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

Show that the relation R on the set Z of all integers defined by (x, y) ∈ R ⇔ (x – y) is divisible by 3 is an equivalence relation.

OR

A binary operation on the set A = {0, 1, 2, 3, 4, 5} is defined as

.

Write the operation table for a * b in A.

Show that zero is the identity for this operation * and each element ‘a’ ≠ 0 of the set is invertible with 6 – a, being the inverse of ‘a’.

Given,

OR

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

Evaluate .

OR

Evaluate as the limit of a sum.

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 .