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Complex Numbers and Quadratic Equations

Short Answer Type

61.
Write the following in the form a + ib.

$space space space space space space fraction numerator 1 minus 2 straight i over denominator left parenthesis 2 plus 3 straight i right parenthesis squared end fraction space$

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space space space space space space space space space space space space space space equals space space space space space fraction numerator 1 minus 2 straight i over denominator negative 5 plus 12 straight i end fraction. fraction numerator negative 5 minus 12 straight i over denominator negative 5 minus 12 straight i end fraction equals fraction numerator negative 5 minus 12 straight i plus 10 straight i plus 24 straight i squared over denominator left parenthesis negative 5 right parenthesis squared minus left parenthesis 12 straight i right parenthesis squared end fraction

space space space space space space space space space space space space space space space space space space space space space space space space space equals space space fraction numerator negative 5 minus 2 straight i plus 24 over denominator 25 minus 144 straight i squared end fraction equals fraction numerator negative 29 minus 2 straight i over denominator 25 plus 144 end fraction equals space fraction numerator negative 29 over denominator 169 end fraction minus fraction numerator 2 straight i over denominator 169 end fraction equals space straight a plus ib

Where

space space space space space space space straight a space equals space fraction numerator negative 29 over denominator 169 end fraction comma space straight b space equals space minus fraction numerator negative 2 over denominator 169 end fraction

102 Views

62.

Find the multiplicative inverse of the following complex numbers :

$space space space space space space space space space space space 4 minus square root of negative 9 end root$

110 Views

63.

Find the multiplicative inverse of the following complex numbers :

$space space space space space space space space space space space space space space space space fraction numerator left parenthesis 2 plus 3 straight i right parenthesis space left parenthesis 3 space plus 2 straight i right parenthesis over denominator 5 space plus space straight i end fraction$

112 Views

64.

Find the multiplicative inverse of :

$space space space space space space space space space space space space straight z space equals space cos space straight theta space minus space 1 space sin space straight theta$

107 Views

65.

Find the multiplicative inverse of :

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98 Views

66.

Find the square root of the following complex numbers :

-5 + 12i

108 Views

67.

Find the square root of the following complex numbers :

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

112 Views

68.

Verify that $space space space space space space top enclose left parenthesis 2 plus 3 straight i right parenthesis left parenthesis straight i minus 2 straight i right parenthesis end enclose space equals space top enclose left parenthesis 2 plus 3 straight i right parenthesis space end enclose space. space top enclose left parenthesis 1 minus 2 straight i right parenthesis end enclose space space$

132 Views

69.

Verify that

$space space space space space space space space space space space space space top enclose open parentheses fraction numerator 3 plus 4 straight i over denominator 1 minus 2 straight i end fraction close parentheses end enclose space equals space fraction numerator top enclose open parentheses 3 plus 4 straight i close parentheses end enclose over denominator top enclose open parentheses 1 minus 2 straight i close parentheses end enclose end fraction$