VU Midterm Past Papers

MTH633 Past Papers — Midterm Past Papers & Solved MCQs

43 solved midterm past paper MCQs for MTH633 at Virtual University, each with the correct answer marked. Use them to learn the VU question style and test your recall. An independent study tool — not affiliated with VU.

Prefer a summary first? Read the MTH633 midterm handout.

Q1. Which of the following is the subgroup of {1, -1, i, -i} under multiplication?
- A.{1}✓ Correct
- B.{-1}
- C.{i}
- D.{-i}
Q2. acyclicgroupG=<a>,if|a|=n,then<a^k>=__
- A.<a^gcd(n, k-2)>
- B.<a^gcd(n k-1)>
- C.<a^gcd(n, k-3)>
- D.<a^gcd(n, k)>✓ Correct
Q3. G=<a>with|a|=30.Then|a^6|=_____
- A.6✓ Correct
- B.1
- C.5
- D.30
Q4. acyclicgroupG=<a>,if|a|=n,then ____
- A.|a^k|=n/gcd(n,k+2)
- B.|a^k|=n/gcd(n,k-1)
- C.|a^k|=n/gcd(n,k+1)
- D.|a^k|=n/gcd(n,k)✓ Correct
Q5. Suppose G =< a >with|a|=7 and e is identity. Which of the following is true?
- A.a^4=e
- B.a^3=e✓ Correct
- C.a^5=e
- D.a^2=e
Q6. A subset H of a group G is a subgroup of G if and only if H is non-empty and -------
- A.whenever a, b ∈ H, then (ab)^-1 ∈ H
- B.whenever a, b ∈ H, then ab^-1 ∈ H✓ Correct
- C.whenever a, b ∈ H, then ab ∈ H
- D.whenever a, b ∈ H, then b^-1 ∈ H
Q7. The group Z4 under addition is the finite cyclic group generated by .
- A.2 and 1
- B.1 and 3✓ Correct
- C.2 and 3
- D.1 and 0
Q8. Suppose G =< a >with |a|=30. Then |a^6| =
- A.5✓ Correct
- B.7
- C.6
- D.4
Q9. G=<a> if |a| = ∞, then a^i=a^j if and only if _____
- A.i<j
- B.i=j✓ Correct
- C.i>j
- D.i≠j
Q10. The group Z4 under addition is the finite cyclic group generated by 2;
- A.2 and 1
- B.2 and 3
- C.1 and 3✓ Correct
- D.1 and 0
Q11. If n=2, then the collection of all even permutations of {1, 2, ..., n} forms a subgroup of order of the symmetric group Sn.
- A.n
- B.n!/2✓ Correct
- C.n/3
- D.n!/4
Q12. For each positive integer n, let Un be the multiplicative group of the nth roots of unity in C. Then the elements of Un can be represented geometrically with equally spaced points on
- A.a square
- B.a rectangle
- C.a circle with centre origin✓ Correct
- D.an ellipse with centre origin
Q13. The number of cosets of subgroup of a group is the ... of subgroup.
- A.left cosets
- B.right cosets
- C.indices✓ Correct
- D.none of above
Q14. Let H be a subgroup of G and a, b in G. If a belongs to H then
- A.aH=H✓ Correct
- B.Ha=aH
- C.bH=H
- D.bH=Hb
Q15. ... is also called the octic group.
- A.D3
- B.D4✓ Correct
- C.D5
- D.D6
Q16. G be an abelian group. A direct product of G is
- A.cyclic
- B.non-cyclic
- C.abelian✓ Correct
- D.non-abelian
Q17. H = {e, x, x²}
- A.{e, x, x²}✓ Correct
- B.{e, x², x⁴}
- C.{e, x³, x⁶}
- D.{e, x⁴, x⁸}
Q18. If m divides the order of a finite abelian group G then G has a .... of order m.
- A.cyclic group
- B.subgroup✓ Correct
- C.normal subgroup
- D.Klein 4-group
Q19. For a cyclic group G = <a>, if |a| = ∞, then aⁱ = aʲ if and only if
- A.i>j
- B.i<j
- C.i=j✓ Correct
- D.i≠j
Q20. A cyclic group G = <a>, aᵏ = e implies
- A.all k-1
- B.|a| | k✓ Correct
- C.|a| | k+1
- D.|a| | k+2
Q21. Which of the following is not a group?
- A.(Z, +)
- B.(Q, +)
- C.(R, +)
- D.(N, +)✓ Correct
Q22. The group Z₂₄ under addition is abelian because it is
- A.cyclic✓ Correct
- B.abelian
- C.infinite
- D.finite
Q23. Let φ: G → G' be group homomorphism and a in G then
- A.φ(aⁿ) = φ(a)ⁿ
- B.φ(aⁿ) = (φ(a))ⁿ✓ Correct
- C.φ(aⁿ) = nφ(a)
- D.φ(a⁻¹) = φ(a)⁻¹
Q24. The set of real numbers R is the union of
- A.Q and Q'
- B.Q and I✓ Correct
- C.R⁺ and R⁻
- D.Z and Q
Q25. Alternating group A₄ which has 12 elements has no subgroup of order
- A.3
- B.4
- C.6✓ Correct
- D.12
Q26. The set of all translations in R² forms ... under composition.
- A.abelian group✓ Correct
- B.non-abelian group
- C.Sylow group
- D.normal group
Q27. There are different abelian groups of order 360
- A.1✓ Correct
- B.2
- C.3
- D.4
Q28. In the group {1,-1,i,-i} the order of -1 is
- A.1
- B.2✓ Correct
- C.3
- D.4
Q29. Let f:G→H be a group homomorphism, the kernel of function f is defined as
- A.ker f={g in G: f(g)=e_H}✓ Correct
- B.ker f={h in H: f(h)=e_G}
- C.ker f={g in G: f(g)=e_G}
- D.ker f={h in H: f(h)=e_H}
Q30. Which of the following is a subgroup of {1,-1,i,-i} under multiplication?
- A.{1}✓ Correct
- B.{-1}
- C.{i}
- D.{-i}
Q31. Finite indecomposable abelian groups are exactly cyclic groups with power of
- A.integers
- B.natural numbers
- C.primes✓ Correct
- D.composites
Q32. The map f:Z→Z defined by f(x)=[x] is a homomorphism if
- A.f(xy)=f(x)+f(y)
- B.f(xy)=f(x)f(y)
- C.f(x+y)=f(x)+f(y)✓ Correct
- D.f(x+y)=f(x)f(y)
Q33. Every permutation of a finite set is a _ of disjoint cycles
- A.sum
- B.difference
- C.product✓ Correct
- D.composition
Q34. A permutation is a cycle if at most _ of its orbits is nontrivial
- A.one✓ Correct
- B.two
- C.three
- D.four
Q35. In a cyclic group G=<a>, if |a|=n, then a^m = a^n iff n divides
- A.m✓ Correct
- B.m-n
- C.m+n
- D.mn
Q36. A group which is not isomorphic to direct product of two proper non-trivial subgroups is called
- A.decomposable
- B.indecomposable✓ Correct
- C.endomorphic
- D.automorphic
Q37. Suppose G=<a> with |a|=5 and e is identity. Which of the following is true?
- A.a^0=e✓ Correct
- B.a^1=e
- C.a^2=e
- D.a^3=e
Q38. Which of the following is true?
- A.D4<S4✓ Correct
- B.D4>S4
- C.D4=S4
- D.D4≅S4
Q39. Let G be a finite cyclic group generated by its element a, then
- A.|G|<|a|
- B.|G|>|a|
- C.|G|=|a|✓ Correct
- D.None of these
Q40. Alternating group A4 which has 12 elements has no subgroup of order
- A.1
- B.2
- C.3
- D.all of above✓ Correct
Q41. Let G and G' be groups and let φ: G → G' be a one-to-one function such that _ . Then φ[G] is a subgroup of G' and φ provides an isomorphism of G with φ[G].
- A.φ(xy)=φ(x)/φ(y) for all x,y in G
- B.φ(xy)=φ(x)φ(y) for all x,y in G✓ Correct
- C.φ(xy)=φ(x)+φ(y) for all x,y in G
- D.φ(xy)=φ(x)-φ(y) for all x,y in G
Q42. An abelian group satisfies _ property
- A.Associative
- B.Closure
- C.Commutative✓ Correct
- D.Distributive
Q43. Order of the group {1,-1,i,-i} is
- A.2
- B.3
- C.4✓ Correct
- D.5

VU Midterm Past Papers

MTH633 Past Papers — Midterm Past Papers & Solved MCQs

Prefer a summary first? Read the MTH633 midterm handout.

Q1. Which of the following is the subgroup of {1, -1, i, -i} under multiplication?
- A.{1}✓ Correct
- B.{-1}
- C.{i}
- D.{-i}
Q2. acyclicgroupG=<a>,if|a|=n,then<a^k>=__
- A.<a^gcd(n, k-2)>
- B.<a^gcd(n k-1)>
- C.<a^gcd(n, k-3)>
- D.<a^gcd(n, k)>✓ Correct
Q3. G=<a>with|a|=30.Then|a^6|=_____
- A.6✓ Correct
- B.1
- C.5
- D.30
Q4. acyclicgroupG=<a>,if|a|=n,then ____
- A.|a^k|=n/gcd(n,k+2)
- B.|a^k|=n/gcd(n,k-1)
- C.|a^k|=n/gcd(n,k+1)
- D.|a^k|=n/gcd(n,k)✓ Correct
Q5. Suppose G =< a >with|a|=7 and e is identity. Which of the following is true?
- A.a^4=e
- B.a^3=e✓ Correct
- C.a^5=e
- D.a^2=e
Q6. A subset H of a group G is a subgroup of G if and only if H is non-empty and -------
- A.whenever a, b ∈ H, then (ab)^-1 ∈ H
- B.whenever a, b ∈ H, then ab^-1 ∈ H✓ Correct
- C.whenever a, b ∈ H, then ab ∈ H
- D.whenever a, b ∈ H, then b^-1 ∈ H
Q7. The group Z4 under addition is the finite cyclic group generated by .
- A.2 and 1
- B.1 and 3✓ Correct
- C.2 and 3
- D.1 and 0
Q8. Suppose G =< a >with |a|=30. Then |a^6| =
- A.5✓ Correct
- B.7
- C.6
- D.4
Q9. G=<a> if |a| = ∞, then a^i=a^j if and only if _____
- A.i<j
- B.i=j✓ Correct
- C.i>j
- D.i≠j
Q10. The group Z4 under addition is the finite cyclic group generated by 2;
- A.2 and 1
- B.2 and 3
- C.1 and 3✓ Correct
- D.1 and 0
Q11. If n=2, then the collection of all even permutations of {1, 2, ..., n} forms a subgroup of order of the symmetric group Sn.
- A.n
- B.n!/2✓ Correct
- C.n/3
- D.n!/4
Q12. For each positive integer n, let Un be the multiplicative group of the nth roots of unity in C. Then the elements of Un can be represented geometrically with equally spaced points on
- A.a square
- B.a rectangle
- C.a circle with centre origin✓ Correct
- D.an ellipse with centre origin
Q13. The number of cosets of subgroup of a group is the ... of subgroup.
- A.left cosets
- B.right cosets
- C.indices✓ Correct
- D.none of above
Q14. Let H be a subgroup of G and a, b in G. If a belongs to H then
- A.aH=H✓ Correct
- B.Ha=aH
- C.bH=H
- D.bH=Hb
Q15. ... is also called the octic group.
- A.D3
- B.D4✓ Correct
- C.D5
- D.D6
Q16. G be an abelian group. A direct product of G is
- A.cyclic
- B.non-cyclic
- C.abelian✓ Correct
- D.non-abelian
Q17. H = {e, x, x²}
- A.{e, x, x²}✓ Correct
- B.{e, x², x⁴}
- C.{e, x³, x⁶}
- D.{e, x⁴, x⁸}
Q18. If m divides the order of a finite abelian group G then G has a .... of order m.
- A.cyclic group
- B.subgroup✓ Correct
- C.normal subgroup
- D.Klein 4-group
Q19. For a cyclic group G = <a>, if |a| = ∞, then aⁱ = aʲ if and only if
- A.i>j
- B.i<j
- C.i=j✓ Correct
- D.i≠j
Q20. A cyclic group G = <a>, aᵏ = e implies
- A.all k-1
- B.|a| | k✓ Correct
- C.|a| | k+1
- D.|a| | k+2
Q21. Which of the following is not a group?
- A.(Z, +)
- B.(Q, +)
- C.(R, +)
- D.(N, +)✓ Correct
Q22. The group Z₂₄ under addition is abelian because it is
- A.cyclic✓ Correct
- B.abelian
- C.infinite
- D.finite
Q23. Let φ: G → G' be group homomorphism and a in G then
- A.φ(aⁿ) = φ(a)ⁿ
- B.φ(aⁿ) = (φ(a))ⁿ✓ Correct
- C.φ(aⁿ) = nφ(a)
- D.φ(a⁻¹) = φ(a)⁻¹
Q24. The set of real numbers R is the union of
- A.Q and Q'
- B.Q and I✓ Correct
- C.R⁺ and R⁻
- D.Z and Q
Q25. Alternating group A₄ which has 12 elements has no subgroup of order
- A.3
- B.4
- C.6✓ Correct
- D.12
Q26. The set of all translations in R² forms ... under composition.
- A.abelian group✓ Correct
- B.non-abelian group
- C.Sylow group
- D.normal group
Q27. There are different abelian groups of order 360
- A.1✓ Correct
- B.2
- C.3
- D.4
Q28. In the group {1,-1,i,-i} the order of -1 is
- A.1
- B.2✓ Correct
- C.3
- D.4
Q29. Let f:G→H be a group homomorphism, the kernel of function f is defined as
- A.ker f={g in G: f(g)=e_H}✓ Correct
- B.ker f={h in H: f(h)=e_G}
- C.ker f={g in G: f(g)=e_G}
- D.ker f={h in H: f(h)=e_H}
Q30. Which of the following is a subgroup of {1,-1,i,-i} under multiplication?
- A.{1}✓ Correct
- B.{-1}
- C.{i}
- D.{-i}
Q31. Finite indecomposable abelian groups are exactly cyclic groups with power of
- A.integers
- B.natural numbers
- C.primes✓ Correct
- D.composites
Q32. The map f:Z→Z defined by f(x)=[x] is a homomorphism if
- A.f(xy)=f(x)+f(y)
- B.f(xy)=f(x)f(y)
- C.f(x+y)=f(x)+f(y)✓ Correct
- D.f(x+y)=f(x)f(y)
Q33. Every permutation of a finite set is a _ of disjoint cycles
- A.sum
- B.difference
- C.product✓ Correct
- D.composition
Q34. A permutation is a cycle if at most _ of its orbits is nontrivial
- A.one✓ Correct
- B.two
- C.three
- D.four
Q35. In a cyclic group G=<a>, if |a|=n, then a^m = a^n iff n divides
- A.m✓ Correct
- B.m-n
- C.m+n
- D.mn
Q36. A group which is not isomorphic to direct product of two proper non-trivial subgroups is called
- A.decomposable
- B.indecomposable✓ Correct
- C.endomorphic
- D.automorphic
Q37. Suppose G=<a> with |a|=5 and e is identity. Which of the following is true?
- A.a^0=e✓ Correct
- B.a^1=e
- C.a^2=e
- D.a^3=e
Q38. Which of the following is true?
- A.D4<S4✓ Correct
- B.D4>S4
- C.D4=S4
- D.D4≅S4
Q39. Let G be a finite cyclic group generated by its element a, then
- A.|G|<|a|
- B.|G|>|a|
- C.|G|=|a|✓ Correct
- D.None of these
Q40. Alternating group A4 which has 12 elements has no subgroup of order
- A.1
- B.2
- C.3
- D.all of above✓ Correct
Q41. Let G and G' be groups and let φ: G → G' be a one-to-one function such that _ . Then φ[G] is a subgroup of G' and φ provides an isomorphism of G with φ[G].
- A.φ(xy)=φ(x)/φ(y) for all x,y in G
- B.φ(xy)=φ(x)φ(y) for all x,y in G✓ Correct
- C.φ(xy)=φ(x)+φ(y) for all x,y in G
- D.φ(xy)=φ(x)-φ(y) for all x,y in G
Q42. An abelian group satisfies _ property
- A.Associative
- B.Closure
- C.Commutative✓ Correct
- D.Distributive
Q43. Order of the group {1,-1,i,-i} is
- A.2
- B.3
- C.4✓ Correct
- D.5