Complex Numbers

Cube roots of unity mean and

Z₁ = 1-2i, Z₂ = 1+i, and Z₃ = 3+4i

(1/Z₁ + 3/Z₂) x (Z₃/Z₂) = (1/1-2i + 3/1+i) x (3+4i/1+i)

= [(1+2i)/5 + 3(1-i)/2] [(3+4i)x(1-i)/2]

Solving this equation one gets,

= (17-11i)/10 + (7+i)/2

= (119+17i-77i+11)/20

= 130-60i/20

= (13/2) - 3i

Any integer that may be expressed as a+ib is referred to be a complex number. Complex numbers, such as +3i and 7+8i, are an example. Here i = -1. This allows us to state that i₂ = 1. Therefore, we may use i = -1 for any equation that does not have a true solution.

A polynomial with two roots or one of degree two is referred to as a quadratic equation. A quadratic equation has the generic form y=ax²+bx+c. The real numbers a ≠ 0, b, and c are present here. 

We are tasked with forming a five-digit number divisible by 3 using the digits 0, 1, 2, 3, 4, and 5 without repetition. A number is divisible by 3 if the sum of its digits is divisible by 3.
Step 1: Total sum of the digits The sum of the digits 0, 1, 2, 3, 4, and 5 is: Since the total sum of the digits is 15, which is divisible by 3, the sum of the digits of any five-digit number formed from these digits will also be divisible by 3, provided that we leave out one of the digits.
Step 2: Choosing one digit to leave out We can leave out any of the six digits (0, 1, 2, 3, 4, 5). If we leave out a digit, the sum of the remaining digits will still be divisible by 3. We need to calculate the number of five-digit numbers that can be formed with the remaining digits.
Step 3: Counting the number of five-digit numbers We must choose 5 digits from the 6 available digits. We have 6 choices for the digit to leave out. For each selection of 5 digits, the number of ways to arrange them is given by the number of permutations of 5 digits. However, the first digit cannot be 0, so we need to adjust the counting. - If 0 is not selected, all 5 digits can be arranged in ways. - If 0 is selected, the first digit cannot be 0, so we must select the first digit from the remaining 4 digits (1, 2, 3, 4, or 5) and arrange the other 4 digits.
Step 4: Calculating the number of arrangements - If 0 is not selected, we have ways. - If 0 is selected, the number of valid arrangements is . Thus, the total number of arrangements is: Thus, the total number of ways to form a five-digit number divisible by 3 is .

We need to find the probability that is divisible by 6, where and are chosen from . We use the factorization , and for this expression to be divisible by 6, either or must be divisible by 2 and 3.
Through calculation, you can verify that the total number of favorable outcomes is 30, and the total number of possible outcomes is 100.
Therefore, the probability is: Thus, the correct answer is .

We are asked to find the number of 5-digit odd numbers greater than 40,000 that can be formed using the digits , such that at least one digit is repeated. To form a 5-digit number, we need to ensure that:
1. The number is odd, meaning the last digit must be one of .
2. The number must be greater than 40,000, so the first digit must be at least 4, i.e., .
3. At least one digit must be repeated.
5
Step 1: Total number of 5-digit odd numbers greater than 40,000:
The first digit can be or (4 choices). The second, third, and fourth digits can be (6 choices each). The last digit can be (3 choices). Thus, the total number of such 5-digit numbers is:
Step 2: Now, we subtract the number of 5-digit odd numbers where no digits are repeated. The first digit can be chosen from (4 choices). The second digit can be chosen from excluding the first digit (5 choices).
The third digit can be chosen from the remaining 4 digits (4 choices). The fourth digit can be chosen from the remaining 3 digits (3 choices). The last digit can be chosen from , excluding the digits used already (2 choices). Thus, the number of such numbers is:
Step 3: Finally, subtract the number of numbers with no repeated digits from the total number of numbers: Therefore, the number of 5-digit odd numbers greater than 40,000, where at least one digit is repeated, is 2352.

Given that and we want to evaluate

Step 1. Rewrite each cosine in exponential form. Recall that Hence, Similarly, and Thus the sum we want is In terms of , notice that Hence

Step 2. Use the identity when A classical identity for any complex numbers is Since we are given , the right side vanishes, giving Thus, Taking the real part does not change the value here, so

Answer:

We are given the complex expression , and we need to find its complex conjugate.
Step 1: First, we simplify the product . Using the distributive property (FOIL method): Since , this becomes:
Step 2: Now, multiply by : Again, using :
Step 3: The result of the multiplication is . The complex conjugate of a complex number is . Therefore, the complex conjugate of is: Thus, the complex conjugate of is .

To solve for the real part of the given expression, we first simplify it: By applying De Moivre's theorem, we know that

:

Thus, the given expression becomes:

Now, to find the real part of this complex number, we multiply both the numerator and denominator by the conjugate of the denominator:

The denominator simplifies to:

Now, for the numerator:

Using the angle addition formulas for cosine and sine:

Thus, the real part of the given expression is: Therefore, the correct answer is , corresponding to option (4)

We are asked to form four-digit numbers using the digits with all digits distinct.
Step 1: Calculating , the total number of four-digit numbers The number of ways to choose the first digit is 7 (since it can be any digit from 1 to 7). The number of ways to choose the second digit is 6 (since one digit is already used). The number of ways to choose the third digit is 5, and the number of ways to choose the fourth digit is 4. Thus, the total number of numbers is: Step 2: Calculating , the number of numbers greater than 3400 For numbers greater than 3400, the first digit must be 3, 4, 5, 6, or 7.
- If the first digit is 3, the second digit must be 4 or greater, which leaves us with 4 choices for the second digit.
The remaining 2 digits can be chosen in ways.
Thus, the number of such numbers is .
- If the first digit is 4, the second digit can be any of the remaining digits (6 choices), and the third and fourth digits can be chosen in ways.
Thus, the number of such numbers is .
- For first digits 5, 6, and 7, the number of possible numbers is similar to the case when the first digit is 4.
Thus, the total number of numbers greater than 3400 is:
Step 3: Finding the ratio The ratio is:
Thus, the ratio is .

We are given a complex number such that:

Step 1: Understanding the Given Condition
Recall that for a complex number, This represents the locus of points that are equidistant from two fixed points and . In our case, - (point on the imaginary axis) - (point on the real axis) The given equation represents the **perpendicular bisector** of the line segment joining these points, which is a circle passing through and with its center on the line joining these points.

Step 2: Determining the Circle's Properties
The line segment joining and has its midpoint at: The radius of the circle is half the distance between these points: Thus, the circle has: - Center at
- Radius
Step 3: Finding the Area in the First Quadrant
Since the circle is symmetric about both axes, the area of the circle is: Since the first quadrant contains one-fourth of the total area,

Final Answer: (2)