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Number Theory Test 31

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Number Theory Test 31
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  • Question 1
    1 / -0
    If $$z + \sqrt {2}|z + 1| + i = 0$$ and $$z = x + iy$$, then
    Solution
    On putting $$z=x+iy$$, We get

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

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

    =>  On comparing real part and imaginary part, We get

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

     So, $$y=-1$$

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

    => $${2(x+1)^2+2} =x^2$$

    => $$x=-2$$


  • Question 2
    1 / -0
    The complex numbers $$z=x+iy$$ which satisfy the equation
    $$\left| \cfrac { z-5i }{ z+5i }  \right| =1$$ lie on:
    Solution
    $$|z-5i|$$$$=|z+5i|$$
    $$|x+iy-5i|$$$$=|x+iy+5i|$$
    $$\sqrt{x^2+(y-5)^2}$$$$=\sqrt{x^2+(y+5)^2}$$
    $$\implies x^2+y^2+25-10y$$$$=x^2+y^2+25+10y$$
    $$\implies y=0$$
    Hence $$x$$ axis.


  • Question 3
    1 / -0
    Let $$P(e^{i\theta_1})$$,  $$Q(e^{i\theta_2})$$  and  $$R(e^{i\theta_3})$$ be the vertices of a triangle PQR in the Argand Plane. The orthocenter of the triangle PQR is 
    Solution
    $$P,Q,R$$ lie on the unit circle and hence origin is the cirumcenter , The centrid is easily found as $$\dfrac{(p+q+r)}{3}$$,Using euler line theorem that centroid divides the line joining the line joining orthocenter and circumcenter in the ratio of $$2:1$$
    Let orthocenter be $$x,y$$

    $$Using \quad section \quad formula;$$
     
    $$Re(\dfrac{(p+q+r)}{3})=\dfrac{(1.x+2.0)}{3}$$
    $$Im(\dfrac{(p+q+r)}{3})=\dfrac{(1.y+2.0)}{3}$$

    we get orthocenter as 
    $$p+q+r\rightarrow$$   $$e^{\theta_1}+$$$$e^{\theta_2}+$$$$e^{\theta_3}$$
  • Question 4
    1 / -0
    The number which have only two factors $$(1$$ and the number itself) are called as ________.
    Solution

  • Question 5
    1 / -0
    The memory of a computer is commonly expressed in terms of Kilobytes or Megabytes. A byte is made up of
    Solution

    The memory of a computer is commonly expressed in terms of Kilobytes or Megabytes. A byte is made up of eight binary digits.

    In mathematics, positive numbers (including zero) are represented as unsigned numbers. ... We have also seen previously that an 8-bit binary number (a byte) can have a value ranging from 0 (000000002) to 255 (111111112), that is 28 = 256 different combinations of bits forming a single 8-bit byte.

    The largest number you can represent with 8 bits is 11111111, or 255 in decimal notation. Since 00000000 is the smallest, you can represent 256 things with a byte. (Remember, a bite is just a pattern. It can represent a letter or a shade of green.)

    Minecraft is a Java program, which means it is 64 bit if you are running 64 bit Java, or 32 bit otherwise. If you are talking about graphics, the textures are 24 bit. ... Color would be 8 bit.

    In mathematics and digital electronics, 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. Each digit is referred to as a bit.

  • Question 6
    1 / -0
    What is the byre capacity of a drum which is $$5$$ inch high, $$10$$ inch diameter, and which has $$60$$ tracks per inch and bit density of $$800$$ bits per inch?
    Solution

    942000 bytes is the byre capacity of a drum which is 5 inch high, 10 inch diameter, and which has 60 tracks per inch and bit density of 800 bits per inch.

    Each bitcoin (1 BTC) can have a fractional part of up to 8 digits so 1 bitcoin can be divided into 100 000 000 units. Each of these bitcoin units (0.00000001 BTC) is called a satoshi. Down to bits. A proposal floated on reddit has called for bitcoin to be broken down into just one smaller unit. That would mean that a single bitcoin would be divided into 1,000,000 individual units – or 'bits'. So, one bit would be worth 0.000001 BTC, which at recent prices would roughly amount

  • Question 7
    1 / -0
    What digits are representative of all binary numbers?
    Solution

    0 and 1 both digits are representative of all binary numbers.

    Binary is a base-2 number system that uses two mutually exclusive states to represent information. A binary number is made up of elements called bits where each bit can be in one of the two possible states. Generally, we represent them with the numerals 1 and 0 .

    A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often the binary number system's 0 and 1. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction.

  • Question 8
    1 / -0
    Computers use the .... number system to store data and perform calculations.
    Solution

    Computers use thebinarynumber system to store data and perform calculations.

    In mathematics and digital electronics, 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. Each digit is referred to as a bit.

    An easy method of converting decimal to binary number equivalents is to write down the decimal number and to continually divide-by-2 (two) to give a result and a remainder of either a “1” or a “0” until the final result equals zero. So for example. Convert the decimal number 29410 into its binary number equivalent.

    Binary numbers use the same rules as decimal - the value of any digit always depends on its position in the whole number. It all gets down to bases. Decimal uses base ten, so that every time a number moves one position to the left in a figure, it increases by a power of ten (eg. 1, 10, 100 etc).

    Binary is a base 2 number system invented by Gottfried Leibniz that is made up of only two numbers: 0 and 1. This number system is the basis for all binary code, which is used to write data such as the instructions that computer processors use, or the digital text you read every day.

  • Question 9
    1 / -0
    What is the highest address possible if $$16$$ bits are used for each address?
    Solution

    65536 is the highest address possible if 16 bits are used for each address.

    Using 16 bits, you can write 65536 addresses (from 0 to 65535, that's 65536 different addresses), and address 65536 bytes. 65536 bytes is 64kB. In computer science, b is bit, B is byte. The byte is the smallest amount of memory you can address.

    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.

    In computer architecture, 16-bit integers, memory addresses, or other data units are those that are 16 bits (2 octets) wide. Also, 16-bit CPU and ALU architectures are those that are based on registers, address buses, or data buses of that size. 16-bit microcomputers are computers in which 16-bit microprocessors were the norm.

  • Question 10
    1 / -0
    The $$0$$ and $$1$$ in the binary numbering system are called binary digits or ____________.
    Solution

    The 0 and 1 in the binary numbering system are called binary digits or Bits.

    A bit (short for binary digit) is the smallest unit of data in a computer. A bit has a single binary value, either 0 or 1. Although computers usually provide instructions that can test and manipulate bits, they generally are designed to store data and execute instructions in bit multiples called bytes.

    A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often the binary number system's 0 and 1. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction, etc.

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