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If z1​=r1​(cosθ1​+isinθ1​) and z2​=r2​(cosθ2​+isinθ2​), then z1​z2​=r1​r2​[cos(θ1​+θ2​)+isin(θ1​+θ2​)] True False Question 19 (4 points) In the polar coördinate system, an equation giving the graph of a circle is a function. True False If AX=B is a solvable system of equations in matrix form, then X=A−1B. True False Question 21 (4 points) A geometric series with common ratio r converges if and only if ∣r∣<1. True False

User Glitch
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1 Answer

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1. Polar coordinates
2. Complex numbers
3. Matrix form
4. Geometric series
5. Convergence
6. Common ratio
7. Matrix inverse

1. For the first question, if z1​=r1​(cosθ1​+isinθ1​) and z2​=r2​(cosθ2​+isinθ2​), then the product of z1​ and z2​ can be found by multiplying their magnitudes and adding their angles.

z1​z2​ = r1​r2​[cos(θ1​+θ2​)+isin(θ1​+θ2​)]

This equation is true, as it follows the rule of multiplying complex numbers in polar form.

Answer: True

2. In the polar coordinate system, an equation giving the graph of a circle is not a function. A circle in polar coordinates is defined by an equation of the form r = constant, where r represents the distance from the origin and the constant determines the radius. For a given value of r, there are multiple possible values of θ that satisfy the equation. Therefore, it does not satisfy the vertical line test required for a function.

Answer: False

3. If AX=B is a solvable system of equations in matrix form, then the solution can be found by multiplying both sides of the equation by the inverse of matrix A.

X = A^(-1)B

This equation is true, as multiplying both sides by the inverse of A allows us to isolate X and solve for it.

Answer: True

4. A geometric series with a common ratio r converges if and only if the absolute value of r is less than 1. In a geometric series, each term is obtained by multiplying the previous term by the common ratio. If the common ratio is between -1 and 1 (excluding -1 and 1), the series converges to a finite value. If the absolute value of the common ratio is greater than or equal to 1, the series diverges and does not have a finite sum.

Answer: True

Please let me know if you need further clarification or have any more questions.

User Leze
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