All questions of Number System & Representation for Electronics and Communication Engineering (ECE) Exam

Explanation:

To solve the equation x3, we need to understand what the exponent 3 represents. In mathematics, an exponent signifies that the base number is multiplied by itself a certain number of times.

So, x3 can be written as x * x * x. This means that x is multiplied by itself three times.

Now, let's evaluate the given options to find the one that equals x3.

a) x6x3:
Here, x6 represents x multiplied by itself six times. Multiplying it by x3 would result in x9, not x3. So, this option is incorrect.

b) x6 x3:
This option is a bit unclear due to the formatting. It could be interpreted as x6 multiplied by x3, which would result in x9. However, if we assume that there is a missing operator between x6 and x3, then it would be x6 * x * x * x, which is equal to x9. Therefore, this option is also incorrect.

c) x6/ x3:
This option represents x6 divided by x3. When dividing two expressions with the same base (x in this case), we subtract the exponents. So, x6 divided by x3 is equal to x(6-3) = x3. Therefore, this option is correct.

d) (x6)3:
This option represents x6 raised to the power of 3. When raising an expression with an exponent to another exponent, we multiply the exponents. So, (x6)3 is equal to x(6*3) = x18, not x3. Therefore, this option is incorrect.

Conclusion:
Among the given options, option 'C' (x6/ x3) is the only one that equals x3. The other options either result in x9 or x18.

Sorry, but I need more information about the subtraction problem in order to provide a specific answer. Could you please provide the numbers you want to subtract?

Conversion of Decimal to Binary
To convert a decimal number to binary, we use the following steps:
1. Divide the decimal number by 2.
2. Write down the remainder (0 or 1).
3. Divide the quotient again by 2 and write down the remainder.
4. Repeat step 3 until the quotient becomes 0.

Conversion of Fractional Part
To convert the fractional part of a decimal number to binary, we use the following steps:
1. Multiply the fractional part by 2.
2. Write down the integer part.
3. Repeat step 1 with the fractional part until the fractional part becomes 0 or the desired number of bits is obtained.

Conversion of (57.375)10 to Binary
1. We divide 57 by 2 to obtain a quotient of 28 and a remainder of 1.
2. We divide 28 by 2 to obtain a quotient of 14 and a remainder of 0.
3. We divide 14 by 2 to obtain a quotient of 7 and a remainder of 0.
4. We divide 7 by 2 to obtain a quotient of 3 and a remainder of 1.
5. We divide 3 by 2 to obtain a quotient of 1 and a remainder of 1.
6. We divide 1 by 2 to obtain a quotient of 0 and a remainder of 1.
7. The integer part of the binary number is obtained by writing the remainders in reverse order: (111001)2.

To obtain the fractional part, we multiply 0.375 by 2 to obtain 0.75. The integer part is 0, so we write down 0. We then multiply 0.75 by 2 to obtain 1.5. The integer part is 1, so we write down 1. We then multiply 0.5 by 2 to obtain 1. The integer part is 1, so we write down 1. The fractional part is 0.011.

Therefore, the binary representation of (57.375)10 is (111001.011)2. The correct answer is option A.

There are no options provided, so it is not possible to determine which of the following is largest among others.

To perform binary subtraction, you need to borrow from the next higher bit if necessary. Here is the step-by-step process for subtracting 101111:

101111
- 1
------
101110

So, subtracting 1 from 101111 gives us the result 101110.

The binary number 111101 is equivalent to the decimal number 61.

Since it is not specified how the division should be done, I will assume that you want to divide it by 2.

When you divide 61 by 2, the quotient is 30 and the remainder is 1.

Therefore, 111101 divided by 2 is equal to 11110 with a remainder of 1.

6 is a whole number that comes after 5 and before 7. It is also an even number and a factor of 12.

Problem:
Find the addition of the binary numbers 11011011010 and 010100101.

Solution:
To add binary numbers, we follow the same rules as adding decimal numbers. We start from the rightmost bit and move towards the left, carrying over any carry values.

Step 1: Write the numbers vertically with the bits aligned.

11011011010
+ 010100101
________________

Step 2: Start adding the bits from right to left.

11011011010
+ 010100101
________________
111011011

Step 3: Carry over any carry values.

11011011010
+ 010100101
________________
111011011

Step 4: Continue adding the remaining bits.

11011011010
+ 010100101
________________
11101111111

Step 5: The addition is complete.

Therefore, the addition of the binary numbers 11011011010 and 010100101 is 11101111111, which corresponds to option 'C'.

It seems like your sentence is incomplete. Could you please provide more information or clarify what you are referring to?

Solution:

To find the result of 1397 x 1397, we will multiply these two numbers together.

Step 1: Multiply the ones place digits
- 7 x 7 = 49

Step 2: Multiply the tens place digits
- 9 x 9 = 81

Step 3: Multiply the hundreds place digits
- 3 x 3 = 9

Step 4: Multiply the thousands place digits
- 1 x 1 = 1

Step 5: Write the individual results
- Result of step 1: 49
- Result of step 2: 81 (write this on the tens place)
- Result of step 3: 9 (write this on the hundreds place)
- Result of step 4: 1 (write this on the thousands place)

Step 6: Add the individual results
- 49 + 81 + 9 + 1 = 140

Therefore, the result of 1397 x 1397 is 140.

However, none of the given options match the correct result. The correct answer should be 140, not any of the options provided.

Let's assume the numerator of the fraction is x. According to the question, the denominator is 3 more than the numerator, so it will be x + 3.

According to the second part of the question, when the denominator is decreased by 2 and the numerator is increased by 7, we get a new fraction equal to 2.

The new numerator will be x + 7, and the new denominator will be (x + 3) - 2 = x + 1.

So, we have the equation (x + 7) / (x + 1) = 2.

To solve this equation, we can cross-multiply:

2(x + 1) = x + 7

2x + 2 = x + 7

2x - x = 7 - 2

x = 5

Now that we have found the value of x, we can substitute it back into the equation to find the denominator:

Denominator = x + 3 = 5 + 3 = 8

The numerator is 5 and the denominator is 8. The sum of the numerator and denominator is 5 + 8 = 13.

Therefore, the correct answer is option B, 13.

Explanation:

To multiply the binary numbers (10.10) and (01.01), we can use the method of binary multiplication similar to decimal multiplication.

Step 1: Write the numbers vertically and align the decimal points.
```
10.10
x 01.01
```

Step 2: Multiply the rightmost digits of the second number by the first number.
```
10.10
x 01.01
------------
10.10 (Multiply 1st digit of 2nd number by 10.10)
```

Step 3: Multiply the second rightmost digits of the second number by the first number, but shifted one place to the left.
```
10.10
x 01.01
------------
10.10 (Multiply 1st digit of 2nd number by 10.10)
+ 10.10 (Multiply 2nd digit of 2nd number by 10.10, shifted 1 place left)
------------
110.10
```

Step 4: Add the partial products.
```
10.10
x 01.01
------------
10.10 (Multiply 1st digit of 2nd number by 10.10)
+ 10.10 (Multiply 2nd digit of 2nd number by 10.10, shifted 1 place left)
------------
110.10 (Add the partial products)
```

Step 5: Adjust the decimal point of the result by counting the total number of digits after the decimal point in both numbers.
In this case, there are 2 digits after the decimal point in each number, so we need to shift the decimal point 4 places to the left in the result.
```
10.10
x 01.01
------------
110.10 (Add the partial products)
```
After shifting the decimal point 4 places to the left, we get the final result as 0.11010.

Conversion to Decimal:
To convert the binary result to decimal, we multiply each digit by the corresponding power of 2 and sum them up.
```
0.11010 = (0 * 2^-1) + (1 * 2^-2) + (1 * 2^-3) + (0 * 2^-4) + (1 * 2^-5)
= 0 + 0.25 + 0.125 + 0 + 0.03125
= 0.40625
```

Therefore, the correct answer is option C) 0.0110010 in binary or 0.40625 in decimal.

-3.0

In 8-bit, the range of values that can be represented is from 0 to 255. Therefore, the largest positive value that can be represented by an 8-bit number is 255.

The number "2" is a whole number that comes after 1 and before 3. It is an even number and is the smallest prime number. It is also the only even prime number. In mathematics, 2 is the base of the binary number system, which is widely used in computer science and digital electronics. Additionally, 2 is often used to represent a couple or a pair of items, such as in the phrase "two of a kind."

The smallest integer that can be represented by an 8-bit number in 2's complement form is -128.

Understanding the Variables
Given the expressions:
- x = a(b - c)
- y = b(c - a)
- z = c(a - b)
These define how x, y, and z relate to the variables a, b, and c.
Objective
We aim to find the value of (x/a)³ + (y/b)³ + (z/c)³.
Substituting Values
Start by substituting x, y, and z into the expression:
- (x/a) = (b - c)
- (y/b) = (c - a)
- (z/c) = (a - b)
Thus, we need to evaluate:
(b - c)³ + (c - a)³ + (a - b)³.
Using the Identity
Utilize the algebraic identity for the sum of cubes:
- a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc).
In our case:
- Let a = b - c
- Let b = c - a
- Let c = a - b
This leads to:
(b - c)³ + (c - a)³ + (a - b)³ = 3(b - c)(c - a)(a - b).
Final Calculation
Now, we can express (b - c)(c - a)(a - b):
xyz = a(b - c) * b(c - a) * c(a - b).
Thus, the expression simplifies to:
(b - c)³ + (c - a)³ + (a - b)³ = 3xyz/abc.
Conclusively, we find:
Answer
The value of (x/a)³ + (y/b)³ + (z/c)³ = 3xyz/abc, which confirms that the correct answer is option 'A'.

To find the range of the exponent E in IEEE 754 Double Precision (Binary64) format, we need to consider the bias and the number of bits used to represent the exponent.

In IEEE 754 Double Precision format, the exponent E is represented using 11 bits. The bias for the exponent is 1023.

The range of the exponent E can be calculated as follows:

Minimum exponent = 1 - bias = 1 - 1023 = -1022
Maximum exponent = (2^11) - 1 - bias = 2047 - 1023 = 1024

Therefore, the range of the exponent E in IEEE 754 Double Precision format is -1022 to 1024.

The number 100101 is a binary number. In decimal form, it is equal to 37.

To find the remainder when 636 is divided by 215, we can use the method of long division.

Step 1: Divide the first digit of the dividend (6) by the divisor (2). The quotient is 3.
Step 2: Multiply the divisor (215) by the quotient obtained in Step 1 (3), and subtract the result from the first three digits of the dividend (636). The result is 636 - (3 * 215) = 636 - 645 = -9.
Step 3: Bring down the next digit of the dividend (3) and append it to the remainder obtained in Step 2 (-9), resulting in -93.
Step 4: Divide the first two digits of the new dividend (-9) by the divisor (2). The quotient is -4.
Step 5: Multiply the divisor (215) by the quotient obtained in Step 4 (-4), and subtract the result from the first two digits of the new dividend (-9). The result is -9 - (-4 * 215) = -9 + 860 = 851.
Step 6: Bring down the next digit of the dividend (6) and append it to the remainder obtained in Step 5 (851), resulting in 8516.
Step 7: Divide the first three digits of the new dividend (851) by the divisor (2). The quotient is 425.
Step 8: Multiply the divisor (215) by the quotient obtained in Step 7 (425), and subtract the result from the first three digits of the new dividend (8516). The result is 8516 - (425 * 215) = 8516 - 91375 = -82859.
Step 9: Bring down the next digit of the dividend (0) and append it to the remainder obtained in Step 8 (-82859), resulting in -828590.
Step 10: Divide the first four digits of the new dividend (-8285) by the divisor (2). The quotient is -4142.
Step 11: Multiply the divisor (215) by the quotient obtained in Step 10 (-4142), and subtract the result from the first four digits of the new dividend (-828590). The result is -828590 - (-4142 * 215) = -828590 + 890230 = 61640.
Step 12: Bring down the next digit of the dividend (9) and append it to the remainder obtained in Step 11 (61640), resulting in 616409.
Step 13: Divide the first five digits of the new dividend (61640) by the divisor (2). The quotient is 30820.
Step 14: Multiply the divisor (215) by the quotient obtained in Step 13 (30820), and subtract the result from the first five digits of the new dividend (616409). The result is 616409 - (30820 * 215) = 616409 - 6632300 = -6015891.
Step 15: Bring down the next digit of the dividend (1) and append it to the remainder obtained in Step 14 (-6015891), resulting in -60158911.

Since the dividend (-60158911) is less than the divisor (215), we cannot continue the division process. The remainder is the last value obtained in the process, which is -60158911.

1's Complement of (10011)2
To find the 1's complement of a binary number, we simply flip all the bits in the number. In this case, we have the binary number (10011)2.

Steps to find the 1's complement:
- Flip all the bits in the given binary number
- (10011)2 becomes (01100)2
Therefore, the 1's complement of (10011)2 is (01100)2.
So, the correct answer is option 'D' - (01100)2.

Performing binary addition:
To perform binary addition, we follow a set of rules similar to decimal addition. The rules are as follows:

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (carry 1 and write 0)

Given binary numbers: 101101 and 011011

Let's perform the addition step by step:

Step 1: Start from the rightmost bit and add the corresponding bits.
1
0 1 1 0 1 0 1 (101101)
+ 0 1 1 0 1 1 (011011)
----------------
1 0 0 1 0 0 0 (1001000)

Step 2: Carry over any 1s to the next column (if applicable).
1
0 1 1 0 1 0 1 (101101)
+ 0 1 1 0 1 1 (011011)
----------------
1 0 0 1 0 0 0 (1001000)

Since there are no more columns to add, we have our final result:

101101 + 011011 = 1001000

Therefore, the correct answer is option 'D' (1001000).

In binary addition, each bit represents a power of 2, where the rightmost bit is 2^0, the second rightmost bit is 2^1, the third rightmost bit is 2^2, and so on. By adding the corresponding bits, we calculate the sum of each power of 2. If there is a carry (1), it is added to the next column.

In this case, we have a carry in the fourth column, so we carry over the 1 to the next column. The final result is obtained by combining all the bits together, from left to right.

Explanation:


To find the smallest 4-bit negative number stored in its 2s complement representation, we need to understand how 2s complement works. In 2s complement, the most significant bit (MSB) represents the sign of the number, where 0 is positive and 1 is negative. The rest of the bits represent the magnitude of the number.

To convert a number to its 2s complement representation, we follow these steps:
1. Invert all the bits (change 0 to 1 and 1 to 0).
2. Add 1 to the result of step 1.

Now, let's find the smallest 4-bit negative number stored in its 2s complement representation.

Step 1: Invert all the bits of 0000 (smallest 4-bit positive number).
0000 -> 1111

Step 2: Add 1 to the result of step 1.
1111 + 1 = 10000 (5-bit number)

Since we are limited to 4 bits, we discard the MSB and get the 4-bit 2s complement representation of -8, which is 0000. But this is not the answer since we are looking for the smallest negative number.

To get the smallest negative number, we need to increment the result of step 1 by 1 before discarding the MSB.

Step 1: Invert all the bits of 0000 (smallest 4-bit positive number).
0000 -> 1111

Step 2: Add 1 to the result of step 1.
1111 + 1 = 10000 (5-bit number)

Step 3: Increment the result of step 2 by 1.
10000 + 1 = 10001 (5-bit number)

Discard the MSB and we get the smallest 4-bit negative number stored in its 2s complement representation, which is 1000 (equals to -8 in decimal).

Therefore, the correct answer is option A.

2 is a positive integer and a whole number. It is the smallest prime number and the only even prime number. It is also the base of the binary number system, which is widely used in computer science and digital technology. In mathematics, 2 is an important number in many areas, including algebra, geometry, and number theory. It is also commonly used in measurements and calculations, such as in time (24 hours in a day) and in angles (360 degrees in a circle).

The binary number 01001 is equivalent to the decimal number 9.

If you want to perform multiplication with another binary number, please provide the second binary number.

Explanation:

Finding the largest 4-digit number divisible by 66:
To find the largest 4-digit number that is exactly divisible by 66, we need to start by considering the highest 4-digit number, which is 9999.

Divisibility rule for 66:
- A number is divisible by 66 if it is divisible by both 6 and 11.

Finding the divisibility by 6:
- The sum of the digits of 9999 is 9+9+9+9 = 36, which is divisible by 6. Therefore, 9999 is divisible by 6.

Finding the divisibility by 11:
- Alternating the sum of the digits of 9999, we get (9-9) + (9-9) = 0. Since the result is 0, 9999 is divisible by 11.

Conclusion:
- Since 9999 is divisible by both 6 and 11, it is divisible by 66.

Checking the next highest number:
- The next highest 4-digit number is 9998, which is not divisible by 66 as it does not meet the divisibility criteria for both 6 and 11.

Therefore, the largest 4-digit number divisible by 66 is:
- 9966 (Option D)

Final Answer:
- The largest 4-digit number that can be exactly divisible by 66 is 9966.

To perform binary subtraction, we need to align the numbers vertically and then subtract each corresponding bit.

101101
- 101101
---------
0

Therefore, the binary subtraction of 101101 - 101101 is equal to 0.

In binary: (-20)10 = (-10100)2

In hexadecimal: (-20)10 = (-14)16

's complement notation. To subtract Y from X using two's complement notation, we can follow the following steps:

1. Find the two's complement of Y:
- Invert all the bits in Y: 11001 becomes 00110.
- Add 1 to the inverted value: 00110 + 1 = 00111.
- The two's complement of Y is 00111.

2. Add X and the two's complement of Y:
- X = 01110
- Two's complement of Y = 00111
- X + two's complement of Y = 01110 + 00111 = 10101

3. Ignore the carry-out from the leftmost bit (if any), and the result is 0101.

Therefore, X - Y in two's complement notation is 0101.

To divide the binary number (011010000) by (0101), we need to perform binary division. Let's go through the steps to find the quotient.

Step 1: Set up the division
Write the dividend and divisor in binary format, aligning them properly:

10010 (dividend)
/0101 (divisor)

Step 2: Perform the first division
Start by dividing the leftmost digits of the dividend (01) by the leftmost digit of the divisor (0). Since we cannot divide 1 by 0, we bring down the next digit (1) from the dividend.

10010 (dividend)
/0101 (divisor)
-

101 (new dividend)

Step 3: Perform the second division
Now, we divide the new dividend (101) by the divisor (0101).

10010 (dividend)
/0101 (divisor)
- 1001
----
101

Since the divisor is larger than the new dividend, we bring down the next digit (0) from the original dividend.

Step 4: Perform the third division
We now divide the new dividend (1010) by the divisor (0101).

10010 (dividend)
/0101 (divisor)
- 1001
----
1010
- 1010
----
0

Since the new dividend is now zero, we stop the division process.

Step 5: Determine the quotient
The quotient is obtained by combining the remainders from each division step. In our case, the remainders are 1001 and 1010, which gives us the binary quotient: 10011010.

Step 6: Convert the quotient to decimal
To convert the binary quotient to decimal, we can use the positional notation. Starting from the rightmost digit, we assign powers of 2 to each digit and sum them up:

1 * 2^7 + 0 * 2^6 + 0 * 2^5 + 1 * 2^4 + 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 128 + 16 + 8 + 2 = 154

Therefore, the quotient of (011010000) divided by (0101) is 10011010 in binary or 154 in decimal.

The correct answer is option 'B' (101001).

To solve this problem, let's assume the number that the boy mistakenly multiplied by 45 is 'x'.

Let's break down the problem into steps:

Step 1: Calculate the correct answer
If the boy had multiplied the number 'x' by 25, the correct answer would have been 25x.

Step 2: Calculate the answer the boy obtained
The boy multiplied the number 'x' by 45, so the answer he obtained is 45x.

Step 3: Find the difference between the two answers
According to the problem, the answer the boy obtained is 200 more than the correct answer. Mathematically, we can express this as:
45x = 25x + 200

Step 4: Solve the equation
To find the value of 'x', we need to solve the equation we obtained in the previous step. Let's simplify it:
45x - 25x = 200
20x = 200
x = 200/20
x = 10

Therefore, the number that the boy mistakenly multiplied by 45 instead of 25 is 10. Hence, the correct answer is option 'B'.

We are given that the sum of the squares of three consecutive positive numbers is 365. Let's assume the three consecutive numbers as x, x+1, and x+2.


We can represent the sum of the squares of these three numbers as an equation:
x^2 + (x+1)^2 + (x+2)^2 = 365

Expanding the equation:
x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 365
3x^2 + 6x + 5 = 365
3x^2 + 6x - 360 = 0
Divide the equation by 3:
x^2 + 2x - 120 = 0


We need to factorize the quadratic equation x^2 + 2x - 120 = 0 to find the values of x.

(x + 12)(x - 10) = 0

From the equation, we have two possible values for x:
x + 12 = 0 or x - 10 = 0

If x + 12 = 0, then x = -12, which is not a positive number. Hence, we discard this solution.

If x - 10 = 0, then x = 10. This gives us the first number as 10.


Using the value of x = 10, we can calculate the other two consecutive numbers:
First number: x = 10
Second number: x + 1 = 10 + 1 = 11
Third number: x + 2 = 10 + 2 = 12


The sum of the three numbers is:
10 + 11 + 12 = 33

Therefore, the correct answer is option 'B', which is 33.