Examine the following functions for maxima and minima:
a) 
b) 
7) Write short notes on following:
a) Maximin or Minimax Principle
b) Riemann Sum
c) Russell's Paradox
d) Chi square test of goodness of fit

If P (A  B  C) = 0, show that P [(A  B)/ C] = P(A/C) + P (B/C)
5) a) Discuss the nature of the following time path: 
b) Solve equation:  for 

Ram and Rahim play for a prize of Rs. 99. The prize is to be won by the player who first throws a 3 with a single die. Ram throws first and if he fails Rahim throws it and if Rahim fails Ram throws it again and this process continues. Find their respective expectations.

Using Simplex method solve the problem:
Maximise z = 4x + 8y + 2k
Subject to:

Also, minimize the same for z

 Given the input matrix and the final demand vector:

A = [[0.10, 0.15, 0.12],
[0.20, 0, 0.30],
[0.25, 0.40, 0.20]]

d = [[100],
[200],
[300]]

a) Explain the economic meaning of the elements 0.30,0 and 200
b) Explain the economic meaning of (if any) of the third column sum
c) Explain the economic meaning of (if any) of the third row sum
d) Write out the specific input-output matrix equation for this model
e) Find the solution output levels of the three industries using Cramer's rule.

Given the input matrix and the final demand vector:

A = [[0.10, 0.15, 0.12],
[0.20, 0, 0.30],
[0.25, 0.40, 0.20]]

d = [[100],
[200],
[300]]

Consider the following utility maximization problem, in which the utility function of the consumer is given by U(x,y)=x^0.7y^0.3. The consumer income is I=200 and p x=3 and py=2. Find the utility maximizing x and y when the budget constraint of the consumer may or may not bind. (Apply Kuhn-Tucker method)

 Explain the theorem of second order optimum.

List the different types of international financial markets. Highlight the importance of international financial markets and instruments.

What is capital account convertibility (CAC)? How are capital account and current account convertibility different?

Explore the effects of tariff on volume of trade and terms of trade for a small country and a large country. Explain with the help of a diagram.

Examine the impact of environmental standards and logistics cost on the location of the production and international trade.

How would you apply technological gap model to explain the pattern of international trade? Explain.

What factors prompted the development of intra-industry trade? Explain. How does this theory contribute to our understanding of international trade?

Using the Ricardian theory of international trade as a framework, illustrate how nations derive benefits from engaging in trade.

Why is 'free trade' between two countries is considered to be better than no trade? Explain with the help of a diagram.

Snowball sampling iv. Content analysis

Canonical co-relation

Maximum Likelihood method as method of estimation

What types of data will you need to assess the performance of Indian economy? Explain the various sources of such data.