Find , if

Does the following trigonometric equation have any solutions? If Yes, obtain the solution(s):

OR

Determine whether the operation * define below on Q is binary operation or not.

a * b = ab+1

If yes, check the commutative and the associative properties. Also check the existence of identity element and the inverse of all elements in.

Find the value of x, y and z, if A =satisfies A’ = A¹

OR

Verify: A(adjA) = (adjA)A = |A|I for matrix A =

Find the shortest distance between the line x - y + 1 = 0 and the curve y² = x.

Define skew lines. Using only vector approach, find the shortest distance between the following two skew lines: