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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

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

 $space space space space space space space space space space space space straight r space cos space straight theta space equals space minus 1$ ... (i)
$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight r space sin space straight theta space equals space 1$ ...(iii)

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

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

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space space space space space space space space space straight r squared space equals space 2 space rightwards double arrow space straight r space equals space square root of 2$

∴ from (ii), $space space space space space space space space space space space space space space space cosθ space space equals space minus fraction numerator 1 over denominator square root of 2 end fraction comma space and space from space left parenthesis iii right parenthesis comma space sin space straight theta space equals space fraction numerator 1 over denominator square root of 2 end fraction$

Since $space space sin space straight theta space$ is + ve and $space space space space space cos space straight theta$ is -ve

∴ $space space space space sin space theta space space equals space fraction numerator 1 over denominator square root of 2 end fraction space equals space space sin space 45 degree space equals sin space left parenthesis space 180 degree space minus space 45 degree right parenthesis space equals space sin space left parenthesis 135 degree right parenthesis space equals space sin open parentheses 135 space x space straight pi over 180 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space sin open parentheses fraction numerator 3 straight pi over denominator 4 end fraction close parentheses$

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

Putting in (i), we get $space space space space space space space space space space space space space space space space space space space space space space straight z space equals straight r space left parenthesis cos space straight theta space plus space straight i space sin space straight theta right parenthesis space equals space square root of 2 open square brackets cos open parentheses fraction numerator 3 straight pi over denominator 4 end fraction close parentheses plus space straight i space sin open parentheses fraction numerator 3 straight pi over denominator 4 end fraction close parentheses close square brackets$

which is the required polar form of z.

Hence. $space space space space space space space space space space space space open vertical bar straight z close vertical bar space equals space straight r space equals space square root of 2 comma space arg space left parenthesis straight z right parenthesis space equals space straight theta space equals space fraction numerator 3 straight pi over denominator 4 end fraction$