Complex Numbers

The complex number is
Its inverse is
Hence, in ordered pair form:

Given . We know and . Substitute these into the equation: Rearrange the terms: Factor out from LHS and from RHS: This doesn't directly lead to the options easily. Let's try another rearrangement from the original equation: Consider the given equation again: . . This is the same. Let's group terms with z and w: If (since are distinct). The original equation was . If we divide by (since are non-zero): If , we can divide by : . Let . Then . We know , so . . This is . This seems complicated. Let's go back to . This form is not useful unless is real. Alternative approach from : . This is the same as before. If , then . If this is true, then LHS = . RHS = . So . This means is a possible solution, provided are distinct and non-zero. If , then . So . Let's check if this satisfies the original equation: This requires . . This is true. So is a valid solution. This matches option (c).

The equation is . Multiply the first term by and the second term by . Denominator common factor: . Numerator of first term: . Numerator of second term: . So the equation becomes: Multiply by 10: Group real and imaginary parts: Real part: (since RHS real part is 0) --- (1) Imaginary part: --- (2) We have a system of two linear equations: 1) 2) Multiply equation (2) by 2: --- (3) Subtract equation (3) from equation (1): . Substitute into equation (2): . So, and . We need to find . . This matches option (d).

Properties of cube roots of unity: 1. 2. From (1), we can derive: Consider the first term: . Substitute : . Since , then . So, the first term is . Consider the second term: . Substitute : . Since , then . So, the second term is . Now, sum the two terms: . From , we have . So, the sum is . This matches option (d).

Combine the terms inside the sum: Note: But the full sum of all th roots of unity is 0: Thus, the total sum:

We are given: Compute : Thus, comparing with , we get: Now compute:

Given: Let’s compute powers of up to using binomial expansions or software. Alternatively, we recognize a trick: Let’s denote: Then, if , we can evaluate using the identity So, since is complex and is real, we can compute just the real part and double it. Using algebraic expansion or substitution (or known evaluation), one finds:

Let us simplify the expression by multiplying numerator and denominator of each term by the conjugate of the denominator: Adding the two: Multiply both sides by 10: Expand: Equating real and imaginary parts: Real:
Imaginary:
Solve: From and , adding both:
Hence, .

We know:
So equation becomes: Only integer solution:

We are given the equation: This equation represents the ratio of the distance between and 2 to the distance between and the origin being equal to 2. We will solve for . Step 1: Express the equation in a more manageable form: Multiplying both sides by , we get: Step 2: Let , where and are real numbers. This gives: and Substituting into the equation: Step 3: Square both sides: Expanding and simplifying: Step 4: Rearranging to find the value of and , we get: Thus, the greatest value of is . % Final Answer

Step 1: Use section formula in complex numbers.
Let , . If divides in the ratio , then Step 2: Find argument of

We are given .
Squaring both sides, we get: $ .
So, .
Equating the real and imaginary parts, we have: \) .
We know that .
We also know the double angle formulas for sine and cosine: \) : \) is the magnitude, .
Now we have: \) .
We know that .
Also, .
Let .
\) : \) $ So, and .
Since and , lies in the third quadrant.
In the third quadrant, , which implies .
In this quadrant, is negative.
Therefore, .
Finally, .

Now simplify:

We are given that the roots of the equation are , , and . These roots are located on the complex plane. The curves and represent circles centered at and , respectively, with radius . The distance between and is: Thus, the two circles intersect at exactly one point, since their radius is half the distance between the centers. Therefore, the number of common points is 1. Thus, the correct answer is 1.

We are given , and we need to find the modulus of the conjugate of . Step 1: First, compute the denominator: Step 2: Now, the conjugate of is: Step 3: To find the modulus of the conjugate of , we use the formula for the modulus of a complex number , which is . The modulus of the numerator is: The modulus of the denominator is: Step 4: Therefore, the modulus of the conjugate of is: Thus, the correct answer is .

Step 1: Let , where .
Then, the multiplicative inverse of , denoted by , satisfies: Step 2: Multiply numerator and denominator by conjugate of : Hence, the multiplicative inverse of is:

Step 1: Properties of equilateral triangle in complex plane
In an equilateral triangle, the distances from the circumcenter to each vertex are equal. So: Step 2: Use symmetry and equal moduli
Take moduli of the expression: This holds true as all terms are of equal magnitude from the circumcenter.

The general formula for the -th roots of a complex number is given by: For , we have and . We need to find one of the 15th roots of , so we apply the formula with : For , we get: Thus, the correct answer is , which corresponds to option (3).

We express in polar form: Using De Moivre's Theorem, we get: The product of the four values is the magnitude raised to the power 4: % Final Answer

We are given the equation .
Step 1: Factor the equation.
Step 2: Solve for .
The solutions to this equation are found by setting each factor equal to zero: For , we get: Thus, the solutions are .
Step 3: Conclusion.
Therefore, the number of possible solutions is 3.

We are given and , and we are asked to find the amplitude of the expression .
Step 1: Compute and .
Step 2: Find the amplitude of . The amplitude of a complex number is given by the argument of the resulting complex number. The argument of is: The argument of is , and the argument of is . Thus, the total amplitude is:

From the given: Equating real and imaginary parts: and Substituting into second: Then, Now, Then, Finally,

We are given that is a purely imaginary number for , and a purely real number for . To solve for and , we analyze the conditions for the real and imaginary parts of under these values of . - When is purely imaginary, the real part must be zero, and when is purely real, the imaginary part must be zero.

We are given a recurrence involving cube roots of unity and need to find the value of based on the conditions. The roots of unity satisfy certain properties that we use to simplify the given expression and find the value of . This involves solving the relation step by step.

We are given the sum of the eccentricities for two conics and need to calculate the result. The formula for the eccentricities involves the distance between the roots, and we apply this to get the desired value. By solving the conditions, we find the correct eccentricity sum.

We are given the complex numbers: We need to evaluate the expression: Step 1: First, compute : Step 2: Next, compute : Step 3: Now, substitute these values into the expression: Step 4: Simplify the expression: Thus, the correct answer is .

The amplitude of the complex number is given by , and the amplitude is always equal to . Therefore, we have: Squaring both sides: Thus, the locus of the point is given by . Therefore, the correct answer is .

Step 1: Compute
Step 2: Identify region
The inequality describes a closed disk of radius 5 centered at . Step 3: Test all options
- Option (2): ⇒ lies inside ✅ - Others exceed the radius or lie outside ⇒ ❌

The equation is: Let , , where is a cube root of unity: Now compute:

Step 1: Use the property of complex conjugates.
If and are conjugates, then:
Step 2: Compare real and imaginary parts
Equating real parts: Equating imaginary parts: From (1): Substitute into (2): Solve this quadratic in :

Step 1: Compute

Step 2: Find and

Step 3: Find and
Using the identity :
Step 4: Substitute into the expression

Step 5: Simplify using trigonometric identity
Recall the half-angle identity: Since , is positive.
Step 6: Match with the given options
The simplified form matches option 3. Verification:
Let : This confirms option 3 is correct. Conclusion: The correct answer is .

Let . Then, Equating: So, the locus is , i.e., the Y-axis.

We are given the expression .
Let's first evaluate the term .
$ is a complex cube root of unity, we know that and .
So, .
\) and : For : For : The sum becomes: \) : \) , we have and .
\) $ Sum for and : For : For : The sum is .
The expression is .
This still doesn't match the options.
Let me check if I misinterpreted the question.
Perhaps the question intended a different range for .
Let's assume there was a typo and the result should lead to one of the options.
Consider the possibility that the sum was intended to be something else.
If the result of the sum was such that is a multiple of , , etc.
, we could get the given options.
Let's review the properties of cube roots of unity again.
If the question was , the answer would be .
For this, we would need for some integer .
This means .
Our sum is 10, which doesn't fit this form.
Let's consider if there's a simplification I missed.
Rethink the problem statement: "If is a complex cube root of unity.
.
.
" This implies either or .
The result is independent of the choice of .
Given the correct answer is , we need , which implies , or , which is not possible for integer .
There must be an error in the question or the provided correct answer.
However, following the steps correctly, the argument of the cosine is , and is not equal to or .
Assuming the correct answer provided is indeed , there might be a subtlety in the problem statement I am overlooking, or a typo.
If the sum was over a different range of , or if the fraction involved 10 instead of the sum.
Let's assume the question intended .
Then , which means , or , which is false.
Given the discrepancy, and strictly following the problem as stated, the answer should be , which is not among the options.
However, since I must choose one of the provided correct answers, and there seems to be an issue with the question as transcribed or my understanding, I cannot definitively arrive at .
However, if there was a mistake and the sum resulted in a multiple of 175, e.
g.
, if the sum was 175, then .
But our sum is 10.
Let's proceed with the next question, acknowledging the potential issue with this one.

First, let's simplify the expression for : $ in polar form: .
Then, .
Now, let's write in polar form: .
So, .
The argument of is .
However, the square root has two values.
Let , where is an integer.
Then .
For , .
For , .
So the two values of are: \) $ The arguments are and .
We have , which satisfies the given condition.
Then, .

The roots of the cubic equation are given as , and .
For a cubic equation with roots , the equation can be written as .
In our case, the roots are .
So the equation is: $ with and : \) : \) , we have: \) : \) $

Step 1: First, compute the argument of the complex fraction. The complex number is in the form .
To find the argument, multiply both numerator and denominator by the conjugate of the denominator. For , multiply both by , and for , multiply both by . Step 2: After simplifying the expressions, calculate the resulting arguments of both fractions.
Step 3: Adding the two arguments results in the range of values between and , confirming the argument lies within this interval.

We are asked to find the imaginary part of the expression: Step 1: Expanding We start by expanding . Use the binomial expansion for : Now calculate the terms: Thus: Step 2: Expanding Next, we expand using distributive property: Now calculate the terms: Thus: Step 3: Dividing the two expressions Now, we divide the two expressions: To simplify this, multiply both the numerator and denominator by the conjugate of the denominator : First, simplify the denominator: Now, expand the numerator: Thus, the expression becomes: Step 4: Identifying the imaginary part The expression is , so the imaginary part is: Thus, the imaginary part of is .

We are given , which is the polar form of a complex number, and we are asked to evaluate: Using De Moivre's theorem, we know: Step 1: Substituting for Substituting into the expression: Step 2: Simplifying the expression Multiply the numerator and the denominator by the conjugate of the denominator: Simplifying the denominator: The numerator simplifies to: Thus, the value of the given expression is .

We are asked to find the square root of the complex number . To find , we use the fact that the square root of a complex number can be expressed as , where and . Here, has and . Step 1: Finding the magnitude The magnitude is: Step 2: Finding the argument The argument is: Step 3: Applying the square root formula Using the formula for the square root of a complex number, the square root of is: After calculating, we find that: Thus, the square root of is .

We are given the complex number and its conjugate . The given expression is: Step 1: Expand the given expression Since , we rewrite: Let this expression be purely imaginary, say , meaning the real part must be zero: Step 2: Rationalizing Multiplying numerator and denominator by the conjugate of the denominator: Expanding the numerator: For the expression to be purely imaginary, the real part must be zero: Since for real , we get: Step 3: Excluding the singularity at The denominator of the original fraction must not be zero: Thus, the required locus is: Thus, the correct answer is:

We are given the set: This represents the set of complex numbers such that the distance from to the point in the complex plane is 1 unit. This is the equation of a circle in the complex plane. To clarify, let's rewrite the equation: Let , where and are real numbers. Then, the equation becomes: The magnitude of a complex number is given by . Therefore: Squaring both sides: This is the equation of a circle with centre at and radius 1 unit. Thus, the correct answer is:

Substitute into the expression and simplify using complex arithmetic or recognize patterns of complex conjugates and powers. The result evaluates to 9.