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.
z1z2 = r1r2[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.