## NOUN EExams Practice Questions: MTH211 - Abstract Algebra

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### NOUN EExams Practice Questions: MTH211 - Abstract Algebra

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)

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