View of /trunk/NationalProblemLibrary/Rochester/setComplexNumbers/ur_cn_1_32.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('complex','imaginary','root')
    6 ## Tagged by cmd6a 4/20/06
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Complex Variables')
   10 ## DBsection('Complex Numbers')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('Complex Analysis')
   15 ## EditionText1('3')
   16 ## AuthorText1('Saff, Snider')
   17 ## Section1('1.5')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGauxiliaryFunctions.pl",
   28 "PGcomplexmacros.pl"
   29 );
   30 
   31 
   32 TEXT(beginproblem());
   33 $showPartialCorrectAnswers = 1;
   34 
   35 $s1 = random(-1,1,2);
   36 if ($s1 == 1) { $pm1 = '+';}
   37 else { $pm1 = '-';}
   38 
   39 $s2 = random(-1,1,2);
   40 if ($s2 == 1) { $pm2 = '+';}
   41 else { $pm2 = '-';}
   42 
   43 $r = random(2,10,1);
   44 
   45 BEGIN_TEXT
   46 Find all the values of the following :
   47 $PAR
   48 (1) \( (1 $pm1 \sqrt{3}i)^\frac{1}{3} \)
   49 $PAR
   50 Place all answers in the following blank, separated by commas:
   51 $PAR
   52 \{ans_rule(80)\}
   53 $PAR
   54 (2) \( (i $pm2 1)^\frac{1}{2} \)
   55 $PAR
   56 Place all answers in the following blank, separated by commas:
   57 $PAR
   58 \{ans_rule(80)\}
   59 $PAR
   60 (3) \( \displaystyle \left(\frac{$r i}{1+i}\right)^\frac{1}{6} \)
   61 $PAR
   62 Place all answers in the following blank, separated by commas:
   63 $PAR
   64 \{ans_rule(80)\}
   65 END_TEXT
   66 
   67 $a = new Complex( 2**(1/3)*cos($s1*$PI/9) , 2**(1/3)*sin($s1*$PI/9));
   68 $b = new Complex( 2**(1/3)*cos(($s1+6)*$PI/9) , 2**(1/3)*sin(($s1+6)*$PI/9));
   69 $c = new Complex( 2**(1/3)*cos(($s1+12)*$PI/9) , 2**(1/3)*sin(($s1+12)*$PI/9));
   70 $d = new Complex( 2**(1/4)*cos((4+$s2)*$PI/8) , 2**(1/4)*sin((4+$s2)*$PI/8));
   71 $f = new Complex( 2**(1/4)*cos((12+$s2)*$PI/8) , 2**(1/4)*sin((12+$s2)*$PI/8));
   72 $g = new Complex();
   73 $g = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI/24);
   74 $h = new Complex();
   75 $h = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI*(1 + 1 * 8)/24);
   76 $j = new Complex();
   77 $j = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI*(1 + 2 * 8)/24);
   78 $k = new Complex();
   79 $k = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI*(1 + 3 * 8)/24);
   80 $l = new Complex();
   81 $l = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI*(1 + 4 * 8)/24);
   82 $m = new Complex();
   83 $m = ($r*$r/2)**(1/12)*2.71828182845905**(i()*$PI*(1 + 5 * 8)/24);
   84 
   85 ANS(multi_cmp( [ cplx_cmp($a), cplx_cmp($b), cplx_cmp($c) ] ));
   86 ANS(multi_cmp( [ cplx_cmp($d), cplx_cmp($f) ] ));
   87 ANS(multi_cmp( [ cplx_cmp($g), cplx_cmp($h), cplx_cmp($j), cplx_cmp($k), cplx_cmp($l), cplx_cmp($m)] ));
   88 
   89 ENDDOCUMENT();        # This should be the last executable line in the problem.
   90