Evaluate .

OR

Evaluate as the limit of a sum.

Given,

OR

Find the inverse of the matrix , by using elementary row transformations.

Using integration, find the area of the region: {(x, y): 0 ≤ 2y ≤ x² , 0 ≤ y ≤ x, 0 < x < 3}.

Find the vector equation of the line passing through (1, 2, 3) and parallel to each of the planes and . Also, find the point of intersection of the line thus obtained with the plane .

A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of B requires 1 g of silver and 2 g of gold. The company can use at most 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs40 and that of type B Rs50, find the number of units of each type that the company should produce to maximize the profit. Formulate and solve graphically the LPP and find the maximum profit.