8.8+OFSA+Solutions


 * OFSA SOLUTIONS for Complex Numbers**

This page contains peer generated solutions and error explanations to OFSA questions. As you read or view the solutions, be critical: check for accuracy, but also for more efficient solution strategies. If you have a better method or different idea/answer, post a discussion and monitor the responses.


 * Quick Directions**
 * Post answers, solutions and error explanations to each OFSA question below.
 * For each "distractor" or incorrect answer choice, explain the error that would lead to that incorrect answer choice.
 * You may either do the above in typed format or using a pencast.
 * Separate each question with a section bar.
 * After each solution, provide a hyperlink back to the corresponding OFSA page.
 * Follow example below.
 * Click here to refer to solution format in 7.7

math B.\indent (\frac {6\sqrt {3}} {2})(cis \frac {23\pi} {12}) math A is incorrect because r1 and r2 are multiplied together, while the thetas are added
 * Question 1**
 * __Solution 1__**
 * __Error Explanation 1__**

C is incorrect because the opposite function is being conducted (x/y)

D is incorrect because of an algebraic error

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math A.\indent (\frac {\sqrt {3}} {3})(cis\frac {17\pi} {12}) math
 * Question 2**
 * __Solution 2__**

A is the correct answer because in this case we are dividing the r from x into y and subtracting the theta from x from the theta from y. math B.\indent (\frac {3} {2})(cis\frac {-17\pi} {12}) math Here, the square root of two isn't included in the final product.
 * __Error Explanation 2__**

math C.\indent (\frac {3\sqrt {2}} {2})(cis\frac {-17\pi} {12}) math In this case, instead of y/x, the expression is x/y. Remember to read carefully! math D.\indent (\dfrac {\sqrt {3}} {3})(cis\frac {23\pi} {12})\\ math Here, the theta values have been subtracted erroneously.

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math A.\indent z^{4} = 1^{4} (cis(\frac {2\pi} {3})) math
 * Question 3**
 * __Solution 3__**


 * __Error Explanation 3__**

math B.\indent z^{4} = 4(cis(\frac {2\pi} {3})) math B is wrong because 1 is to the fourth power, NOT multiplied by four.

math C.\indent z^{4} = 1^{4} (cis(\frac {4\pi} {3})) math C is wrong because the theta values weren't added up correctly.

math D.\indent z^{4} = 4(cis(\frac {4\pi} {3})) math D is incorrect because of both B and C.

Most errors in these types of problems arise from not multiplying/adding pi correctly; draw out the unit circle if you're having problems!

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math C: z=2^{\frac{1}{6}}cis(\frac{\pi}{12}) \indent z=2^{\frac{1}{6}}cis(\frac {3\pi} {4} )\indent z=2^{\frac{1}{6}}cis(\frac{3\pi}{4}) \\ \text {C is correct because it successfully accounts for the correct radius and the right correct angles. It also accounts for the three separate solutions there needs to be because z was cubed} math math A: z=2^{\frac{1}{6}}cis(\frac{\pi}{12}) \text{ This is wrong because it only gets one solution and we need to have three solutions because it is z cubed in the question} math math B: z=1^{\frac{1}{6}}cis(\frac{\pi}{12}) \text{ This is wrong because it both only accounts for once solution as above and it assumes 1 is the radius. A student may have chosen this because they were panicking } math math D: z= 2^{\frac{1}{6}} cis(\frac{\pi}{6}), z=2^{\frac{1}{6}}cis(\frac{\pi}{4}) z=2^{\frac{1}{6}}cis(\frac{2\pi}{3}) \text { A student may have picked this if they were running out of time. The might have known there would be three solutions by looking at the exponent but did not have enough time to actually get the answers} math
 * Question 4**
 * __Solution 4__**
 * __Error Explanation 4__**

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math \text{How many solutions are there to this equation and how many of those solutions are expected to be real?} x^4 = 16 \\ math A If you graph the solutions, you see that they are each spaced pi/2 apart. The "first" solution is on the real axis with no imaginary component at 2cis(o). If you continue finding solutions, there is only one other place that has no imaginary component, which is at 2cis(pi). Click here to return to OFSA.
 * Question 5**
 * __Solution 5__**
 * __Error Explanation 5__**

math \text{Which of the following is not a complex solution to this equation?} x^4=4+4i \\ math B There are 4 solutions spaced evenly apart starting from pi/4 all with the radius 4sqrt(2). Adding pi/2 to the angle gets the other solutions. The only solution whose angle doesn't match the others in terms of spacing is B since there is no way to get pi/2 from adding pi/2 to pi/4. Click here to return to OFSA.
 * Question 6**
 * __Solution 6__**
 * __Error Explanation 6__**

math \text{z and w are two complex numbers, and w = 7 + 7sqrt(3)i. If the product of z and w is 140cis(pi/2), what is z in rectangular form} \\ math B This problem is essentially asking the quotient of 140cis(pi/2) and 7 + 7sqrt(3)i. If you convert w to polar form, you get 14cis(pi/3). From there, you can eliminate any answer choice that doesn't have a theta of pi/6 in polar form, which is A(pi/3), C(pi/2) and D(-pi/2). Click here to return to OFSA.
 * Question 7**
 * __Solution 7__**
 * __Error Explanation 7__**

The answer is option D. Both A and B are false because they are not in the correct form, it must be fully simplified. The angle can still be reduced due to the unit circle angles repeating every 2 pi. C is false because DeMoivre's theorem is used in reverse. Click here to return to OFSA.
 * Question 8**
 * __Solution 8__**
 * __Error Explanation 8__**

math \text { The correct answer is B: } 1cis(\frac{\pi} {4} ) \text{ because the radius would become 1 because z would make a 45 - 45 triangle and } \frac{\sqrt{2}}{2} * \sqrt {2} = 1. \text {Since it is a 45-45 triangle and both a and b are positive. The angle is }\frac {\pi}{4}} math Click here to return to OFSA. math \text { Answer A is incorrect because it uses the radius of } \frac\sqrt{2}}{2} } \\ math math \text { Answer C is incorrect because it uses the incorrect angle. A student may have picked this if they were in a hurry } \\ math math \text{ Answer D is incorrect because it uses both the incorrect angle and radius but it contains common numbers from the orginal z. A student may have picked this out of desperation. } math
 * Question 9**
 * __Solution 9__**
 * __Error Explanation 9__**

math \text{ The correct answer is D because } x^{3}=27 + 0i \text { which becomes } x^{3} = 27cis(0)} \\ \text {With Demoivre's rule, we know that } z= 3 \indent z=\frac {-3}{2} + \frac{3\sqrt{3}}{2}i \indent z=\frac {3}{2} - \frac{3\sqrt{3}}{2} \\ math math \text{ Answer A is incorrect because it ignores all the complex solutions. A person could easily be tricked into doing this } \\ math math \text{ Answer B is incorrect because it ignores the fractions in the actual results and uses the radii of the resulting solutions. A person that is rushing may pick this easily } \\ math math \text{ Answer C is incorrect because it places the negative signs incorrectly. It is difficult to differentiate between this and answer option D making it easy to be picked} \\ math Click here to return to OFSA.
 * Question 10**
 * __Solution 10__**
 * __Error Explanation 10__**

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 * Question 11**
 * __Solution 11__**
 * __Error Explanation 11__**

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 * Question 12**
 * __Solution 12__**
 * __Error Explanation 12__**

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 * Question 13**
 * __Solution 13__**
 * __Error Explanation 13__**

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 * Question 14**
 * __Solution 14__**
 * __Error Explanation 14__**

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 * Question 15**
 * __Solution 15__**
 * __Error Explanation 15__**

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 * Question 16**
 * __Solution 16__**
 * __Error Explanation 16__**

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 * Question 17**
 * __Solution 17__**
 * __Error Explanation 17__**

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 * Question 18**
 * __Solution 18__**
 * __Error Explanation 18__**

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 * Question 19**
 * __Solution 19__**
 * __Error Explanation 19__**

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 * Question 20**
 * __Solution 20__**
 * __Error Explanation 20__**

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 * Question 21**
 * __Solution 21__**
 * __Error Explanation 21__**

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 * Question 22**
 * __Solution 22__**
 * __Error Explanation 22__**

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 * Question 23**
 * __Solution 23__**
 * __Error Explanation 23__**

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 * Question 24**
 * __Solution 24__**
 * __Error Explanation 24__**