Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.

Consider the functions

I. $${e^{ - x}}$$
II. $${x^2} - \sin x$$
III. $$\sqrt {{x^3} + 1} $$

Which of the above functions is/are increasing everywhere in [0,1]?

Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.

Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,

I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) $$ \le $$ rank(A) + rank(B)
IV. det(A + B) $$ \le $$ det(A) + det(B)

Which of the above statements are TRUE?