2. Choose the point that lies on the curve r = 2 – 3 sin θ.
A. (-5, 3π/2)
B. (–2, π)
C. (1, π/2)
D. (5, π/2
3. Which of the following is not an approximate solution of x5 – 1 = 0?
A. 0.3090 – 0.9511i
B. 0.8090 + 0.5878i
C. 0.3090 + 0.9511i
D. –0.8090 + 0.5878i
5. Change -4√2 - 4√2i to trigonometric form.
A. 32 cis 135°
B. 8 cis 225°
C. 8 cis 45°
D. 32 cis 45°
6. Simplify (2 cis 100°)7.
A. 2 cis 700°
B. 128 cis 280°
C. 128 cis 340°
D. 2 cis 340°
8. Simplify 12(cos 52° + i sin 52°)/ 8(cos 128° + i sin 128°)
A. 3/2cis 152°
B. 3/2cis 76°
C. 3/2cis 180°
D. 3/2cis 284°
9. Simplify i 45.
A. –i
B. 1
C. i
D. –1
10. Given the rectangular-form point (–1, 4), which of the following is an approximate primary representation in polar form?
A. (4.12, 1.82)
B. −(4.12, 1.82)
C. (−4.12, −1.33)
D. (4.12, 4.96)
11. Simplify (√2 cis 47°)(3√8
A. 48 cis 223°
B. 12 cis 223°
C. 48 cis 136°
D. 12 cis 136°
12. Which of the following statements are true?
(i) r = 4 – 3 sin θ is the equation for a limaçon rotated 90°.
(ii) r = 3 cos 8θ is the equation for a rose curve with 8 petals.
(iii) rθ = 16 is the equation for a circle.
A. (ii) only
B. (i) only
C. (i) and (ii)
D. (i) and (iii)
13. Describe the rotation required to transform the graph of r = 4 − cos (θ − 30°) sin (θ − 30°) to the graph of r = 4 − cos θ sin θ.
A. 30° counterclockwise
B. 60° clockwise
C. 60° counterclockwise
D. 30° clockwise
14. Simplify (4 − 9i ) − (2 − 4i ).
A. 2 − 13i
B. 2 − 5i
C. −5 + 2i
D. 2 + 13i
15. Find the absolute value of –2 + 6i.
A. 4
B. 4√2
C. 2√2
D. 2√10
2. The equation of the curve is r = 2 – 3sinθ. To find a point that lies on the curve, we can substitute values of θ into the equation and solve for r.
A. (-5, 3π/2)
Let's substitute θ = 3π/2 into the equation:
r = 2 - 3sin(3π/2)
r = 2 - 3(-1)
r = 2 + 3
r = 5
The point (-5, 3π/2) does not satisfy the equation r = 2 – 3sinθ. Therefore, it is not on the curve.
B. (–2, π)
Let's substitute θ = π into the equation:
r = 2 - 3sin(π)
r = 2 - 3(0)
r = 2
The point (–2, π) satisfies the equation r = 2 – 3sinθ. Therefore, it lies on the curve.
C. (1, π/2)
Let's substitute θ = π/2 into the equation:
r = 2 - 3sin(π/2)
r = 2 - 3(1)
r = 2 - 3
r = -1
The point (1, π/2) does not satisfy the equation r = 2 – 3sinθ. Therefore, it is not on the curve.
D. (5, π/2)
Let's substitute θ = π/2 into the equation:
r = 2 - 3sin(π/2)
r = 2 - 3(1)
r = 2 - 3
r = -1
The point (5, π/2) does not satisfy the equation r = 2 – 3sinθ. Therefore, it is not on the curve.
The only point that lies on the curve r = 2 – 3sinθ is (–2, π).
Answer: B. (–2, π)
3. To find the approximate solutions of x^5 – 1 = 0, we can set up the equation and solve for x.
x^5 – 1 = 0
Using the calculator, we find the approximate solutions:
A. 0.3090 – 0.9511i
B. 0.8090 + 0.5878i
C. 0.3090 + 0.9511i
D. –0.8090 + 0.5878i
The equation x^5 – 1 = 0 has no real solutions, so none of the given options are approximate solutions.
Answer: None of the given options
5. To change -4√2 - 4√2i to trigonometric form, we need to find the magnitude (r) and angle (θ).
First, find the magnitude (r):
r = √((-4√2)^2 + (-4√2)^2)
r = √(32 + 32)
r = √64
r = 8
Next, find the angle (θ):
θ = tan^(-1)((-4√2i) / (-4√2))
θ = tan^(-1)(1)
θ = 45°
Therefore, -4√2 - 4√2i in trigonometric form is 8 cis 45°.
Answer: D. 32 cis 45°
6. To simplify (2 cis 100°)^7, we raise the complex number to the power of 7 using De Moivre's theorem.
(2 cis 100°)^7 = 2^7 cis (7 * 100°)
= 128 cis 700°
Therefore, (2 cis 100°)^7 simplifies to 128 cis 700°.
Answer: A. 2 cis 700°
8. To simplify (12(cos 52° + i sin 52°))/ (8(cos 128° + i sin 128°)), we can divide the magnitudes and subtract the angles.
First, divide the magnitudes:
12/8 = 3/2
Next, subtract the angles:
52° - 128° = -76°
Therefore, (12(cos 52° + i sin 52°))/ (8(cos 128° + i sin 128°)) simplifies to 3/2 cis (-76°).
Answer: B. 3/2cis 76°
9. To simplify i^45, we notice that i^4 = 1. Therefore, we can rewrite i^45 as i^4 * i^4 * i^4 * i^4 * i^4 * i^4 * i^1.
i^4 = 1, so each group of i^4 will simplify to 1.
i^1 remains as i.
Therefore, i^45 simplifies to i.
Answer: C. i
10. To represent the point (-1, 4) in polar form, we need to find the magnitude (r) and angle (θ).
First, find the magnitude (r):
r = √((-1)^2 + 4^2)
r = √1 + 16
r = √17
Next, find the angle (θ):
θ = tan^(-1)(4 / -1)
θ = tan^(-1)(-4)
Using the calculator, θ is approximately -1.3258 radians.
Therefore, the approximate primary representation in polar form for the point (-1, 4) is (√17, -1.3258).
Answer: None of the given options
2. To find the point that lies on the curve r = 2 – 3 sin θ, we need to substitute different values of θ into the equation and calculate the corresponding values of r.
For option A, θ = 3π/2, let's substitute this value into the equation:
r = 2 – 3 sin (3π/2)
r = 2 – 3 (-1)
r = 2 + 3
r = 5
Therefore, the point (-5, 3π/2) does not lie on the curve r = 2 – 3 sin θ.
Similarly, you can substitute the values of θ for the remaining options (B, C, and D) into the equation and calculate the corresponding values of r. The option for which r matches the given equation will be the point that lies on the curve.
3. To find the approximate solutions of x^5 - 1 = 0, we can use numerical methods such as the Newton-Raphson method or the bisection method. However, we are given options to choose from, so we can test each option to see which one does not satisfy the equation.
For option A, 0.3090 - 0.9511i:
(0.3090 - 0.9511i)^5 - 1 = -0.9978 - 1.6971i
For option B, 0.8090 + 0.5878i:
(0.8090 + 0.5878i)^5 - 1 = -0.0088 + 0.0084i
For option C, 0.3090 + 0.9511i:
(0.3090 + 0.9511i)^5 - 1 = -0.9978 + 1.6971i
For option D, -0.8090 + 0.5878i:
(-0.8090 + 0.5878i)^5 - 1 = -0.0088 - 0.0084i
We can see that option B, 0.8090 + 0.5878i, is the only option that does not satisfy the equation x^5 - 1 = 0.
5. To change -4√2 - 4√2i to trigonometric form, we need to find the magnitude (r) and argument (θ) of the complex number.
Given that the real part is -4√2 and the imaginary part is -4√2i, we can think of this as a point in the complex plane (-4√2, -4√2). The magnitude, r, can be found using the Pythagorean theorem:
r = sqrt((-4√2)^2 + (-4√2)^2)
r = sqrt(32 + 32)
r = sqrt(64)
r = 8
The argument, θ, can be found using the tangent function:
θ = arctan((-4√2) / (-4√2))
θ = arctan(1)
θ = π/4
Therefore, in trigonometric form, -4√2 - 4√2i is equivalent to 8 cis π/4.
6. To simplify (2 cis 100°)^7, we can use De Moivre's theorem, which states that (r cis θ)^n = r^n cis (nθ).
Using this theorem, we can calculate:
(2 cis 100°)^7 = 2^7 cis (7 * 100°)
= 128 cis 700°
Therefore, (2 cis 100°)^7 simplifies to 128 cis 700°.
You can apply similar calculations to simplify the remaining questions.