Ques 1.

Consider the following two matrices.

(i) Find the rank of ‘A’ and ‘B’
(ii) Show that (AB)^-1 = B^-1 A ^-1
(iii) Show that (A^-1)^-1= A
(iv) Show that (B^-1)^-1= B

Ques 2.

An individual consumer consumes two commodities X1 & X2. The utility function is 

The price of commodity one is P1 = Rs.3.00, the price of commodity two is P2 = Rs.4.00, the individual’s income per period is Rs.108. Determine the utility maximizing level of X1 & X2 and derive the demand curves for the two commodities.

Ques 3.

Let Z = f(x,y) = 3x³ -5y² -225x + 70y + 23.

(i) Find the stationary points of z.
(ii) Determine if at these points the function is at a relative maximum, relative
minimum, infixion point, or saddle point.

Ques 4.

Solve the following differential equation

given Y(0) = 4

Ques 5.

If Z = f(x,y) = xy

Find the maximum value for f(x,y) if x and y are constrained to sum to 1 (That is, x +y = 1). Solve the problem in two ways: by substitution and by using the Lagrangian multiplier method.

Ques 6.

Adjugate of a matrix

Ques 7.

Decomposable matrix

Ques 8.

Singular matrix

Ques 9.

Evaluate  

Ques 10.

Explain the concept of maximum value function.

Ques 11.

Let the production function by a b Q AL^a K^b  . Find the elasticity of production with respect to labour (L).

Ques 12.

Denote by a, b and c the column vectors

Calculate 2a - 5b, 2a- 5b +c, a ^/ .b,