Algebra Of Complex Numbers

To solve the problem, we are given a function such that:

1. Use the Property of the Function:
The functional equation is the definition of a linear function. Specifically, a function of the form satisfies this equation for all real numbers.

2. Determine the Constant:
We are given . If , then:
So, the function is

3. Evaluate the Summation:
We are to find: We know: Therefore:

Final Answer:
The value of is

To solve the problem, we need to evaluate , where is a square matrix of order 3.

1. Recall Key Properties:
Let be an matrix. The following results are useful:

• if is invertible.
• only when is invertible, and for determinant purposes, we use composition rules properly.

2. Step-by-Step Determinant Computation:
We are given is a 3x3 matrix, i.e., .
Start with . It’s also a 3x3 matrix, so: Now, apply the adjugate again:

Final Answer:
The value of is

We want to find the values of for which the binomial expansion of is valid.
The binomial expansion is valid when .
This means:

$ x (-c, c) c = \frac{7}{5} 5c + 7 5c = 5 \cdot \frac{7}{5} = 7 5c + 7 = 7 + 7 = 14 {14}$.

To solve the problem, we need to find the quadratic equation whose roots are given by the limits and .

1. Evaluate the Limit for :
Notice that .
Thus, .

2. Evaluate the Limit for :
We have .
Rewrite using , so .

3. Form the Quadratic Equation:
The roots of the quadratic equation are and .
The equation is , which expands to:
.

Final Answer:
The quadratic equation is .

We are given that

$ (x+1)(2x^2+3) x^2 x B = -2A A = -\frac{1}{5} B B = \frac{2}{5} B + C = 3 C A - B + C A - B + C {2}$.

To find the product of all possible values of for the line that is perpendicular to and forms a triangle with the coordinate axes of area 3, we proceed as follows:

1. Determining the Slope Condition:
Rewrite the line in slope-intercept form:


The slope is . Rewrite the line :


The slope is . Since the lines are perpendicular, the product of their slopes is -1:

2. Equation of the Line:
With , the line is . Rewrite in slope-intercept form:

3. Finding the Intercepts:
The x-intercept occurs when :



The y-intercept occurs when :

4. Area of the Triangle:
The triangle formed by the x-intercept and y-intercept with the coordinate axes has area:


Given the area is 3:

5. Product of Possible Values:
The possible values of are 6 and -6. Their product is:

Final Answer:
The product of all possible values of is .

We are given the cubic equation: with roots such that . Step 1: Find the Roots Using the Rational Root Theorem, we check possible rational roots among . Checking : Thus, is a factor of the polynomial. Performing synthetic division on by :

The quotient polynomial is: Factoring: Thus, the roots are: Step 2: Compute the Given Expression Substituting the values: Solving for : Final Answer:

Step 1: Identify the roots of the equation. The equation can be rewritten using the quadratic formula: The roots are: and are complex conjugates and also represent the cube roots of unity, and , where and .

Step 2: Simplify the expressions using properties of roots. Since and similarly for higher powers due to the periodicity: Calculate each term: Therefore:

Step 3: Sum all terms. However, considering that the correct answer should be as given, it's important to revise the previous step's simplification or check for misinterpretation or miscalculations. The periodicity and properties suggest reevaluating the intermediate sums or assumptions about the calculations.

We are given the cubic equation: with roots such that and they form a geometric progression.

Step 1: Relationship Between Roots Since are in geometric progression, we let: Using Vieta’s formulas: 1. Sum of roots: Substituting values: Multiplying both sides by : Factoring: 2. Product of roots: Substituting values:

Step 2: Finding the Extreme Value of The given quadratic expression: Since the coefficient of is , the extreme value occurs at: Substituting and : The extreme value is: Substituting and solving, we get:

Final Answer:

We are given the equation: and need to determine the value of in terms of .

--- Step 1: Factorize the Denominator Factorizing : Comparing coefficients: Solving: Thus, the denominator factors as:

--- Step 2: Partial Fraction Decomposition The given expression can be rewritten as: Multiplying both sides by : Expanding: Expanding further: Grouping terms: Comparing coefficients: 1. For : 2. For : 3. For : Substituting : 4. Constant term: Substituting : Solving the system: 1. 2. Multiplying the first equation by 2: Adding both equations: Substituting into :

--- Step 3: Compute Since , we conclude:

Final Answer:

Step 1: The given expression is . To find the extreme value, we begin by differentiating: Setting , we solve for : Therefore, .

Step 2: Determine the extreme value . Substitute into the expression for : Thus, .

Step 3: The equation is , where . Using Vieta's formulas, the sum of the squares of the roots is given by: For this quadratic, the sum of the roots is and the product is . Therefore, the sum of the squares of the roots is .