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Praxis II Grab Bag Review by Jonathan

Praxis II Grab Bag Review by Jonathan. November 20, 2007. Hint: (cos(x) + i sin(x)) k = cos(kx) + i sin(kx). A. cos(2) + i sin(2) B. 5 1/3 cos(2) + 5 1/3 i sin(2) C. 5 1/3 + 5 1/3 i D. Undefined. If z = 5cos(6) + 5 i sin(6) . Find z 1/3. Answer is B. z = 5cos(6) + 5 i sin(6)

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Praxis II Grab Bag Review by Jonathan

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  1. Praxis IIGrab Bag Reviewby Jonathan November 20, 2007

  2. Hint: (cos(x) + isin(x))k = cos(kx) + isin(kx) A. cos(2) + isin(2) B. 51/3cos(2) + 51/3isin(2) C. 51/3 + 51/3i D. Undefined If z = 5cos(6) + 5isin(6).Find z1/3.

  3. Answer is B z = 5cos(6) + 5isin(6) = 5[cos(6) + isin(6)] So, z1/3 = 51/3[cos(6) + isin(6)]1/3 Using the hint gives: z1/3 = 51/3[cos((1/3)6) + isin((1/3)6)] z1/3 = 51/3[cos(2) + isin(2)] = 51/3cos(2) + 51/3isin(2)

  4. De Moivre’s Formula [cos(x) + i sin(x)]k = cos(kx) + i sin(kx)

  5. If 8 ≡ x (mod 7). What is x? A. 1 B. 2 C. 5 D. 9

  6. Modular Arithmetic Review • a ≡ b (mod n) • Read “a is congruent to b mod n” • Deals with remainders: b is the remainder of a/n • a – b ≡ x (mod n) • Where x is a multiple of n • EX: 6 ≡ 2 (mod 4) 6 – 2 ≡ 4 (mod 4)

  7. Some operations If a1≡ b1 (mod n) and a2 ≡ b2 (mod n) Then, (a1 + a2) ≡ (b1 + b2) (mod n) (a1 - a2) ≡ (b1 - b2) (mod n) (a1 a2) ≡ (b1 b2) (mod n)

  8. If 8 ≡ 3 (mod 5) and 72 ≡ 3x (mod 5). What is x? A. 3 B. 11 C. 7 D. 4

  9. If 8 ≡ 3 (mod 5) and 72 ≡ 3x (mod 5). What is x? First, recognize we can use (a1 a2) ≡ (b1 b2) (mod n) So, (72) ≡ (3x) (mod 5) Then, a1 = 8 and b1 = 3 a2 = 72/8 = 9 and b2 = x Therefore 9 ≡ x (mod 5)  9/5 = 1 r4 x = 4

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