Complex Numbers Alpha, Round 1 Test #123 1. Write your 6-digit ID# in the I.D. NUMBER grid, left-justified, and bubble. Check that each column has only one number darkened. 2. In the EXAM NO. grid, write the 3-digit Test # on this test cover and bubble. 3. In the Name blank, print your name; in the Subject blank, print the name of the test; in the Date blank, print your school name (no abbreviations). 4. Scoring for this test is 5 times the number correct + the number omitted. 5. You may not sit adjacent to anyone from your school. 6. TURN OFF ALL CELL PHONES OR OTHER PORTABLE ELECTRONIC DEVICES NOW. 7. No calculators may be used on this test. 8. Any inappropriate behavior or any form of cheating will lead to a ban of the student and/or school from future national conventions, disqualification of the student and/or school from this convention, at the discretion of the Mu Alpha Theta Governing Council. 9. If a student believes a test item is defective, select “E) NOTA” and file a Dispute Form explaining why. 10. If a problem has multiple correct answers, any of those answers will be counted as correct. Do not select “E) NOTA” in that instance. 11. Unless a question asks for an approximation or a rounded answer, give the exact answer. Alpha Complex Numbers 2013 ΜΑΘ National Convention Note: For all questions, answer “(E) NOTA” means none of the above answers is correct. Furthermore, assume that i 1, cis cos i sin , and Re(z ) and Im(z ) are the real and imaginary parts of z, respectively, unless otherwise specified. Assume all answer choices have the correct units. Good luck! 1. Compute 1 3 11 61. (A) i 2013 (B) 2013 (C) 2013 (D) i 2013 (E) NOTA 2. Compute i2013. (A) i (B) 1 (C) i (D) 1 (E) NOTA 34 i 3. Compute . 5 12i 5 25 13 169 (A) (B) (C) (D) (E) NOTA 13 169 5 25 iii 4. Compute . 444 1 34 1 14 (A) 2 cis kk , 0,1 (B) 2 cis 2kk , 0,1 28 28 1 12 1 14 (C) 2 cis kk , 0,1 (D) 2 cis kk , 0,1 (E) NOTA 28 28 45 Re cis((2n 1) ) n1 5. Compute 45 . Im cis(2(2n 1) ) n1 (A) 21 (B) 212 (C) 212 (D) 2 (E) NOTA Page 1 of 7 Alpha Complex Numbers 2013 ΜΑΘ National Convention k1 k 2k3 k 4 6. Let a i i i i , where each ki is randomly chosen from the set {1,2,3,4}. What is the probability that a 0? 7 9 37 39 (A) (B) (C) (D) (E) NOTA 64 64 256 256 6 7. The solutions to the equation z 729 can be written in the form zik3(cos k sin k ) where k {1,2,3,4,5,6} and 01 2 3 4 5 6 2 . Compute the value of |zz36 |. 33 (A) (B) 3 (C) 33 (D) 6 (E) NOTA 2 8. Consider two complex numbers z a bi and w c di, a,,,, b c d as well as the vectors v1 Re( z ),Im( z ) and v2 Re( w ),Im( w ) . Which of the following is equal to vv12 ? (A) Re(zw ) (B) Rezw (C) Rezw (D) Rezw (E) NOTA n n! 9. The binomial coefficient, , can be written as r r!! n r n n( n 1)( n 2) ( n r 1) r r! i in order to accommodate for all complex n. Using this, compute . 4 5 9 9 5 (A) (B) (C) (D) (E) NOTA 12 24 24 12 10. Define the operation as cis( ) cis( ) cos cis . There exist real numbers 0 90 and 0 90 (both in degrees) such that cis +cis 2 cis43 cis35 . Compute , ignoring units in your answer. (A) 78 (B) 624 (C) 1248 (D) 1505 (E) NOTA Page 2 of 7 Alpha Complex Numbers 2013 ΜΑΘ National Convention 11. Region R is bounded by the set of all complex numbers on the Argand plane where a and b are real numbers satisfying the equation a2212 a 13 6 6 b b . Compute the area of R. (A) 16 (B) 32 (C) 64 (D) 128 (E) NOTA 12. A complex number z satisfies the equation zz2. Given that Re(za ) , determine the value of z in terms of a. 1 aa2 1 aa2 1 aa2 4 (A) (B) (C) (D) (E) NOTA 1 a 2 2 2 3 6 13. Consider two vectors v1 x, y and v2 1 i ,3 2 i , where xy,. Given that z a bi v12 v 5 6 i , compute xy . (A) (B) 4 (C) (D) 7 (E) NOTA 32 14. Consider the matrix A . Determine all possible values of such that the 41 10 determinant of BAI is 0, where I . 01 (A) 12 i (B) 23 i (C) 34 i (D) 56 i (E) NOTA 15. The fourth roots of unity are the solutions to the equation x4 1 0. Graphing these roots on the complex plane gives a square which we call S1. Form a sequence of squares Si , where the nth square Sn is formed by connecting the midpoints of Sn1. For example, by connecting the midpoints of S1, we obtain S2. Now, let fxn () be the polynomial with roots equal to the vertices of Sn . Compute fn (0). n1 4 1 1 4 (A) (B) (C) (D) (E) NOTA 5 3 3 5 Page 3 of 7 Alpha Complex Numbers 2013 ΜΑΘ National Convention 16. Consider Eli’s function, f( x ) 9 x22 5 x k , where x and k are real. Determine the interval for such that the graph of fx() does not intersect the -axis. 5 5 55 55 (A) , (B) , (C) , (D) ,, (E) NOTA 6 6 66 66 2012 n 17. Consider Patrick’s function, f( x ) x 1 1 nx . f has 2013 roots; one of which n0 is equal to 1, while the other 2012 are imaginary. Let S2 equal the sum of the products of the roots of taken two at a time. Compute 4021 1 1 4021 (A) (B) (C) (D) (E) NOTA 2012 2012 2012 2012 18. Consider Laura’s imaginary, six-sided, fair dice. The sides on the first die show the numbers i,2 i ,3,4,5,6, i while the sides of the second die show the numbers 1,2,3,4i ,5 i ,6. i Laura takes rolls both the dice once and multiplies the numbers shown. If the result is real, the player gets the absolute value of the result in dollars. If the result is imaginary, the player loses the absolute value of the result in dollars. What is the expected value of this game, in dollars? (A) $2.25 (B) $1.25 (C) $1.25 (D) $2.25 (E) NOTA 2013 19. Consider Caleb’s favorite expression, Ci1 3 . Evaluate C. (A) 22013 (B) 22013 (C) 22012 1 i 3 (D) 22012 1i 3 (E) NOTA nth 20. Consider Will’s geometric series. He doesn’t care what the first term is, as long as it’s S2. not zero. He would, however, like a common ratio such that at some point in the geometric series, the term equals the first term. What is the smallest value of n such that there are exactly 50 possible common ratios for the series located in the second quadrant of the Argand plane? Do not consider points on the axes. (A) 200 (B) 201 (C) 202 (D) 203 (E) NOTA Page 4 of 7 Alpha Complex Numbers 2013 ΜΑΘ National Convention xx24 21. The cosine function can be approximated by the function fx( ) 1 . Evaluate 2! 4! fi( ). 11 13 37 (A) (B) (C) (D) (E) NOTA 24 24 24 2013 22. Calculate cis . 1 (A) cis 1006 2013 (B) cis 1007 2013 (B) cis 2012 2013 (D) cis 2013 2014 (E) NOTA 1 k 1 k 23. A sequence of points zik is plotted on the Argand plane. A bug begins at z1 2 and travels along the segments z1 z 2, z 2 z 3 , , znn z 1 ,..., z 2012 z 2013 . Let D be the total distance the bug travels. Find the remainder when is divided by100. (A) 18 (B) 19 (C) 90 (D) 91 (E) NOTA 1 13 3 24. Consider the matrix A , and the complex vector, z . Calculate 2 31 4i Az37 . 1 3 4i 3 1 3 4i 3 1 3 4i 3 1 3 4i 3 (A) (B) (C) (D) (E) NOTA 2 3 3 4i 2 3 3 4i 2 3 3 4i 2 3 3 4i k x ik 25. Given that ex , compute the value of . k k0 k! k 0 nn( !) 1 n1 (A) iei i (B) iei i (C) iei i (D) iei i (E) NOTA Page 5 of 7 Alpha Complex Numbers 2013 ΜΑΘ National Convention 26. Let f( x ) x2013 x 2012 x 1. Denote Rn() as the remainder when is divided fx()2013 by xn . Compute the remainder when Ri()k is divided by 1000. k0 (A) 0 (B) 42 (C) 52 (D) 56 (E) NOTA 21n j 27. Consider a sequence of functions fn (). x x Let the set contain2013 all arguments j0 02 such that cis is a2012 solution to fxn ( ) 0 for a given Determine the smallest positive integer such that the sum of the entries in is greater than or equal to 2013 .