Math 135 Prelim #2 – July 24, 2006This exam has 6 problems and 7 numbered pages.Name: Instructor: Michael KozdronYou have 75 minutes to complete this exam. Show all work neatly and in order, and clearlyindicate your final answers. Answers must be justified whenever possible in order to earn fullcredit.Unless otherwise specified, no credit will be given for unsupported answers,even if your final answer is correct. Points will be deducted for incoherent, incorrect,and/or irrelevant statements. A formula page will be provided, and calculators are permitted,but no other aids are allowed.You are allowed to use standard notation. However, any new notation or abbreviations thatyou introduce must be clearly defined.This examination consists of 6 problems and is worth 100 total points. You must a nswerall of the questions in the space provided.Good luck!Problem Score123456TOTAL:Formula PageThe numerical equivalents of the letters are as follows:A B C D E F G H I J K L M0 1 2 3 4 5 6 7 8 9 10 11 12N O P Q R S T U V W X Y Z13 14 15 16 17 18 19 20 21 22 23 24 25A =a bc ddet(A) = ad − bcA−1= det(A)−1d −b−c aI =n0(n0− 1) + · · · + n25(n25− 1)n(n − 1)k =0.0265n(0.065 − I) + n (I − 0.0385)# digits in binary representation of x =ln xln 2+ 1A B C D E F G H I J K L M N O P Q R S T U V W X Y ZA A B C D E F G H I J K L M N O P Q R S T U V W X Y ZB B C D E F G H I J K L M N O P Q R S T U V W X Y Z AC C D E F G H I J K L M N O P Q R S T U V W X Y Z A BD D E F G H I J K L M N O P Q R S T U V W X Y Z A B CE E F G H I J K L M N O P Q R S T U V W X Y Z A B C DF F G H I J K L M N O P Q R S T U V W X Y Z A B C D EG G H I J K L M N O P Q R S T U V W X Y Z A B C D E FH H I J K L M N O P Q R S T U V W X Y Z A B C D E F GI I J K L M N O P Q R S T U V W X Y Z A B C D E F G HJ J K L M N O P Q R S T U V W X Y Z A B C D E F G H IK K L M N O P Q R S T U V W X Y Z A B C D E F G H I JL L M N O P Q R S T U V W X Y Z A B C D E F G H I J KM M N O P Q R S T U V W X Y Z A B C D E F G H I J K LN N O P Q R S T U V W X Y Z A B C D E F G H I J K L MO O P Q R S T U V W X Y Z A B C D E F G H I J K L M NP P Q R S T U V W X Y Z A B C D E F G H I J K L M N OQ Q R S T U V W X Y Z A B C D E F G H I J K L M N O PR R S T U V W X Y Z A B C D E F G H I J K L M N O P QS S T U V W X Y Z A B C D E F G H I J K L M N O P Q RT T U V W X Y Z A B C D E F G H I J K L M N O P Q R SU U V W X Y Z A B C D E F G H I J K L M N O P Q R S TV V W X Y Z A B C D E F G H I J K L M N O P Q R S T UW W X Y Z A B C D E F G H I J K L M N O P Q R S T U VX X Y Z A B C D E F G H I J K L M N O P Q R S T U V WY Y Z A B C D E F G H I J K L M N O P Q R S T U V W XZ Z A B C D E F G H I J K L M N O P Q R S T U V W X Y1. (16 points) Suppose that A =−2 1−3 2and B =−1 23 3.(a) Compute AB MOD 26.(b) Let C = AB MOD 26 be the matrix that you computed in (a). Determine C−1MOD 26.(continued)12. (16 points)(a) Convert the number with base twenty-six representation ELVES to decimal (base ten).(b) Convert the number with binary (base two) representation 11011001 to decimal.(c) Convert the number with decimal representation 123 to octal (base eight).(d) Let a = 110110 and b = 10101 be two binary numbers. Compute the binary numbersa + b and a − b.(continued)23. (16 points)(a) Encipher the message GANDALF THE GREY using the Vigen`ere method with the keywordBILBO.(b) Suppose that the Vigen`ere encipherment produced the ciphertext PKSFIH QDNB whenthe three-letter key string XV was used. (The last letter of the key string is not yetknown.) Decipher as much of the plaintext as possible, and based on the plaintext youobtain, determine the missing plaintext letters and the third letter of the key string.(continued)34. (16 points)(a) The ciphertext ELPF resulted from a Hill encipherment with the key matrix A =4 33 1.Decipher the message.(b) The ciphertext BAOI resulted from a Hill encipherment of the plaintext BASE. Basedon this information, determine the key matrix A.(continued)45. (20 points) A message was enciphered using the Vigen`ere method with a keywordof a certain length. The ciphertext is shown b elow, and certain repeated letter groups areunderlined.YYFHS WZBJG KFFWV JVZYS SBWQU XLBGS WKVHG PPGHJJETRF YYSGK FITOC WUGLB YYSLF MRZOG TWGWC SVBLBJWCUA TIHDZ RVBGC TDSGH TUWHC SVTRF YYSGO WBZRFIFBKW XUOUY YYFRB JZBWV JCOQR TWARF IFFZV JISWVJJVDR TNGOW JFBHF NEUWC WLZHH MVADZ QFBHF NEUWCKZBGH MVARB JIWQU YFPUW SXHKS RRZOO SUWQH MVRDFPESVG GZBGH MVALB …
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