MTH211 TMA

Q1 Which of the following is divisible by 17 for all positive integer n
$$7^{n}+2$$
$$6^{n}+2$$
$$2.7^{n}+3.5^{n}-5$$
$$3.5^{2n+1}+2^{3n+1}$$

Q2 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 $$f_{X}$$
$$f^{-1}_{X}(a,b)=(3a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{5}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+4b)$$

Q3 Find all the real number that satisfy the inequality $$ 1/x < x^{2} $$
$$ \left \{x: x < 0 \; or \; x > 1 \right \}$$
$$ \left \{x: x < 2 \; or \; x > 1 \right \}$$
$$ \left \{x: x < -1 \; or \; x > 1 \right \}$$
$$ \left \{x: x < -1 \; or \; x >2 \right \}$$

Q4 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?
m+ n
3n + 4n
mn
$$mn^{e}$$

Q5 What is the generator of (Z, +) cyclic group?
$$\infty$$
2
0
1..ans

Q6 If G is a cyclic group of order 4 generated by a, and let $$H= <a^{2}>$$
$${e, e^2} \; and \; {a, a^3}$$
$${e, a^2} \; and \; {a, a^3 }$$
$${a, a^2} \; and \; {a, a^3 }$$
$${e, a^3} \; and \;{e, a^2 }$$

Q7 Find all x in Z satisfying the equation 5x=1 (mod 6)
{�?�.. ,�??1,5,11, …..}..ans
{�?�.. ,�??2,4,10, …..}
{�?�.. ,0,7,12, …..}
{�?�.. �??6,0,6, ….}

Q8 What is addition of 3 and 5 under modulo 7
15
8
5
1..ans

Q9 What is 3 multiply by 4 under modulo 12
12
0..ans
7
4

Q10 Which of the following multiplication tables defined on the set G = {a,b,c,d} form a group? <grp1>
(i) A group (ii) not a group
(i) Not a group (ii) Not a group
(i) Not a group (ii) A group
both are group

Q11 Which of the following is divisible by 17 for all positive integer n
$$7^{n}+2$$
$$6^{n}+2$$
$$2.7^{n}+3.5^{n}-5$$
$$3.5^{2n+1}+2^{3n+1}$$

Q12 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 $$f_{X}$$
$$f^{-1}_{X}(a,b)=(3a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{5}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+4b)$$

Q13 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
Only unto
only injective
bijective
no solution

Q14 Which of the following pair of functions has f o g = g o f
$$f(y)=y^{3} \; and \; g(y)=\sqrt[3]{y}$$
$$f(y)=y^{5} \; and \; g(y)=3y+7$$
$$f(y)=y^{2} \; and \; g(y)=y+7$$
$$f(y)=y^{2} \; and \; g(y)=3y+7$$

Q15 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
, f1 and g1
f2 and g1
f2 and g2

Q16 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?
(AuB)= A
(A\B)n(B\A)= 0
(A\B)n(B\A)= Z
, (A\B)u(B\A)= empty set

Q17 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 $$V^{l}$$
$$\left \{ y\epsilon R:1 < y \; or\; y> 3 \right \}$$
$$\left \{ y\epsilon R:-1 < y \; or\; y> 1 \right \}$$
$$\left \{ y\epsilon R:0\leq y \; or\; y> 3 \right \}$$
$$\left \{ y\epsilon R:0 < y \; or\; y> 1 \right \}$$

Q18 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
$$\left \{ y\epsilon R:2 < y < 6 \right \}$$
$$\left \{ y\epsilon R:2\leq y < 6 \right \}$$
$$\left \{ y\epsilon R:2\leq y\geq 6 \right \}$$
$$\left \{ y\epsilon R:2 < y\geq 6 \right \}$$

Q19 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 only symmetric
is only reflexive and transitive
is only symmetric and Transitive
Is reflexive, symmetric and transitive

Q20 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
140
120u
140h
20h

Q21 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 $$f_{X}$$
$$f^{-1}_{X}(a,b)=(3a-b,\; -2a+b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{5}_{X}(a,b)=(2a-b,\; -7a+b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -2a+3b)$$

Q22 Find all x in Z satisfying the equation 5x=1 (mod 6)
$$\left \{�?�.. ,�??1,5,11,….\right \}$$
$$\left \{�?�.. ,�??2,4,10, ….\right \}$$
$$\left \{�?�.. ,0,7,12, …..\right \}$$
$$\left \{�?�.. ,�??6,0,6, ….\right \}$$

Q23 What is addition of 3 and 5 under modulo 7
15
8
5
1…ans

Q24 What is 3 multiply by 4 under modulo 12
12
0…ans
7
4

Q25 Which of the following multiplication tables defined on the set G = {a,b,c,d} form a group? <grp1>
(i) A group (ii) not a group
(i) Not a group (ii) Not a group
(i) Not a group (ii) A group
both are group

Q26 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?
(AuB)= A
(A\B)n(B\A)= empty set
(A\B)n(B\A)= Z
, (A\B)u(B\A)= empty set

Q27 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 $$V^{l}$$
$$\left \{ y\epsilon R:1 < y \; or\; y> 3 \right \}$$
$$\left \{ y\epsilon R:-1 < y \; or\; y> 1 \right \}$$
$$\left \{ y\epsilon R:0\leq y \; or\; y> 3 \right \}$$
$$\left \{ y\epsilon R:0 < y \; or\; y> 1 \right \}$$

Q28 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
$$\left \{ y\epsilon R:2 < y < 6 \right \}$$
$$\left \{ y\epsilon R:2\leq y < 6 \right \}$$
$$\left \{ y\epsilon R:2\leq y\geq 6 \right \}$$
$$\left \{ y\epsilon R:2 < y\geq 6 \right \}$$

Q29 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 only symmetric
is only reflexive and transitive
is only symmetric and Transitive
Is reflexive, symmetric and transitive

Q30 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
140
120u
140h
20h

Q31 Which of the following is divisible by 17 for all positive integer n
$$7^{n}+2$$
$$6^{n}+2$$
$$2.7^{n}+3.5^{n}-5$$
$$3.5^{2n+1}+2^{3n+1}$$

Q32 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 $$f_{X}$$
$$f^{-1}_{X}(a,b)=(3a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{5}_{X}(a,b)=(2a-b,\; -5a+3b)$$
$$f^{1}_{X}(a,b)=(2a-b,\; -5a+4b)$$

Q33 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
Only unto
only injective
bijective
no solution

Q34 Which of the following pair of functions has f o g = g o f
$$f(y)=y^{3} \; and \; g(y)=\sqrt[3]{y}$$
$$f(y)=y^{5} \; and \; g(y)=3y+7$$
$$f(y)=y^{2} \; and \; g(y)=y+7$$
$$f(y)=y^{2} \; and \; g(y)=3y+7$$

Q35 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
, f1 and g1
f2 and g1
f2 and g2

Q36 Find the order of element -1 in the multiplicative group $$\left\{1,-1,- i, (-i) \right\}$$
0
3
1
2…ans

Q37 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 $$a^{e} = n$$
only (ii)
(i) and (ii)…ans
(ii) and (iii)
(i) and (iii)

Q38 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
Quarterion’s theorem
Unit group theorem
Lagrange�??s theorem
Polymorphism

Q39 Find all the real number that satisfy the inequality $$1 < x^{2} < 4$$
$$ \left \{x:-1 < x < 2 \; or \; -2 < x < -1\right \}$$
$$ \left \{x:1 < x < 2 \; or \; -2 < x < 1\right \}$$
$$ \left \{x:-1 < x < 2 \; or \; -2 < x < -1\right \}$$
$$ \left \{x:1 < x < 2 \; or \; -2 < x < -1\right \}$$

Q40 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)
(i) only
(ii) and (iii)
(i) and (ii)

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