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Section A.
Answer all SIX questions of this section (60 marks in total)
1. Determine the values of parameters 
and 
such that the following system of equations has infinitely many solutions and, using a matrix method, determine all solution when and have these values.
.
2. Given that the function 
is homogeneous of degree 2, determine expressions for 
in terms of .
3. Solve the following differential equation 
4. Find and classify the critical points of the function 
.
5. Determine the following integrals: 
, 
6. Solve the differential equation: 
.
SECTION B
Answer two questions from this section (20 marks each)
7. a) Expand 
in ascending power of x as far as the term in 
. State the range of values of x for which the series is valid.
b) Suppose that consumer demand depends upon the price-trend according to the formula 
. Suppose also that the quantity available is determined in a similar way, by the formula 
. Write down the condition for equilibrium and determine the equilibrium-price path 
if 
and 
. Is the solution stable?
8. a)Find the inverse of the matrix 
b) A house-buyer takes out a loan of amount 
with a bank. The bank’s interest rate is fixed at 5% per annum and payments of 
are made at the end of each year. Let 
be the amount of loan outstanding after t repayments of P have been made.
Explain why 
By solving this difference equation, find .
Suppose that the loan is to be completely repaid after 20 payments. Find an expression for P in terms L.
9. a) A consumer’s ‘utility function’ for goods X and Y is 
where x and y denote the amounts of X and Y consumed. The consumer spends a total of 150 dollars on X and Y, and the prices of each (per unit) are, respectively, 3 and 5 dollars. Find the quantities of X and Y that the consumer buys to maximize the utility function.
b) Solve the following differential equation 
10. a) A sequence of numbers 
is obtained as follows: 
and every other number in the sequence is obtained by multiplying the previous number by 1/2 and adding 1. Find a general expression, in as simple a form as possible, for .
b)Given the objective function 
and the constraint function 
.
Form the Lagrangian function and find point(s) that satisfy the first-order condition.
Classify stationary points using the second-order sufficient condition.
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| Mathematics II | | | Цели, задачи и основные проблемы реализации региональной политики в Российской Федерации |