1. Which of the following options are true?

• (Z,×) is a group

• (Z,+) is not a group

• (Q,×) is a group

• (Z,−) is not a group

• (R,+) is a group

Correct Answer :(Q,×) is a group

(Z,−) is not a group

(R,+) is a group

2. Let T be a group of 100 elements, what is the largest possible subgroup of T other than T itself?

• 50

• 75

• 25

• 20

Correct Answer : 50

3. The generating function for odd partitions of x is:

• 1/(1−x) × 1/(1−x^2) ×1/(1−x^3) ×……

• 1/x x 1/(1−𝑥) × 1 (1−𝑥^3)×1/(1−𝑥^5)×……

• 1/𝑥×1/(1−𝑥)×1/(1−𝑥^2)×1/(1−𝑥^3)×……

• 1/(1−𝑥)×1/(1−𝑥^3)×1/(1−𝑥^5)×……

Correct Answer :1/(1−𝑥)×1/(1−𝑥^3)×1/(1−𝑥^5)×……

4. For a path graph of P7, in how many ways can you color this graph with 4 colors so that no two adjacent nodes have the same color?

• • 8748

  • 2916

  • 16384

  • 65536

Correct Answer :2916

5. What are the elements of Z7, i.e., integer modulo 7?

• • {1, 2, 3, 4, 5, 6, 7}

  • {1, 2, 3, 4, 5, 6}

  • {0, 1, 2, 3, 4, 5, 6}

  • {0, 1, 2, 3, 4, 5, 6, 7}

Correct Answer :{0, 1, 2, 3, 4, 5, 6}

6. Which of the following statement(s) is/are true?

I) The operations of a group and its subgroup can be different.

II) (Q, +) is a subgroup of (R, +)

III) A group under addition modulo n, where n is prime, cannot have a subgroup except for a trivial subgroup identity element alone.

• II and III

• I and II

• I and III

• I II and III

Correct Answer : II and III

7. State whether true/false:

𝑃𝑑(𝑛) is the coefficient of 𝑥^𝑛 in the function

(1+𝑥)(1+𝑥2)(1+𝑥3)……..(1+𝑥𝑛)

• True

• False

Correct Answer : True

8. Let A be a group with 70 elements. Then A can not have a subgroup of cardinality

• • • 5

    • 10

    • 20

    • 35

Correct Answer : 20

9. (Z6,+) is a group, what is the inverse of 4 in this group?

• 1

• 2

• 3

• 4

Correct Answer : 2

10. Given a complete graph with 11 vertices, in how many ways can you color this graph such that no two adjacent vertices have the same color? You are given 26 colors.

• 𝐶(𝐾11)=26×25×24×…×1

• 𝐶(𝐾11)=26×25×24×…×14

• 𝐶(𝐾11)=26×25×24×…×16

• C(K11)=26×25×24×…×15

Correct Answer :𝐶(𝐾11)=26×25×24×…×16

WEEK-12 DETAILED SOLUTION