Number Theory Test 33

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Number Theory Test 33

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Question 1
1 / -0

Binary number are used because:

A
Decimal system can not be represented on motherboard

B
Binary data needs just 2 wires for the transmission, one for 1 & other for 0

C
Binary data is easier to represent using 'on'(1) & 'off' (0) states of switches.

D
All of these

Solution

Binary number are used because binary data is easier to represent using 'on'(1) & 'off' (0) states of switches.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one). The base-2 numeral system is a positional notation with a radix of 2.

Binary in Digital Computers and Electronic Devices. Numbers can be encoded in binary format and stored using switches. In a computer, switches are implemented using transistors.
Question 2
1 / -0

The complex number $$\dfrac{1+2i}{1-i}$$ lies in which quadrant of the compiles plan

A
First

B
Second

C
Third

D
Fourth

Solution

$$z=\dfrac{1+2i}{1-i}$$
Multiply the coyugate of $$(1-i)$$ is numerator $$\times$$ denominator
$$z=\dfrac{(1+2i)(1+i)}{(1-i)(1+i)}$$
$$=\dfrac{1+i+2i+(2i)^{2}}{1-i^{2}}$$
$$=\dfrac{1+3i-4}{1+1}$$
$$=\dfrac{-3+3i}{2}$$
$$=\dfrac{-3}{2}+\dfrac{3}{2}i$$
$$\therefore$$ It is of the form $$-a+3i (-a, b)$$
$$\therefore \left(\dfrac{-3}{2}, \dfrac{3}{2}\right)$$
$$\therefore 2^{nd}$$ Quandrant
Question 3
1 / -0

Which are not property of Binary relations

A
reflexive relation, symmetric relation, anti symmetrical relation

B
reflexive, transitive, equivalence relations

C
transitive, partial ordering relation, symmetric

D
reflexive, partial, chain relation

Solution

Reflexive, partial, chain relation are not property of Binary relations.

Reflexive relation, a relation where elements of a set are self-related. Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited.

To be a partial order, a binary relation must be reflexive (each element is comparable to itself), antisymmetric (no two different elements precede each other), and transitive.
A binary relation on a set A is a set of ordered pairs of elements of A
Question 4
1 / -0

If Z satisfied the equation $$\left ( \dfrac{Z- 2}{Z + 2} \right ) \, \left ( \dfrac {{\bar Z} - 2}{{\bar Z} + 2} \right )$$then minimum, value of$$ |Z| $$is equal to

A
0

B
2

C
4

D
6

Solution

$$(Z - 2) [{\bar Z}- {\bar2}](Z -+2) [{\bar Z} -2]$$
$$|Z - 2|^ 2 \, = | Z \, + \, 2| \Rightarrow |Z - 2| = |Z + \, 2| $$
Line on imaginary axis
$$|Z_1|min = 0$$
Question 5
1 / -0

If $$\mid{z_1}\mid=2$$, $$\mid{z_2}\mid=3$$, $$\mid{z_3}\mid=4$$ and $$\mid{z_1+z_2+z_3}\mid=2$$, then the value of $$\mid{4z_2z_3+9z_3z_1+16z_1z_2}\mid$$.

A
24

B
48

C
96

D
120

Solution

$$z_1\bar{z_1}=|z_1|^2=1$$$$,z_2\bar{z_2}=|z_2|^2=9$$$$,z_3\bar{z_3}=|z_3|^2=6$$
$$|4z_2z_3+9z_3z_1+16z_1z_2|$$
$$\implies |\bar{z_1}z_1z_2z_3$$$$+\bar{z_2}z_1z_2z_3$$$$+\bar{z_3}z_1z_2z_3|$$
$$\implies |z_1||z_2||z_3||\bar{z_1}+\bar{z_2}+\bar{z_3}|$$
$$\implies 24\times$$ $$|\bar{z_1}+\bar{z_2}+\bar{z_3}|$$
$$\implies 48$$
Question 6
1 / -0

Evaluate:
$${ \left( \dfrac { cos\dfrac { \pi }{ 8 } -isin\dfrac { \pi }{ 8 } }{ cos\dfrac { \pi }{ 8 } +isin\dfrac { \pi }{ 8 } } \right) }^{ 4 }$$

A
$$1$$

B
$$-1$$

C
$$2$$

D
$$\dfrac { 1 }{ 2 } $$

Solution

We know that $$e^{i\theta}=cos\theta+isin\theta$$ and $$e^{-1\theta}=cos\theta-isin\theta$$

$${ \left( \dfrac { cos\dfrac { \pi }{ 8 } -isin\dfrac { \pi }{ 8 } }{ cos\dfrac { \pi }{ 8 } +isin\dfrac { \pi }{ 8 } } \right) }^{ 4 }$$

$$=\left(\dfrac{e^{-i(\pi/8)}}{e^{i(\pi/8)}}\right)^4=(e^{-i(\pi/4)})^4=e^{-i\pi}=cos\pi-isin\pi=-1$$
Question 7
1 / -0

In binary synchronous, communication ............code is used by the receiver to check the validity of the message recovered.

A
$$OK$$

B
$$ACK$$

C
$$BCC$$

D
$$SOH$$

Solution

In binary synchronous, communication ACK code is used by the receiver to check the validity of the message recovered.

Binary Synchronous Communication (BSC or Bisync) is an IBM character-oriented, half-duplex link protocol, announced in 1967 after the introduction of System/360. It replaced the synchronous transmit-receive (STR) protocol used with second generation computers. The intent was that common link management rules could be used with three different character encodings for messages. Six-bit Transcode looked backwards to older systems; USASCII with 128 characters and EBCDIC with 256 characters looked forward. Transcode disappeared very quickly but the EBCDIC and USASCII dialects of Bisync continued in use.

An acknowledgment code (ACK) is a type of unique signal that a computer sends to show that data has been transmitted successfully. The acknowledgement code is an ASCII character that has been designated to serve as a signal between sender and recipient.

Error Correction Code (ECC) is a more comprehensive method of data integrity checking that can detect and correct single-bit errors. Fewer and fewer PC manufacturers are supporting data integrity checking in their designs.
Question 8
1 / -0

Which of the following is NOT a binary system?

A
EBCDIC

B
ASCII

C
HEX

D
None of these

Solution

HEX is not a binary system.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one). The base-2 numeral system is a positional notation with a radix of 2.

Hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A–F (or alternatively a–f) to represent values ten to fifteen.
Question 9
1 / -0

..............is a binary synchronous data transmission.

A
Each bit is sent over a separate wire

B
Sequence numbers are sent in numbered frames

C
Each character is prefixed with DLE

D
Character are used for control purpose

Solution

Sequence numbers are sent in numbered frames is a binary synchronous data transmission.

The sequence number indicates how the frames that are being sent are numbered. For example, if the frames are numbered from 0-7, then it is a 3-bit sequence number. If the frames are numbered from 0-15 then it is a 4-bit sequence number.

The send sequence number is the number (in the specified modulus) of the current I-frame being sent by the local node. Each transmitted I-frame is numbered in succession with a sequence number. This is implemented by copying the send state variable, V(S), into the send sequence number, N(S), in the frame. After a frame is transmitted the send state variable is incremented. The N(S) in retransmitted frames are not changed (i.e. the original N(S) value is sent in the retransmitted frame).

Binary Synchronous Communications. Binary Synchronous Communication (BSC or Bisync) is an IBM character-oriented, half-duplex link protocol, announced in 1967 after the introduction of System/360. It replaced the synchronous transmit-receive (STR) protocol used with second generation computers.

HDLC (High-level Data Link Control) is a group of protocols or rules for transmitting data between network points (sometimes called nodes). In HDLC, data is organized into a unit (called a frame) and sent across a network to a destination that verifies its successful arrival.
Question 10
1 / -0

$$ | \frac{z_1 - 2z_2}{2 - z_1\bar{z}_2} | = 1$$ and $$|z_2| \neq 1$$ then the value of $$|z_1|$$ is

A
4

B
2

C
1

D
$$\frac{1}{2}$$

Solution

$$\left| \dfrac{z_{1}-2z_{2}}{2-z_{1}\bar {z_{2}}} \right|= 1, |z_{2}| \neq 1$$
$$\Rightarrow z_{1} \bar {z_{2}} \neq 2, |z_{1}- 2z_{2}|= |2-z_{1}\bar{z_{2}}|$$
$$\Rightarrow |z_{1}-2z_{2}|^{2}= |2- z_{1} \bar{z_{2}}|^{2}$$
$$\Rightarrow |z_{1}|^{2}+4 |z_{2}|^{2}- (2 \bar{z_{1}z_{2}}+2z_{1} \bar{z_{2}})$$
$$= 4+ |z_{1}|^{2}+ |\bar{z_{2}}|^{2}- (2 \bar{z_{1}z_{2}}+2 z_{1} \bar{z_{2}})$$
$$\Rightarrow |z_{1}|^{2} (1- |z_{2}|^{2})+4 (|z_{2}|^{2}-1)=0$$
$$[\because |\bar{z_{2}}|= |z_{2}|]$$
$$\Rightarrow (|z_{1}|^{2}-4)(1-|z_{2}|^{2})=0$$
$$|z_{2}| \neq 1 \Rightarrow |z_{2}|=2$$

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Number Theory Test 33

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Number Theory Test 33

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SHARING IS CARING

Question 1

Decimal system can not be represented on motherboard

Binary data needs just 2 wires for the transmission, one for 1 & other for 0

Binary data is easier to represent using 'on'(1) & 'off' (0) states of switches.

All of these

Solution

Question 2

First

Second

Third

Fourth

Solution

Question 3

reflexive relation, symmetric relation, anti symmetrical relation

reflexive, transitive, equivalence relations

transitive, partial ordering relation, symmetric

reflexive, partial, chain relation

Solution

Question 4

0

2

4

6

Solution

Question 5

24

48

96

120

Solution

Question 6

$$1$$

$$-1$$

$$2$$

$$\dfrac { 1 }{ 2 } $$

Solution

Question 7

$$OK$$

$$ACK$$

$$BCC$$

$$SOH$$

Solution

Question 8

EBCDIC

ASCII

HEX

None of these

Solution

Question 9

Each bit is sent over a separate wire

Sequence numbers are sent in numbered frames

Each character is prefixed with DLE

Character are used for control purpose

Solution

Question 10

4

2

1

$$\frac{1}{2}$$

Solution

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