Number Theory Test 32

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Number Theory Test 32

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Weekly Quiz Competition

Question 1
1 / -0

If $${z}_{1}=-3+5i;{z}_{2}=-5-3i$$ and $$z$$ is a complex number lying on the line segment joining $${z}_{1}$$ and $${z}_{2}$$, then $$arg(z)$$ can be:

A
$$-\cfrac { 3\pi }{ 4 } $$

B
$$-\cfrac { \pi }{ 4 } $$

C
$$\cfrac { \pi }{ 6 } $$

D
$$\cfrac { 5\pi }{ 6 } $$
Question 2
1 / -0

Which of the following is not a binary number?

A
001

B
101

C
202

D
110

Solution
Question 3
1 / -0

Find a complex number z satisfying the equation $$z+\sqrt{2}|z+1|+i=0$$

A
$$2-i$$

B
$$-2-i$$

C
$$\sqrt{2}-i$$

D
None of these

Solution

Let $$z=x+iy$$. Then,

$$z+\sqrt{2}|z+1|+i=0$$

$$x+iy+\sqrt{2}|(x+1)+iy|+i=0$$

$$x+\sqrt{2}\sqrt{(x+1)^2+y^2}+(y+1)i=0$$

$$x+\sqrt{2(x+1)^2+2y^2}+(y+1)i=0$$

$$x+\sqrt{2(x+1)^2+2}=0$$ and $$y=-1$$

$$\sqrt{2(x+1)^2+2}=-x$$ and $$y=-1$$

$$2(x+1)^2+2=x^2$$ and $$y=-1$$

$$x^2+4x+4=0$$ and $$y=-1$$

$$(x+2)^2=0$$ and $$y=-1$$

$$x=-2$$ and $$y=-1$$

Hence, $$z=x+iy=-2-i$$
Question 4
1 / -0

If $${ a }^{ 2 }+{ b }^{ 2 }=1$$, then $$\dfrac {\left( 1+b+ia \right) }{\left( 1+b-ia \right)} $$ is

A
$$1$$

B
$$2$$

C
$$b+ia$$

D
$$a+ib$$

Solution

$$a^2+b^2=1$$

$$\implies 1-a^2=b^2$$

$$\dfrac{1+b+ia}{1+b-ia}$$

$$=\dfrac{1+b+ia}{1+b-ia}\times \dfrac{1+b+ia}{1+b+ia}$$

$$=\dfrac{(1+b)^2-a^2+2ia(1+b)}{(1+b)^2+a^2}$$

$$=\dfrac{b^2+2b+1-a^2+2ia(1+b)}{1+2b+b^2+a^2}$$

$$=\dfrac{2b^2+2b+2ia(1+b)}{2+2b}$$

$$=\dfrac{(2b+2ia)(1+b)}{2(1+b)}$$

$$=b+ia$$

Answer-(C)
Question 5
1 / -0

Multiplication of $$111_2$$ by $$101_2$$ is?

A
$$110011_2$$

B
$$100011_2$$

C
$$111100_2$$

D
$$000101_2$$

E
None of the above

Solution

Multiplication of 1112 by 1012 is 1000112

A binary multiplier is an electronic circuit used in digital electronics, such as a computer, to multiply two binary numbers. The lattice method is an alternative to long multiplication for numbers. In this approach, a lattice is first constructed, sized to fit the numbers being multiplied. If we are multiplying an -digit number by an -digit number, the size of the lattice is .
Question 6
1 / -0

If $$\dfrac {lz_{2}}{mz_{1}}$$ is purely imaginary number, then $$\left |\dfrac {\lambda z_{1} + \mu z_{2}}{\lambda z_{1} - \mu z_{2}}\right |$$ is equal to

A
$$\dfrac {l}{m}$$

B
$$\dfrac {\lambda}{\mu}$$

C
$$\dfrac {-\lambda}{\mu}$$

D
$$1$$

Solution

$$\cfrac { { lz }_{ 2 } }{ m{ z }_{ 1 } } =ki\\ \cfrac { { z }_{ 2 } }{ { z }_{ 1 } } =\cfrac { mki }{ l } \\ \implies |\cfrac { { \lambda z }_{ 1 }+\mu { z }_{ 2 } }{ { \lambda z }_{ 1 }-\mu { z }_{ 2 } } |\\ \implies |\cfrac { \lambda +\mu { z }_{ 2 }/{ z }_{ 1 } }{ \lambda -\mu { z }_{ 2 }/{ z }_{ 1 } } |\\ \implies |\cfrac { \lambda +\mu \cfrac { mki }{ l } }{ \lambda -\mu \cfrac { mki }{ l } } |\\ \implies|\cfrac { \lambda l+\mu mki }{ \lambda l-\mu mki } |$$
$$\implies|\cfrac{re^{i\theta}}{re^{-i\theta}}|$$
$$\implies |e^{2i\theta}|$$
$$\implies 1$$
Question 7
1 / -0

An output device that converts data from a binary format in main storage to coded hole patterns punched into a paper tape is?

A
Paper tape punch

B
Punched paper tape

C
Magnetic disk

D
Magnetic tape

E
None of the above

Solution

An output device that converts data from a binary format in main storage to coded hole patterns punched into a paper tape is Paper tape punch

Punched tape or perforated paper tape is a form of data storage, consisting of a long strip of paper in which holes are punched to store data. Now effectively obsolete, it was widely used during much of the twentieth century for teleprinter communication, for input to computers of the 1950s and 1960s, and later as a storage medium for minicomputers and CNC machine tools.
Question 8
1 / -0

The $$2$$'s complement number of $$110010$$ is?

A
$$001101$$

B
$$110011$$

C
$$010011$$

D
All of the above

E
None of the above

Solution

The 2's complement number of 110010 is None of the above.

2's complement number of (110010)=001110.
Question 9
1 / -0

Instructions and memory addresses are represented by.

A
Character codes

B
Binary codes

C
Binary word

D
Parity bit

E
None of the above

Solution

Instructions and memory addresses are represented by binary codes.

In computing, a memory address is a reference to a specific memory location used at various levels by software and hardware. Memory addresses are fixed-length sequences of digits conventionally displayed and manipulated as unsigned integers.

ASCII code. The American Standard Code for Information Interchange (ASCII), uses a 7-bit binary code to represent text and other characters within computers, communications equipment, and other devices. Each letter or symbol is assigned a number from 0 to 127.
Question 10
1 / -0

The real part of $$(1-\cos\theta +2i \sin\theta)^{-1}$$ is?

A
$$\displaystyle\frac{1}{3+5\cos\theta}$$

B
$$\displaystyle\frac{1}{5-3\cos\theta}$$

C
$$\displaystyle\frac{1}{3-5\cos\theta}$$

D
$$\displaystyle\frac{1}{5+3\cos\theta}$$

Solution

$$(1-\cos\theta +2i \sin\theta)^{-1}$$
$$=\dfrac{1}{(1-\cos\theta +2i \sin\theta)}$$
$$=\dfrac{1}{(2\sin^2\dfrac{\theta}{2} +4i \sin\dfrac{\theta}{2}.\cos \dfrac{\theta}{2})}$$
$$=\dfrac{1}{2\sin\dfrac{\theta}{2}}\dfrac{1}{(\sin\dfrac{\theta}{2} +2i \cos\dfrac{\theta}{2})}$$
$$=\dfrac{1}{2\sin\dfrac{\theta}{2}}\dfrac{\left(\sin\dfrac{\theta}{2}-2i\cos \dfrac{\theta}{2}\right)}{(\sin^2\dfrac{\theta}{2} +4 \cos^2\dfrac{\theta}{2})}$$
$$=\dfrac{1}{\sin\dfrac{\theta}{2}}\dfrac{\left(\sin\dfrac{\theta}{2}-2i\cos \dfrac{\theta}{2}\right)}{(2\sin^2\dfrac{\theta}{2} +8 \cos^2\dfrac{\theta}{2})}$$
$$=\dfrac{\left(1-2i\cot \dfrac{\theta}{2}\right)}{\{1-\cos \theta+4(1+\cos \theta)\}}$$
$$=\dfrac{\left(1-2i\cot \dfrac{\theta}{2}\right)}{5+3\cos \theta}$$
So the real part is $$\dfrac{1}{5+3\cos \theta}$$..

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Number Theory Test 32

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Number Theory Test 32

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Question 1

$$-\cfrac { 3\pi }{ 4 } $$

$$-\cfrac { \pi }{ 4 } $$

$$\cfrac { \pi }{ 6 } $$

$$\cfrac { 5\pi }{ 6 } $$

Question 2

001

101

202

110

Solution

Question 3

$$2-i$$

$$-2-i$$

$$\sqrt{2}-i$$

None of these

Solution

Question 4

$$1$$

$$2$$

$$b+ia$$

$$a+ib$$

Solution

Question 5

$$110011_2$$

$$100011_2$$

$$111100_2$$

$$000101_2$$

None of the above

Solution

Question 6

$$\dfrac {l}{m}$$

$$\dfrac {\lambda}{\mu}$$

$$\dfrac {-\lambda}{\mu}$$

$$1$$

Solution

Question 7

Paper tape punch

Punched paper tape

Magnetic disk

Magnetic tape

None of the above

Solution

Question 8

$$001101$$

$$110011$$

$$010011$$

All of the above

None of the above

Solution

Question 9

Character codes

Binary codes

Binary word

Parity bit

None of the above

Solution

Question 10

$$\displaystyle\frac{1}{3+5\cos\theta}$$

$$\displaystyle\frac{1}{5-3\cos\theta}$$

$$\displaystyle\frac{1}{3-5\cos\theta}$$

$$\displaystyle\frac{1}{5+3\cos\theta}$$

Solution

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