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Complex Numbers and Quadratic Equations

Short Answer Type

11.

Find the multiplicative inverse of each of the complex number.

4 - 3i

175 Views

12.

Find the multiplicative inverse of each of the complex number.

$space space space space space space space space space space space space square root of 5 plus space 3 straight i$

126 Views

13.

Express the following in a + ib form.

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

120 Views

14.

Find the modulus and the arguments of each of the complex numbers.

$space space space space space space space space space space minus 1 minus straight i square root of 3$

132 Views

15.

Convert each of the complex numbers given in the Polar form.

$space space space space space space space space space space space space space square root of 3 plus space straight i$

114 Views

16.

Convert each of the complex numbers given in the Polar form.

-1 + i

124 Views

17.

Convert each of the complex numbers given in the Polar form.

-3

122 Views

18.
Convert each of the complex numbers given in the Polar form.
1 - i

Here, z = 1 - i =

Let z = 1 - i = $space space space space space space space space space space straight r open parentheses cos space straight theta space plus space straight i space sin space straight theta close parentheses$ ..(i)

$rightwards double arrow$ $space space space space space space space space space space space space space space space straight r space cos space straight theta space equals space 1 space space space space space space space space space space space space space space space space straight r space sin space straight theta space equals space minus 1$ ..(ii)
 .. (iii)

Squaring and adding (ii) adn (iii), we get

$space space space space space space space space space space space space space space space straight r squared left parenthesis sin squared straight theta space plus space cos squared straight theta right parenthesis space equals space left parenthesis 1 right parenthesis squared plus space left parenthesis negative 1 right parenthesis squared space equals space 1 space plus space 1 space equals space 2$

or $space space space space space space space space space space straight r space equals space square root of 2$

From (ii), $space space space space space space space space space space space space space space space cos space straight theta space equals space fraction numerator 1 over denominator square root of 2 end fraction$

From (iii), $space space space space space space space space space sin space straight theta space equals space fraction numerator negative 1 over denominator square root of 2 end fraction$

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

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

Putting in (i), we get $space space space space space space space space space space space straight z space equals space square root of 2 open square brackets cos open parentheses fraction numerator negative straight pi over denominator 4 end fraction close parentheses plus straight i space sin space open parentheses fraction numerator negative straight pi over denominator 4 end fraction close parentheses close square brackets$
Which is the required polar form.

Hence, $space space space space space space space space space space space space space space space space space space space space space space open vertical bar straight z close vertical bar space space equals space square root of 2 comma space arg space left parenthesis straight z right parenthesis space equals space fraction numerator negative straight pi over denominator 4 end fraction$