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| 1 : | jj | 143 | ##DESCRIPTION |
| 2 : | ## Some simple complex number algebra | ||
| 3 : | ##ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | |||
| 6 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 7 : | |||
| 8 : | loadMacros( | ||
| 9 : | "PG.pl", | ||
| 10 : | "PGbasicmacros.pl", | ||
| 11 : | "PGchoicemacros.pl", | ||
| 12 : | "PGanswermacros.pl", | ||
| 13 : | "PGauxiliaryFunctions.pl", | ||
| 14 : | "PGcomplexmacros.pl" | ||
| 15 : | ); | ||
| 16 : | |||
| 17 : | TEXT(beginproblem()); | ||
| 18 : | $showPartialCorrectAnswers = 1; | ||
| 19 : | |||
| 20 : | $a = new Complex( 0, $PI ); | ||
| 21 : | $b = new Complex( -2*sqrt($one = random(2, 3, 1 )), -($two = non_zero_random( 1, 5, 1 )) ); | ||
| 22 : | $c = (new Complex( 1, -1 ) ) * ( new Complex( -sqrt($three=random(2,3,1)), 1 ) ); | ||
| 23 : | $d = (new Complex( sqrt($four=random(2,3,1)), ($five = random (-5, -1, 1 )) ) )**2; | ||
| 24 : | $e = ( new Complex( ($six = random(-5, -1, 1 )), sqrt($seven = random(2,3, 1))))/( new Complex( | ||
| 25 : | ($eight = random(1, 5, 1)), ($nine = random( 1, 5, 1 ) ))); | ||
| 26 : | $f = -sqrt(7) * ( new Complex( 1, 1 ))/( new Complex( sqrt($ten = random( 2, 3, 1)), 1 ) ); | ||
| 27 : | |||
| 28 : | |||
| 29 : | BEGIN_TEXT | ||
| 30 : | |||
| 31 : | Write the following numbers in the polar form \(re^{i \theta} \), \(-\pi < \theta \le \pi\): $PAR | ||
| 32 : | (a) \( \pi i\) $PAR | ||
| 33 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 34 : | (b) \( -2\sqrt{$one}\ -\ $two i \) $PAR | ||
| 35 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 36 : | (c) \( ( 1\ -\ i)( - \sqrt{$three}\ +\ i)\) $PAR | ||
| 37 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 38 : | (d) \((\sqrt{$four}\ $five i)^2\) $PAR | ||
| 39 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 40 : | (e) \( \displaystyle \frac{$six\ +\ \sqrt{$seven}i}{$eight\ +\ $nine i}\) $PAR | ||
| 41 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 42 : | (f) \( \displaystyle \frac{-\sqrt{7}(1\ +\ i)}{\sqrt{$ten}\ +\ i}\) $PAR | ||
| 43 : | \( r = \) \{ans_rule(15)\}, \( \theta = \) \{ans_rule(15)\}, $PAR | ||
| 44 : | |||
| 45 : | $PAR $BR | ||
| 46 : | END_TEXT | ||
| 47 : | |||
| 48 : | |||
| 49 : | ANS(num_cmp( $a -> rho )); | ||
| 50 : | ANS(num_cmp( $a -> theta)); | ||
| 51 : | ANS(num_cmp( $b -> rho )); | ||
| 52 : | ANS(num_cmp( $b -> theta)); | ||
| 53 : | ANS(num_cmp( $c -> rho )); | ||
| 54 : | ANS(num_cmp( $c -> theta)); | ||
| 55 : | ANS(num_cmp( $d -> rho )); | ||
| 56 : | ANS(num_cmp( $d -> theta)); | ||
| 57 : | ANS(num_cmp( $e -> rho )); | ||
| 58 : | ANS(num_cmp( $e -> theta)); | ||
| 59 : | ANS(num_cmp( $f -> rho )); | ||
| 60 : | ANS(num_cmp( $f -> theta)); | ||
| 61 : | |||
| 62 : | |||
| 63 : | |||
| 64 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 65 : |
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