Paper 2 CS · UNIT 01 · MOCK TEST

Mock 7 – Groups & Algebraic Structures

Unit 1: Discrete Structures and Optimisation · 28 questions · about 33 minutes. Press End Mock any time to see your score and the correct answers.

All 14 topic-wise mocks in this unit

0 / 28 answered 00:00

Q1. Which of the following additional properties is supposed to be satisfied by a group (G, *) to be an abelian group?

ACommutative BIdentity CInverse DAssociativity

Q2. Let and

be subgroups of group . Which of the following is False?

AH1.H2 = {h1*h2 : h1 ∈ H1 and h2 ∈ H2} is a subgroup. BH1.H2 = {h1*h2 : h1 ∈ H1 and h2 ∈ H2} is a normal subgroup given that H1 and H2 are normal subgroups. Cis a subgroup. Dis a subgroup.

Q3. In a group of 120 people: 65 eat Rice, 45 eat bread, 42 eat curd, 20 eat both Rice and bread, 25 eat both Rice and curd, 15 eat both bread and curd, and 8 eat all three items. Which of the following is the number of people who eat at least one of the three items?

A56 B100 C92 D65

Q4. Choose the correct statement for a group G:

AIf for all x, y ∈ G, (xy)² = x²y², then G is commutative. BIf for all x ∈ G, x³ = 1, then G is commutative (1 is the identity element of G). CIf for all x ∈ G, x⁵ = 1, then G is commutative (1 is the identity element of G). DIf G is commutative, a subgroup of G need not be commutative.

Q5. If x and y are elements in a group G and if x⁵ = y³ = e, where e is the identity of G, then the inverse of x²yx⁴y² must be

Ay²xy²x⁴ Byxy²x³ Cyx⁶y⁶x³ Dx⁴y²x²y

Q6. Arrange the following in ascending order:
(a) Remainder of 49¹⁶ when divided by 17
(b) Remainder of 2⁴⁴⁶ when divided by 9
(c) Remainder of 155¹⁷ when divided by 17
(d) Last digit of the number 7⁴⁵
Choose the correct answer from the options given below:

A(a), (b), (c), (d) B(a), (b), (d), (c) C(a), (c), (b), (d) D(d), (c), (b), (a)

Q7. Statement (I): If H is a non-empty finite subset of a group G and ab ∈ H for all a, b ∈ H, then H is also a group.
Statement (II): There is no homomorphism from (Z, +) to (Q, +), where Z is the set of integers and Q the set of rationals.
In the light of the above statements, choose the most appropriate answer from the options given below:

ABoth Statement I and Statement II are correct BBoth Statement I and Statement II are incorrect CStatement I is correct but Statement II is incorrect DStatement I is incorrect but Statement II is correct

Q8. Let (Z, +) denote the group of all integers under addition. Then the number of automorphisms of (Z, +) is

A1 B2 C3 D4

Q9. Let G = R \ {0} (non-zero reals) and H = {−1, 1} be groups under multiplication. Then the map φ : G → H defined by φ(x) = x/|x| is

ANot a homomorphism BA one-one homomorphism, which is not onto CAn onto homomorphism, which is not one-one DA homomorphism

Q10. Consider the following statements:
A. The identity is unique in any monoid.
B. A monoid is a group if every element of the monoid has an inverse.
C. A semigroup has closure, associative and identity properties.
D. A quasigroup has the closure property.

AA, B and D only BB, C and D only CA, B and C only DA, C and D only

Q11. If A = {4n + 2 | n is a natural number} and B = {3n | n is a natural number}, which of the following is correct for A ∩ B?

A{12n² + 6n | n is a natural number} B{24n − 12 | n is a natural number} C{60n + 30 | n is a natural number} D{12n − 6 | n is a natural number}

Q12. Z is the set of integers. ⟨Z, *⟩ forms a group if * is:

Aa * b = a + b + 2 Ba * b = 2a + b Ca * b = (a + b)/2 Da * b = a² + b

Q13. Consider the following statements:
A. A group can have zero elements.
B. A group cannot have any idempotent element except the identity element.
C. An isomorphism from a group ⟨G, *⟩ to ⟨G, *⟩ is called an automorphism.
D. If every element in a group is its own inverse, then the group must be abelian.
Choose the correct answer about the four statements given above.

AB, C and D are correct BOnly A and C are correct COnly B and C are correct DAll A, B, C and D are correct

Q14. Let ({a, b}, *) be a semigroup where a * a = b. Consider:
(A) a * b = b * a
(B) b * b = b
Choose the most appropriate answer from the options given below:

A(A) only true B(B) only true CBoth (A) and (B) true DNeither (A) nor (B) true

Q15. A fair share scheduler computes the priority of a process based on which of the following?
(1) Processor utilization by the individual process.
(2) Weight assigned to the user/group.
(3) Processor utilization by the user/group.

AOnly (1) and (2) BOnly (1) and (3) C(1), (2) and (3) DOnly (2) and (3)

Q16.

Choose the correct statement for a group G :

AIf for all x, y ∈ G, (xy)² = x²y², then G is commutative. BIf for all x ∈ G, x³ = 1, then G is commutative (1 is the identity of G). CIf for all x ∈ G, x⁵ = 1, then G is commutative (1 is the identity of G). DIf G is commutative, a subgroup of G need not be commutative.

Q17.

Z is set of integers, <Z,*> forms group if * in :

A*(a, b)=a+b+2 B*(a, b)=2a+b C*(a, b)=(a+b) / 2 D*(a, b)=a2+b

Q18.

Arrange the following in ascending order: (a) remainder of 49¹⁶ divided by 17 (b) remainder of 24⁴⁶ divided by 9 (c) remainder of 155¹⁷ divided by 17 (d) last digit of 7⁴⁵

A(a), (b), (c), (d) B(a), (b), (d), (c) C(a), (c), (b), (d) D(d), (c), (b), (a)

Q19.

Given below are two statements :

Statement (I): If H is a nonempty finite subset of ∈ ∀ ∈ a group G and ab H a,b H then H is also a group

Statement (II): There is no homomorphism existing from (Z,+) to (Q,+); where Z is a set of integers and Q is a set of rational numbers. In the light of the above statements, choose the most appropriate answer from the options given below:

Q20.

Let G = ℝ − {0} and H = {−1, 1} be groups under multiplication. Then the map φ : G → H defined by φ(x) = x/|x| is

ANot a homomorphism BA one-one homomorphism, which is not onto CAn onto homomorphism, which is not one to one DAn homomorphism

Q21.

Consider the following statements.

A. The identity is unique in any monoid.

B. A monoid is a group if there exists inverse of each element of monoid.

C. Semi group has closure, associative and identity properties.

D. Quasi group has closure property.

AA, B and D only BB, C and D only CA, B and C only DA, C and D only

Q22.

Consider: A. A group can have zero elements. B. A group cannot have any idempotent element except the identity. C. An isomorphism from a group ⟨G, *⟩ to ⟨G, *⟩ is called an automorphism. D. If every element in a group is its own inverse, the group must be abelian.

AB,C and D are correct BOnly A and C are correct COnly B and C are correct DAll A, B, C and D are correct.

Q23.

Let ({a, b}, *) be a semigroup where a * a = b. Which hold? (A) a * b = b * a (B) b * b = b

A(A) only true B(B) only true CBoth (A) and (B) true DNeither (A) nor (B) true

Q24.

Statement I: 5 divides n⁵ − n whenever n is a non-negative integer. Statement II: 6 divides n³ − n whenever n is a non-negative integer.

Q25.

Using a^(2c) mod n = (a^c)² mod n and a^(2c+1) mod n = a·(a^c)² mod n, for a = 7, b = 17 and n = 561, what is a^b mod n?

A160 B166 C157 D67 G = ⟨V , E⟩

Q26.

Consider the following statements : S (G, *) a ∈ G

1 : If a group is of order n, and is am = e m ≤ n such that for some integer , then m must divide n. S (G, *)

2 : If a group is of even order, then there a ∈ G a ≠ e must be an element such that and a * a = e . Which of the statements is (are) correct? S

AOnly 1 S BOnly 2 S S CBoth 1 and 2 S S DNeither 1 nor 2

Q27.

Consider the following statements : S n > 1, aϕ(n) ≡ 1(mod n)

1 : For any integer a ∈ Z∗ ϕ(n) for all n , where is Euler's phi function. S p ap ≡ 1(mod p)

2 : If is prime, then for all a ∈ Z∗ p . Which one of the following is/are correct? S

AOnly 1 S BOnly 2 S S CBoth 1 and 2 S S DNeither 1 nor 2

Q28.

Which of the following statements are true?

(i) Every logic network is equivalent to one using just NAND gates or just NOR gates.

(ii) Boolean expressions and logic networks correspond to labelled acyclic digraphs.

(iii) No two Boolean algebras with n atoms are isomorphic.

(iv) Non-zero elements of finite Boolean algebras are not uniquely expressible as joins of atoms.

A(i) and (iv) only B(i), (ii) and (iii) only C(i) and (ii) only D(ii), (iii) and (iv) only ≤ <

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