MCQ on Riemann integral(B.Sc final year)
(B.sc course)
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 (1 point)
◯ L(P,f,α) ≤ L(P*,f,α) ≤ U(P*,f,α) ≤ U(P,f,α)
◯ L(P,f,α) ≤ U(P*,f,α) ≤ L(P*,f,α) ≤ U(P,f,α)
◯ U(P,f,α) ≤ L(P*,f,α) ≤ U(P*,f,α) ≤ L(P,f,α)
◯ L(P,f,α) ≤ L(P*,f,α) ≤ U(P,f,α) ≤ U(P*,f,α)
6. If f is Riemann integral over [a,b], and P1 and P2 be two partition of [a,b] then
L(P1,f,α) ≤U(P2,f,α)
(1 point)
◯ True
◯ False
7. If f is continuous on [a,b] .then f∈ℜ(α) (1 point)
◯ True
◯ False
8. If f is bounded function on [a, b] ,then f∈ℜ(α) (1 point)
◯ True
◯ False
9. If f is bounded real function on [a,b] and α is monotonic increasesing function
on [ a, b] then E= {L(P, f ,α) : P is a partition of [a, b]}. Is bounded above.
(1 point)
◯ True
◯ False
10. If f is monotonic on [a,b] and α is continuous on [a, b] ,then f∈ℜ(α) (1 point)
◯ True
◯ False
11. U(P,- f ,α)=________. (1 point)
◯ L(P, f ,α)
◯ U(P, f ,α)
◯ -U(P, f ,α)
◯ -L(P, f ,α)
12. If f(x)=0 for all irrational x, f(x)=1 for all rational x,then f∈ℜ(α) on [a,b] (1 point)
◯ True
◯ False
13. Every bounded function is Riemann integral on [0,1]. (1 point)
◯ True
◯ False
14. f is bounded real function on [a, b], and f²∈ℜ(α) on [a,b] then f∈ℜ(α) on [a,b] (1 point)
◯ True
◯ False
15. Let f is a bounded real function on [a, b] and f∈ℜ(α) on [a,b] then f³∈ℜ(α) on
[a,b]
(1 point)
◯ True
◯ False
16. Every Riemann integrable function define on [0,1] is monotonicallyi ncreasing.
(1 point)
◯ True
◯ False
17. Suppose f>0 ,f is continuous on [a,b], and ₐ∫ᵇf(x) = 0. then f(x)=0. (1 point)
◯ For only x=(a+b)/2
◯ For only x=a
◯ For only x=b
◯ For all x∈[a,b].
18. If f is Riemann integrable on [a,b] such that ₐ∫ᵇ f(x)=0.Then f(x)=0 for all x∈
[a,b].
(1 point)
◯ True
◯ False
Part 2:-
1. If f∈ℜ(α) and α′∈ ℜ(α) then fα′∈ ℜ(α). Then which of the following statementi s true.
(1 point)
◯ ₐ∫ᵇf dα= ₐ∫ᵇf(x)α(x) dx
◯ ₐ∫ᵇf dα= ₐ∫ᵇf′(x)α(x) dx
◯ ₐ∫ᵇf dα= ₐ∫ᵇf(x)α′(x) dx
◯ All are true
2. ₀∫¹x³ dx² =______. (1 point)
◯ 0
◯ 1
◯ 2/5
◯ 1/2
3. if f∈ℜ(α) on [a,b] than f²∈ ℜ(α) on [a,b]. (1 point)
◯ True
◯ False
4. If f(x)=lxl ,and α(x)=x² for x∈[-2,2] then |₋₂∫² f dα | ≤ _______ (1 point)
◯ 16
◯ 8
◯ 32
◯ 4
5. If 5≤f(x)≤10 ,α(x)=x² for x∈[0,2] .Then L(P,f,α) ∈ ________. (1 point)
◯ [20,40]
◯ [5,10]
◯ [10,20]
◯ [11,22]
6. If 5≤f(x)≤10 ,α(x)=x² for x∈[0,2] .Then U(P,f,α) ∈ ________. (1 point)
◯ [20,40]
◯ [5,10]
◯ [10,20]
◯ [11,22]
7. Let f be function f :[0,1] --->R , defined by f(1/2)=1 and f(x)=0, for all x∈ [0,1]\
{1/2} .Then f is Riemann integrable.
(1 point)
◯ True
◯ False
8. If f is riemann integrable on [0,1] such thatₐ ∫ ᵇf(x)dx =0 then f(x)=0 for at least one x∈[0,1].
(1 point)
◯ True
◯ False
9. Let f be function f :[0,1] --->R , defined by f(1/2)=1 and f(x)=0, for all x ∈[0,1]\
{1/2} .Then _______
(1 point)
◯ f is NOT Riemann integrable
◯ f is Riemann integrable
◯ ₀ ∫ ¹f(x)dx =0
◯ B and C both are correct
10. If f is continuous and Riemann integrable on [a,b] ,then there exit a number c
laying between a and b such that ₐ∫ᵇ f(x) dx=(b-a) f(c).(1 point)
◯ True
◯ False
11. If lfl is Riemann integrable, then f is _______. (1 point)
◯ unit step function
◯ Riemann integrable
◯ need not be Riemann integrable
◯ none of above
12. f is bounded function having only finite number of points of discontinuity in
[a,b] is NOT Riemann integrable on [a,b].
(1 point)
◯ True
◯ False
13. If M and m are bounds of an integrable function f over [a,b] then (1 point)
◯ m(b-a) ≤ₐ∫ᵇf dx ≤M(b-a)
◯ M(b-a) ≤ₐ∫ᵇf dx ≤m(b-a)
◯ M(b-a) ≤ₐ∫ᵇ -f dx ≤m(b-a
◯ no one is correct
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