Free Printable MA8551 / ANT

1.

Find all the subgroup of

\left(Z_{12},+\right)

a)

$\left\{0,\ 6\right\},\left\{0,\ 3,\ 6,\ 9\right\},\ \left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

b)

$\left\{0,\ 3,\ 6,\ 9\right\},\ \left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

c)

$\left\{0,\ 6\right\},\left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

2.

When\ the\ function\ f\ is\

called\ an\ isomorphism

a)

$A\ \hom ogeneous\ function\ is\ statifies\$ $both\ one-to-one\ and\ onto$

b)

$function\ is\ statifies\ both\ one-to-one\ and\ onto$

c)

$None\ of\ these$

3.

$Deter\min e\ U_{14},\ the\ group\ of\ units\ of\ the\ ring\ \left(Z_n,\ +,\ \circ\right)$

a)

$\left\{1,3,5,7,9,11,13\right\}$

b)

$\left\{1,3,5,9,11,13\right\}$

c)

$\left\{1,3,7,9,11,13\right\}$

4.

$When\ the\ ring\ R\ is\ called\ commutative\ ring$

a)

$If\ ab=ba\ for\ all\ a,b\in R$

b)

$If\ ab\ne ba\ for\ all\ a,b\in R$

c)

$If\ ab=ba\ for\ all\ a,b\notin R$

5.

$Find\ the\ value\ of\ additive$ $\ identity\ from\ the\ given\ equation$ $x+y=x+y-7,\ x\circ y=x+y-3xy$

a)

$Z=0$

b)

$Z=-1$

c)

$Z=7$

6.

$Find\ \left[100\right]^{-1}\ in\ the\ ring\ Z_{1009}$

a)

110

b)

101

c)

111

7.

$FInd\ the\ multiplicative\ inverse\ of\ 4\ in\ Z_{11}$

a)

4

b)

1

c)

3

8.

$What\ is\ the\ last\ digit\ in\ 3^{55}$

a)

1

b)

5

c)

7

9.

$Howmany\ units\ are\ there\ in\ the\ ring\ Z_8?$

a)

7

b)

4

c)

2

10.

$Which\ one\ is\ the\ correct\ form\ of$ $Division\ a\lg orithm$

a)

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

b)

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

c)

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

11.

How many polynomials are there of degree

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

a)

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

b)

10 (11)

c)

$11^n$

12.

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)

$2x^2+3$

13.

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)

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

14.

For\ f\left(x\right)\in F\left(x\right)\ and\ a\in F,\ then\

x-a\ is\ a\ factor\ f\left(x\right)\

iff\ a\ is\ root\ of\ f\left(x\right)

a)

Remainder Theorem

b)

Fermat's Theorem

c)

Factor Theorem

15.

$Howmany\ polynomials\ are\$ $there\ in\deg ree\ 2\ in\ Z_2\left[x\right]$

a)

2

b)

3

c)

4

16.

$How\ many\ units\ are\ there\ in\$ $the\ ring\ Z_p\left[x\right],\ p\ is\ prime$

a)

$p-1$

b)

$p$

c)

$p+1$

17.

$Give\ the\ characteristic\ for\ Q\left[x\right]$

a)

0

b)

1

c)

any integer

18.

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

19.

The Division Algorithm is contains ---- parts

a)

Two

b)

Three

c)

Infinite

20.

Find the positive factor of f(23)

a)

{1, 3, 23}

b)

{1, 23}

c)

{1, 3, 7, 19, 23}

21.

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

22.

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

23.

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

24.

Express 3014 = ( )_eight.

a)

1244

b)

5706

c)

1028

25.

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

a)

25

b)

23

c)

21

26.

$If\ p\ and\ p^2+8\ are\ primes,\ then\ p^3+4\ is$

a)

prime

b)

composite

c)

None of these

27.

GCD of (6, 141) is

a)

1

b)

6

c)

3

28.

When the LDE is solvable

a)

$\text{d|c}$

b)

$\text{c|d}$

c)

$\text{d|a}$

d)

$\text{d|b}$

29.

In Fibonacci series which one is the 9^th position

a)

21

b)

25

c)

34

d)

45

30.

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)

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

31.

Using casting out nines, find the value of 68464

a)

2

b)

0

c)

5

d)

1

32.

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

33.

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

34.

The solution of sun Tsu's puzzle by iteration is

a)

$x=52-105t$

b)

$x=52+105t$

c)

$x=-52-105t$

35.

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

36.

What is the remainder of 5! Is divided by 15

a)

0

b)

3

c)

1

d)

5

37.

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

38.

Find the remainder when $24^{16}$ is divided by 17.

a)

1

b)

-1

c)

2

d)

3

39.

Is the statement $\left(12+18\right)^{17}\equiv12^{11}+18^{17}$ correct

a)

Yes

b)

No

40.

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)

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

41.

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)

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

42.

What is the canonical decomposition of the number 24

a)

7.3

b)

$2^2.6$

c)

2^3.3

d)

$2^2.3^2$

43.

The Tau and Sigma functions are

a)

Additive

b)

Multiplicative

c)

both a and b

d)

neither a nor b

44.

Solve the congruence

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

a)

$x=1,2,5$

b)

$x=1,5$

c)

$x=2,5$

d)

$x=1,6$

45.

Find the ones digit in the base-seven expansion of

$5^{101}$

a)

3

b)

2

c)

-1

d)

0

MA8551 / ANT - MODEL II

Total questions: 45

Worksheet time: 58mins

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