Free Printable MA8551 / ANT

1.

When the group G is called Abelian group

a)

It's satisfies Closure

b)

It's satisfies Commutative

c)

It's satisfies Associative

2.

For every group G, if $a,b,c\in G$ and $ab=ac$ then $b=c$

a)

Left cancellation law

b)

Right cancellation law

c)

Abelian group

3.

When the set G is written as $\left(ab\right)^{-1}=a^{-1}b^{-1}$ for all $a,b\in G$

a)

G is a group

b)

G is an abelian

c)

G is commutative

4.

Define Subgroup

a)

A empty subset H of a group G is called a sub-group of G whenever H itself a group under the binary operation in G

b)

A non-empty subset H of a group G is called a sub-group of G whenever H itself a group under the binary operation in G

c)

A non-empty subset H of a group G is called a sub-group of G

5.

When $H\cap K$ is called Subgroup of G

a)

H is Subgroup of G

b)

K is subgroup of G

c)

Both H and K are subgroup of G

6.

Which one is the subgroups of $\left(Z_{11}^*,\ \circ\right)$

a)

{1, 10}

b)

{1, 3, 4, 5, 9}

c)

both

7.

Which one is the subgroups of $\left(Z_{11}^*,\ \circ\right)$

a)

{1, 10}

b)

{1, 3, 4, 5, 9}

c)

both

8.

$f_1$ is the identity we can rotate $90\degree$ we get $f_2$ . what is the order of $f_2$

a)

4

b)

2

c)

1

9.

Which one is the properties of group homomorphism

a)

ab=ba

b)

$f\left(a^n\right)=\left[f\left(a\right)\right]^n$

c)

$a+b=ab$

10.

Every subgroup of a cyclic group is

a)

group

b)

subgroup

c)

cyclic

11.

State Lagrange's theorem

a)

If G is a finite group of order n with H is a subgroup of order m, then m divides n

b)

If H is a group of order n then n divides any integer

c)

If G is a finite group of order m with H is a subgroup of order n then m divides n

12.

When the ring is called commutative ring

a)

A ring R satisfies commutative law under multiplication

b)

A ring R satisfies Closure condition

c)

Not defined

13.

The value of $\left[25\right]^{-1}$ in $Z_{72}$ is

a)

41

b)

24

c)

49

14.

For all integers n exactly one of n, 2n-1 and 2n+1 is divisible by

a)

2

b)

3

c)

5

15.

How many zero divisors are there in $Z_{17}$

a)

16

b)

17

c)

0

16.

How many units are there in the ring $Z_7$

a)

6

b)

4

c)

5

17.

How many polynomials are there of degree

$n$ in $Z_{11}\left[x\right]$ ?

a)

$10\left(11\right)^n$

b)

10 (11)

c)

$11^n$

18.

Let $f\left(x\right),g\left(x\right)\in Z_7\left(x\right)$ where $f\left(x\right)=2x^2+x+2$ and $g\left(x\right)=6x+1$ find $f\left(x\right)+g\left(x\right)$

a)

$7x+3$

b)

$2x^2+7x+3$

c)

$2x^2+3$

19.

If $f\left(x\right)=x^2+1,\ g\left(x\right)=x^4+x^3+x^2+x+1,\ f\left(x\right),\ g\left(x\right)\in Z_2\left[x\right],$

find $r\left(x\right)$

a)

1

b)

2

c)

0

20.

If $f\left(x\right),\ g\left(x\right)\in Q\left[x\right],\ f\left(x\right)=x^8+7x^5-4x^4+3x^3+5x^2-4,\ g\left(x\right)=x-3$

find the remainder when $f\left(x\right)$ is divided by $g\left(x\right)$

a)

8060

b)

3

c)

8660

21.

If $f\left(x\right)=x^3+5x^2+2x+6,\ f\left(x\right)\in Z_7\left[x\right]$ , then determine all of the roots in $Z_7$ and write $f\left(x\right)$ as a product of first degree polynomials.

a)

$f\left(x\right)=\left(x-1\right)\left(x-3\right)\left(x-5\right)$

b)

$f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-5\right)$

c)

$\left(x-2\right)\left(x-4\right)\left(x-6\right)$

22.

How many units are there in the ring $Z_5\left[x\right]$

a)

5

b)

4

c)

0

23.

Let $ab=ba=u$ , then $b$ is called

a)

multiplicative inverse of $a$

b)

additive inverse of $a$

c)

Zero element

24.

Every non-zero polynomial of degree is less than or equal to 1 is called

a)

irreducible

b)

reducible

c)

none of the above

25.

When the polynomial is called monic

a)

degree < 1

b)

degree 0

c)

its leading coefficient is 1

26.

If $\left|F\right|=q$ and degree $s\left(x\right)=n$ , then $\frac{F\left[x\right]}{s\left(x\right)}$ contains

a)

$q^n$ element

b)

$q-n$ element

c)

n element

27.

What is the degree of $2x^3+x+1$

a)

2

b)

0

c)

3

28.

How many monic polynomials in $Z_7\left[x\right]$ have degree 4

a)

$7^5$

b)

$7^6$

c)

$7^4$

29.

What is the gcd of $f\left(x\right)=x^2+x-2,\ g\left(x\right)=x^5-x^4+x^3+x^2-x-1$

a)

$x-1$

b)

$x^2-2$

c)

$x-2$

30.

Check whether $S\left(x\right)=x^2+1$ is reducible or not in $Z_2\left[x\right]$

a)

Yes

b)

No

31.

Let $s\left(x\right)=x^4+x^3+1\in Z_2\left[x\right]$ What is the order of the field $\frac{Z_2\left[x\right]}{S\left(x\right)}?$

a)

8

b)

32

c)

16

32.

How many elements in $\frac{Z_p\left[x\right]}{s\left(x\right)}$ generate the multiplicative group of non-zero elements of this field?

a)

$p^n-1$

b)

$p^n$

c)

$p^{n-1}$

33.

Give the characteristic for $Z_{10}\left[x\right]$

a)

$10^n$

b)

$10^2$

c)

10

34.

Construct a finite field of 27 elements

a)

9

b)

3^3

c)

3

35.

A polynomial over R that is the zero element or is of degree 0 is called

a)

Degree 1

b)

Constant Polynomial

c)

Polynomial ring

36.

Which one is remainder theorem

a)

For

$f\left(x\right)\in F\left[x\right]$ and $a\in F,$ the remainder in the division of $f\left(x\right)$ by $x-a$ is $f\left(a\right)$

b)

If $f\left(x\right)\in F\left[x\right]\ and\ a\in F$ , then $x-a$ is a factor of $f\left(x\right)$ if and only if $a$ is a root of $f\left(x\right)$

c)

$g\left(x\right)=q\left(x\right)f\left(x\right)+r\left(x\right)$

37.

The Division Algorithm is contains ---- parts

a)

Two

b)

Three

c)

Infinite

38.

Find the positive factor of f(19)

a)

{1, 3, 19}

b)

{1, 19)

c)

{1, 3, 7, 19}

39.

25 div 5 is

a)

0

b)

5

c)

1

40.

Every non empty set of positive integers has a least number.

a)

a) The well ordering principle

b)

The pigeonhole principle

c)

The inclusion-Exclusion principle

41.

Let f(n) denote the number of positive factors of a positive integer n, Evaluate f(12)

a)

6

b)

4

c)

2

42.

Evaluate $\sum_{\text{d|18}}^{ }\left(\frac{1}{18}\right)$ where d is a positive integer.

a)

$\frac{33}{18}$

b)

$\frac{39}{18}$

c)

39

43.

The sum and the product of any two even integers are ---

a)

Even

b)

Odd

c)

None of these

44.

Whether it is true or false. Let a and b be positive integers such that a|b and b|a. Then a = b

a)

False

b)

True

45.

Express 676 = ( )_eight.

a)

1244

b)

1145

c)

1028

46.

Find the number of ones in the binary representation of $2^5-1$

a)

4

b)

3

c)

5

47.

Express (1101)₂ into octal integer.

a)

15

b)

17

c)

21

48.

Find the positive integer of 1976. It is divisible by 13

a)

135

b)

138

c)

357

49.

Does the conjecture hold for $R_{10}^2$ ?

a)

Yes

b)

No

50.

Using the formula of $\pi\left(n\right)$ find the number primes $\le100$

a)

25

b)

23

c)

21

51.

There are infinitely many primes

a)

Euclid

b)

Euler

c)

Prime

52.

Two positive integers a and b are relatively prime if their gcd is…….

a)

2

b)

1

c)

None of these

53.

In linear combination the values of $\alpha$ and $\beta$ is

a)

Unique

b)

Need not be unique

c)

None of these

54.

Find

$ab$ if $\left[a,b\right]=156$ and a and b are relatively prime.

a)

$ab=1$

b)

$ab=156$

c)

$ab=155$

d)

$ab=157$

55.

Express (28, 12) as a linear combination of 28 and 12

a)

1\left(28\right)+\left(-2\right)12

b)

$\left(-1\right)28+2\left(12\right)$

c)

$\left(-1\right)28+\left(-2\right)12$

d)

$1\left(28\right)+2\left(12\right)$

56.

Express $\left(1101\right)_2$ into a hexadecimal digit.

a)

D

b)

A

c)

15

d)

C

57.

When the LDE is solvable

a)

$\text{d|c}$

b)

$\text{c|d}$

c)

$\text{d|a}$

d)

$\text{d|b}$

58.

The LDE has ---- number of solution

a)

1

b)

2

c)

infinite

d)

None of the above

59.

a\equiv a\left(mod\ m\right)

is called

a)

Reflexive

b)

Transitive

c)

Symmetric

d)

Anti Symmetric

60.

When the linear congruence

$ax\equiv b\left(mod\ m\right)$ has a unique solution

a)

$\left[a,\ m\right]=1$

b)

$\left(a,\ m\right)=1$

c)

$\left(a,\ m\right)\ne1$

d)

$\left(a,\ m\right)=d$

61.

The congruence

$12x\equiv24\left(mod\ 6\right)$ has how many number of incongruent solution

a)

6

b)

4

c)

2

d)

1

62.

Using casting out nines, find the value of 68464

a)

2

b)

0

c)

5

d)

1

63.

check the following statement: The linear system has a more number of solution modulo if and only if

\left(\Delta,\ m\right)=1

a)

Statement is correct

b)

Statement is wrong

64.

A palindrome with an even number of digits is

a)

divisible by 10

b)

Not divisible by 11

c)

divisible by 11

d)

not divisible by 10

65.

What is the remainder of 5! Is divided by 15

a)

0

b)

3

c)

1

d)

5

66.

Whether the LDE

$1076x+2076y=3076$ is solvable

a)

Yes, it is solvable

b)

It is not solvable

c)

Not defined

67.

Whether the number 548152 is divisible by 11

a)

Yes

b)

No

68.

The LDE is of the form

a)

ax-by=c

b)

$ax+by=c$

c)

$ax+by+c=0$

69.

Which one is the cassini's formula

a)

$F_{n-1}F_{n+1}+F_n^2=\left(-1\right)^n$

b)

$F_{n+1}F_{n-1}-F_n^2=\left(1\right)^n$

c)

$F_{n+1}F_{n-1}-F_n^2=\left(-1\right)^n$

70.

Using modulo 9, find the missing digit d in the congruence 7167 - 1776 = 53d1

a)

$either\ d=0\ or\ d=9$

b)

$d=-9$

c)

$d\ne0$

71.

The solution of sun Tsu's puzzle by iteration is

a)

$x=52-105t$

b)

$x=52+105t$

c)

$x=-52-105t$

72.

Every integer n in base b is congruent to the sum of its digits

a)

modulo b+1

b)

modulo -b-1

c)

modulo b - 1

73.

Which one of the following statement is correct

a)

Every odd integer is congruent to 1 or 3 modulo 4

b)

The square of every integer is congruent to 1 modulo 4

c)

Both a and b

d)

None of the above

74.

In CRT the moduli are

a)

Prime

b)

Composite

c)

Pairwise prime

d)

Pairwise relatively prime

75.

Using casting out nines, find the value of 68464

a)

2

b)

0

c)

5

d)

1

76.

The product of four consecutive integers is divisible by

a)

5

b)

10

c)

12

d)

14

77.

In Fibonacci series which one is the 9^th position

a)

21

b)

25

c)

34

d)

45

78.

What is the value of $\tau\left(p\right)$ where p is a prime

a)

3

b)

1

c)

2

d)

0

79.

Is $\sigma\left(p^e\right)=\frac{p^{e+1}+1}{p+1}$ ?

a)

Yes

b)

No

80.

Let n be a positive integer, then the Tau function denotes

a)

The number of positive number of n

b)

The number of prime factor of n

c)

The number of positive factors of n

d)

The least positive factor of n

81.

In multiplicative function the gcd of the two positive integer m and n are

a)

Prime

b)

Relatively Prime

c)

Composite

d)

Either Prime or Composite

82.

Let m be a positive integer and $\phi\left(m\right)$ denotes the number of positive integers is less than or equal to m and relatively prime to is called -----

a)

Euler's Theorem

b)

Multiplication Function

c)

Fermat's Theorem

d)

Euler's Phi Function

83.

Let p be a prime and a any integer such that p does not divide a then -----

a)

$a^{p+1}\equiv1\left(mod\ p\right)$

b)

$a^{p-1}\equiv-1\left(mod\ p\right)$

c)

$a^{p-1}\equiv1\left(mod\ p\right)$

d)

$a^{p+1}\equiv-1\left(mod\ p\right)$

84.

Whether the statement is true or false. If the congruence

$x^2\equiv1\left(mod\ m\right)$ has exactly two solutions, then m is a prime

a)

True

b)

False

85.

Solve the congruence

$x^2\equiv1\left(mod\ 6\right)$

a)

$x=1,2,5$

b)

$x=1,5$

c)

$x=2,5$

d)

$x=1,6$

86.

When a positive integer a is self-invertible.

a)

$a\equiv1\left(mod\ p\right)$

b)

$a\equiv-1\left(mod\ p\right)$

c)

Either a or b

d)

None of the above

87.

The Tau and Sigma functions are

a)

Additive

b)

Multiplicative

c)

both a and b

d)

neither a nor b

88.

What is the value of $\sigma\left(p\right)$ , where $p$ is a prime

a)

2

b)

1

c)

$p-1$

d)

$p+1$

89.

The sum of the positive factor of $pq$ is

a)

$1+p+q$

b)

$1-p+q+pq$

c)

$p+q+pq$

d)

None of the above

90.

Let m be a positive integer and a any integer with (a, m) = 1 then $a^{\phi\left(m\right)-1}$ is

a)

an inverse of a modulo m

b)

an inverse of $a^m$ modulo m

c)

called Fermat's little theorem

d)

Euler's Phi function

91.

What is the canonical decomposition of the number 24

a)

7.3

b)

$2^2.6$

c)

2^3.3

d)

$2^2.3^2$

92.

A number theoretic function f is multiplicative if

a)

f\left(mn\right)=f\left(m\right)f\left(n\right)

b)

$f\left(mn\right)=\frac{f\left(m\right)}{f\left(n\right)}$

c)

$f\left(mn\right)=f\left(m\right)-f\left(n\right)$

d)

$f\left(mn\right)=f\left(n\right)$

93.

Which one is wrong

a)

Let $p$ be a prime and a any positive integer. Then $a^p\equiv a\left(mod\ p\right)$

b)

If p is a prime and p be odd, then $2\left(p-3\right)!\equiv-1\left(mod\ p\right)$

c)

A positive integer $n\ge2$ is a prime if and only if $\left(n-2\right)!\equiv-1\left(mod\ n\right)$

d)

All the above

94.

If $p$ is a prime, then $\left(p-1\right)!\equiv-1\left(mod\ p\right)$ is called

a)

Fermat’s theorem

b)

Wilson’s theorem

c)

Euler’s theorem

d)

Multiplicative function

95.

The value of $\sum_{\text{d|n}}^{ }\phi\left(d\right)$ for $n=9$

a)

7

b)

9

c)

11

d)

13

96.

The sigma function denote the ---- of the positive factors of n

a)

multiple

b)

sum

c)

number

d)

function

97.

Let $\left(a,\ m\right)=1$ and then solution of the linear congruence $ax\equiv b\left(mod\ m\right)$ is

a)

$x\equiv a^{\phi\left(m\right)-1}b\left(mod\ m\right)$

b)

$x\equiv\alpha^{\phi\left(p\right)-1}b\left(mod\ p\right)$

c)

$x\equiv a^{\phi\left(m\right)-1}b\left(mod\ p\right)$

d)

$x\equiv a^{\phi\left(m\right)-1}\left(mod\ m\right)$

98.

How many units are there in the ring $Z_2\times Z_2\times Z_2$

a)

0

b)

2

c)

1

d)

3

99.

Which one is not a group among the following?

a)

$\left(Z,\ +\right)$

b)

$\left(Z,\ \circ\right)$

c)

$\left(Q,\ +\right)$

100.

Which one is an abelian group from the following sets?

a)

$\left(Z,\ +\right)$

b)

$\left(Z,\ \circ\right)$

c)

$\left(Q,\ \circ\right)$

MA8551 / ANT - MODEL

Total questions: 100

Worksheet time: 2hrs 40mins

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