Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test # on ths test cover and bubble. 3. In the Name blank, prnt your name; n the Subject blank, prnt the name of the test; n the Date blank, prnt your school name (no abbrevatons).. Scorng for ths test s 5 tmes the number correct + the number omtted. 5. You may not st 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 ths test. 8. Any napproprate behavor or any form of cheatng wll lead to a ban of the student and/or school from future natonal conventons, dsqualfcaton of the student and/or school from ths conventon, at the dscreton of the Mu Alpha Theta Governng Councl. 9. If a student beleves a test tem s defectve, select E) NOTA and fle a Dspute Form explanng why. 0. If a problem has multple correct answers, any of those answers wll be counted as correct. Do not select E) NOTA n that nstance.. Unless a queston asks for an approxmaton or a rounded answer, gve the exact answer.
03 ΜΑΘ Natonal Conventon Note: For all questons, answer means none of the above answers s correct. Furthermore, assume that, cs cos sn, and Re( z ) and Im( z ) are the real and magnary parts of z, respectvely, unless otherwse specfed.. Compute 3 6. Assume all answer choces have the correct unts. Good luck! 03 03 03 03. Compute 03. 3. Compute 3. 5 5 3 5 69 3 5 69 5. Compute. 3 cs k, k 0, 8 cs k, k 0, 8 cs k, k 0, 8 cs k, k 0, 8 5. Compute 5 n 5 n Re cs((n ) ) Im cs((n ) ). Page of 7
03 ΜΑΘ Natonal Conventon k3 6. Let a k k k, where each k s randomly chosen from the set {,,3,}. What s the probablty that a 0? 7 6 9 6 37 56 39 56 7. The solutons to the equaton 6 z 79 can be wrtten n the form zk 3(cos k sn k ) where k {,,3,,5,6} and 0 3 5 6. Compute the value of z z. 3 6 3 3 3 3 3 6 8. Consder two complex numbers z a b and w c d, a, b, c, d, as well as the vectors v Re( z),im( z) and v Re( w),im( w). Whch of the followng s equal to v v? Re( z w) Re z w Re zw Re z w 9. The bnomal coeffcent, n n!, r r! n r! can be wrtten as n n( n )( n ) ( n r ) r r! n order to accommodate for all complex n. Usng ths, compute. 5 9 9 5 0. Defne the operaton as cs( ) cs( ) cos cs. There exst real numbers 0 90 and 0 90 (both n degrees) such that Compute, gnorng unts n your answer. cs +cs cs3 cs35. 78 6 8 505 Page of 7
03 ΜΑΘ Natonal Conventon. Regon R s bounded by the set of all complex numbers on the Argand plane z a b where a and b are real numbers satsfyng the equaton a a 3 6 6 b b. Compute the area of R. 6 3 6 8. A complex number z satsfes the equaton z z. Gven that Re( z) a, determne the value of z n terms of a. a a a a a a a 3. Consder two vectors v x, y and v,3, where xy,. Gven that v v 5 6, compute x y. 3 6 7 3. Consder the matrx A. Determne all possble values of such that the 0 determnant of B A I s 0, where I. 0 3 3 5 6 5. The fourth roots of unty are the solutons to the equaton x 0. Graphng these roots on the complex plane gves a square whch we call S. Form a sequence of squares S, where the n th square S n s formed by connectng the mdponts of S. n For example, by connectng the mdponts of S, we obtan S. Now, let f ( x ) be the polynomal wth roots equal to the vertces of S n. Compute n f (0). n n 5 3 3 5 Page 3 of 7
03 ΜΑΘ Natonal Conventon 6. Consder El s functon, f ( x) 9x 5 x k, where x and k are real. Determne the nterval for k such that the graph of f( x ) does not ntersect the x -axs. 5, 6 5, 6 5 5, 6 6 5 5,, 6 6 0 n f ( x) x nx. f has 03 roots; one of whch n0 s equal to, whle the other 0 are magnary. Let S equal the sum of the products of the roots of f( x ) taken two at a tme. Compute S. 7. Consder Patrck s functon, 0 0 0 0 0 0 8. Consder Laura s magnary, sx-sded, far dce. The sdes on the frst de show the numbers,,3,,5,6, whle the sdes of the second de show the numbers,,3,,5,6. Laura takes rolls both the dce once and multples the numbers shown. If the result s real, the player gets the absolute value of the result n dollars. If the result s magnary, the player loses the absolute value of the result n dollars. What s the expected value of ths game, n dollars? $.5 $.5 $.5 $.5 C 3. Evaluate C. 9. Consder Caleb s favorte expresson, 03 03 03 0 3 0 3 0. Consder Wll s geometrc seres. He doesn t care what the frst term s, as long as t s not zero. He would, however, lke a common rato such that at some pont n the geometrc seres, the n th term equals the frst term. What s the smallest value of n such that there are exactly 50 possble common ratos for the seres located n the second quadrant of the Argand plane? Do not consder ponts on the axes. 00 0 0 03 Page of 7
03 ΜΑΘ Natonal Conventon. The cosne functon can be approxmated by the functon f( ). x x f( x). Evaluate!! 3 37. Calculate 03 cs. cs006 03 cs007 03 cs0 03 cs03 0 k k 3. A sequence of ponts zk s plotted on the Argand plane. A bug begns at z and travels along the segments zz, zz3,, znzn,..., z0z03. Let D be the total dstance the bug travels. Fnd the remander when D s dvded by00. 8 9 90 9. Consder the matrx A 37 z. 3 A, 3 and the complex vector, 3 z. Calculate 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5. Gven that e x k x, compute the value of k! k 0 k 0 k n k. nn (!) e e e e Page 5 of 7
03 ΜΑΘ Natonal Conventon 6. Let 03 0 f ( x) x x x. Denote Rn ( ) as the remander when f( x ) s dvded by x n. Compute the remander when 03 k R ( ) s dvded by 000. k 0 0 5 56 7. Consder a sequence of functons n j f ( x) x. Let the set contan all arguments n 0 such that cs s a soluton to f ( x) 0 for a gven n. Determne the smallest postve nteger n such that the sum of the entres n s greater than or equal to 03. 0 3 j0 n 8. A Gaussan Integer s a complex number z a b where ab,. Consder the Gaussan Integer z m 3 n. Whch of the followng s not a possble value for 00 0 0 03 z? Use the followng nformaton for Problems 9 and 30: In Problem 8, we defned the Gaussan Integers. Another set of ntegers, called the Esensten Integers, are defned, for ab, ntegers, as z a b; 3 e 3. Because the argument of s, or 60, the Esensten Integers form a trangular 3 lattce on the Argand Plane, whereas the Gaussan Integers form a square lattce. 9. The trangular regon T has vertces located at the ponts z, z 3, and z3. Compute the area of T. (Hnt: Graph the ponts wth shfted axes by what angle would you shft them by?). 3 3 9 3 3 Page 6 of 7
03 ΜΑΘ Natonal Conventon 30. The absolute value of the Gaussan Integers s smply z a b. Ths s not the case for the Esensten Integers. For the general Esensten Integer z a b, deduce the absolute value n terms of a and b. a b ab a b a b a b ab Page 7 of 7