AJAY PATHAK (MATH)

  MCQ on ring theory  1.  (Z,+,•) is ? (1 point)       ◯ field       ◯ Ring       ◯ integral domain       ◯ Ring and integral domain 2.  A integral domain is field if________ (1 point)       ◯ finite       ◯ infinit 3.  If in a ring with unit (xy)²=x²y² ∀x,y∈R,then, (1 point)       ◯ R is commutative ring       ◯ R is an integral domain       ◯ R is a field       ◯ Not of this 4.  Which of following is not a ring? (1 point)       ◯ (N,+,•)       ◯ (Z,+,•)       ◯ (Q,+,•)       ◯ (R,+,•) 5.  In a ring (R,+,•) which of following is correct? (1 point)       ◯ (a+b)²⁼a²+ab+ba+b²       ◯ (a+b)(a-b)=a²-ab+ba-b²       ◯ Both (a) and (b) correct       ◯ Both (a)&(b) are not correct  ...

  MCQ on Riemann integral Part 1:- 1.  If P is common refinement of P₁ and P₂ then  (1 point)       ◯ P₁ ∪P₂⊂P       ◯ P₁ ∪P₂⊂P and P⊂P₁∪P₂       ◯ P₁ ∪P₂⊂P or P⊂P₁∪P₂       ◯ P⊂P₁∪P₂ 2.  If P is refinement of P1 and P2 ,then P is the common refinement of P₁ and P₂ (1 point)       ◯ True       ◯ False 3.  Let f be real and bounded on [ a,b]  α monotonically increasing on [a, b]  and P ={x₁,x₂,...xₙ} be partition of [a b] .then U(P,f,α)=________ (1 point)       ◯ ∑ᵢⁿ₌₁ Mᵢ∆αᵢ       ◯ ∑ᵢⁿ₌₁ mᵢ∆αᵢ       ◯  Mᵢ∆αᵢ       ◯ mᵢ∆αᵢ 4.  __________ is refinement of the partition P= {1,3,5,7,10}  of [1,10]. (1 point)       ◯ P₁={1,2,3,4,.....,10}       ◯ P₂={1,3,10}       ◯ P₃={1,3,5,7,9}       ◯ P₄=∅ 5.  If P* is refinement of P then...