1 Which of the following is divisible by 17 for all positive integer n

$$3.5^{2n+1}+2^{3n+1 }$$

2 A matrix $$ X=\bigl(\begin{pmatrix} 3 & 1\\ 5 & 2\end{pmatrix}\bigr)$$ define a function from

$$\mathbb{R}^{2}\; to\;\mathbb{R}^{2} $$ by $$f_{X}(a,b)=(3a+b,\; 5a+2b)$$ . find the inverse function of $$f_{X}$$

$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$

3 Find all the real number that satisfy the inequality $$ 1/x < x^{2} $$

$$ \left \{x: x < 0 \; or \; x > 1 \right \}$$

4 If H is a group and x and y belongs to H such that xy=yx, given that the order of x is m, the order of y is n, and (m,n)= 1, what is the order of xy?

mn

5 What is the generator of (Z, +) cyclic group?

1

6 If G is a cyclic group of order 4 generated by a, and let $$H= <a^{2}>$$

$${e, a^2} \; and \; {a, a^3 }$$

7 Find all x in Z satisfying the equation 5x=1 (mod 6)

{?.. ,??1,5,11, .....}

8 What is addition of 3 and 5 under modulo 7

1

9 What is 3 multiply by 4 under modulo 12

0

10 Which of the following multiplication tables defined on the set G = {a,b,c,d} form a group? <grp1>

(i) Not a group (ii) A group

11 Which of the following is divisible by 17 for all positive integer n

$$3.5^{2n+1}+2^{3n+1 }$$

12 A matrix $$ X=\bigl(\begin{pmatrix} 3 & 1\\ 5 & 2\end{pmatrix}\bigr)$$ define a function from

$$\mathbb{R}^{2}\; to\;\mathbb{R}^{2} $$ by $$f_{X}(a,b)=(3a+b,\; 5a+2b)$$ . find the inverse function of

$$f_{X}$$

$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$

13 Given a set $$X=\left \{ a,b,c \right \}$$, and a function $$\Psi :X\; \rightarrow \; X $$ define by $$\Psi(a)=b,\;

\Psi (b)=a,\; \Psi (c)=c $$ . the function is

bijective

14 Which of the following pair of functions has f o g = g o f

$$f(y)=y^{2} \; and \; g(y)=3y+7$ $

15 Four relations a to d are defined on sets A and B as in the diagram shown. Which of the relations represent a function from A to B?

f1 and f2

16 For sets A and B , if A and B are subset of Z (the set of Integer) which of the following relations between the two subset is true?

(A\B)n(B\ A)= 0

17 If R (the set of real number) be the universal set and sets $$V=\left \{ y\epsilon R:0 < y\leq 3 \right \}$$ and $$W=\left \{ y\epsilon R:2\leq y < 4 \right \}$$ What is $$V^{l}$$

$$\left \{ y\epsilon R:0\leq y \; or\; y> 3 \right \}$$

18 Let R be the universal set and suppose that $$X=\left \{ y\epsilon R:0 < y\leq 7 \right \}$$ and $$Y=\left \{ y\epsilon R:6\leq y < 12 \right \}$$ find X\Y

$$\left \{ y\epsilon R:2 < y < 6 \right \}$$

19 Consider a relation * defined on $$(a,b),\; (c,d)\; \epsilon \; \Re ^{2}$$ by $$(a,b)\;* (c,d)\; $$ to mean $$2a-b

= 2c-d $$ which of the following is true about *

Is reflexive, symmetric and trans itive

20 Four sets X, Y, V and W has u, 7, h and 20 elements respectively, how many elements has the Cartesian product (Y x V x W) formed from the sets Y, V and W

140h

21 A matrix $$ X=\begin{pmatrix} 1 &2 \\ 2& 5 \end{pmatrix}$$ define a function from $$\mathbb{R}^{2}\; to\;\mathbb{R}^{2} $$ by $$f_{X}(a,b)=(3a+b,\; 5a+2b)$$ . find the inverse function of $$f_{X}$$

$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$

22 Find all x in Z satisfying the equation 5x=1 (mod 6)

$$\left \{?.. ,??1,5,11,....\right \}$$

23 What is addition of 3 and 5 under modulo 7

1

24 What is 3 multiply by 4 under modulo 12

0

25 Which of the following multiplication tables defined on the set G = {a,b,c,d} form a group? <grp1>

(i) Not a group (ii) A group

26 For sets A and B , if A and B are subset of Z (the set of Integer) which of the following relations between the two subset is true?

(A\B)n(B\ A)= empty set

27 If R (the set of real number) be the universal set and sets $$V=\left \{ y\epsilon R:0 < y\leq 3 \right \}$$ and

$$W=\left \{ y\epsilon R:2\leq y < 4 \right \}$$ What is $$V^{l}$$

$$\left \{ y\epsilon R:0\leq y \; or\; y> 3 \right \}$$

28 Let R be the universal set and suppose that $$X=\left \{ y\epsilon R:0 < y\leq 7 \right \}$$ and $$Y=\left \{

y\epsilon R:6\leq y < 12 \right \}$$ find X\Y

$$\left \{ y\epsilon R:2 < y < 6 \right \}$$

29 Consider a relation * defined on $$(a,b),\; (c,d)\; \epsilon \; \Re ^{2}$$ by $$(a,b)\;* (c,d)\; $$ to mean $$2a-b

= 2c-d $$ which of the following is true about *

Is reflexive, symmetric and trans itive

30 Four sets X, Y, V and W has u, 7, h and 20 elements respectively, how many elements has the Cartesian

product (Y x V x W) formed from the sets Y, V and W

140h

31 Which of the following is divisible by 17 for all positive integer n

$$3.5^{2n+1}+2^{3n+1 }$$

32 A matrix $$ X=\begin{pmatrix} 1 &2 \\ 2& 5 \end{pmatrix} $$ define a function from $$\mathbb{R}^{2}\;

to\;\mathbb{R}^{2} $$ by $$f_{X}(a,b)=(3a+b,\; 5a+2b)$$ . find the inverse function of $$f_{X}$$

$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$

33 Given a set $$X=\left \{ a,b,c \right \}$$, and a function $$\Psi :X\; \rightarrow \; X $$ define by $$\Psi(a)=b,\;

\Psi (b)=a,\; \Psi (c)=c $$ . the function is

bijective

34 Which of the following pair of functions has f o g = g o f

$$f(y)=y^{2} \; and \; g(y)=3y+7$ $

35 Four relations a to d are defined on sets A and B as in the diagram shown. Which of the relations represent a

function from A to B?

f1 and f2

36 Find the order of element -1 in the multiplicative group $$\left\{1,-1,- i, (-i) \right\}$$

2

37 Which of the following is/are true about a group G. (i) The order of an element a in G is the least positive

integer n such that $$a^{n} = e$$. (ii) if such integer does not exist then the order of a is greater than one or

infinite (iii) The order of an element a in G is the least positive integer n such that $$a^{e} = n$$

(i) and (i i)

38 The assertion that if H is a subgroup of a finite group G, then the order of H divides the order of G is called

Lagrange??s theo rem

39 Find all the real number that satisfy the inequality $$1 < x^{2} < 4$$

$$ \left \{x:1 < x < 2 \; or \; -2 < x < -1\right \}$$

40 Given that x, y, z be any elements of $$\mathbb{R}, which of the following statement is/are true? (i) if x > y

and y > z, then x > z (ii) if x > y then x + z < y + z (iii) if x > y and z > 0, then zx > zy

(i) and (iii)

## NOUN EExams Practice Questions: MTH211 - Abstract Algebra

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