following statements are true or false and also give the reason in support of your answer:

a) If X and Y are uncorrelated, the V(Rag) reduces to that of Variance of Ratio Estimator of Population mean

b) Among all the members of the General Class of the estimators only Regression estimator is an unbiased estimator of the Population mean.

c) An Incomplete Block Design with the following parameters b=8,k=3,9=8,r=3 is found to be a Balanced Incomplete Block Design.

d) A One-half fractional factorial design of a 2ª full factorial design will be denoted by a 22 Factorial design.

e) The Cluster estimator ỹ, becomes an unbiased estimator of population mean only if M and , are uncorrelated

Explain the terms PATH, CIRCUIT and CYCLES in context of Graphs

Explain and prove the Handshaking Theorem, with suitable example

Determine whether the above graph has a Hamiltonian circuit. If it has, find such a circuit. If it does not have, justify it.

What is a chromatic number of a graph? What is a chromatic number of the following graph?

Prove that the complement of  is G

Find the solution of the recurrences relation 

Find an explicit recurrence relation for minimum number of moves in which the n-disks in tower of Hanoi puzzle can be solved! Also solve the obtained recurrence relation through an iterative method.

Explain inclusion-exclusion principle and Pigeon Hole Principle with example.

What is the probability that a number between 1 and 10,000 is divisible by neither 2, 3, 5 nor 7?

How many words can be formed using letter of DEPARTMENT using each letter at most once?

i) If each letter must be used,

ii) If some or all the letters may be omitted.

There are three Companies, C1, C2 and C3. The party C1 has 4 members, C2 has 5 members and C3 has 6 members in an assembly. Suppose we want to select two persons, both from the same Company, to become president and vice president. In how many ways can this be done?

What is equivalence relation? Explain use of equivalence relation with the help of an example.

Explain Decidable and Undecidable Problems. Give example for each.

If L1 and L2 are context free languages then, prove that L1 U L2 is a context free language.

Write the finite automata corresponding to the regular expression (a + b)*ab

Explain whether function: f(x) = x² posses an inverse function or not.

Draw logic circuit for the following Boolean Expression:

Write the following statements in the symbolic form.

i) Some students can not appear in exam.

ii) Everyone can not sing.