Write the following complex numbers in polar form and hence, fin
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    Complex Numbers and Quadratic Equations

    Multiple Choice QuestionsShort Answer Type

    21.

    solve each of the following equations:

    x2 + 3 = 0

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    Answer
    22.

    solve each of the following equations:

    2x2 + x + 1 = 0

    120 Views
    Answer
    23.

    solve each of the following equations:

    space space space space space space square root of 2 space straight x squared space plus space straight x space plus space square root of 2 space equals space 0

    115 Views
    Answer
    24.

    Solve each of the following equations:

     space space space space space space space space space space straight x squared space plus space straight x space plus space fraction numerator 1 over denominator square root of 2 end fraction equals space 0

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    Answer
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    25.

    Write the following complex numbers in polar form and hence, find their modulus and argument.


    space space space space space space space space space space space space space 1 half minus fraction numerator square root of 3 over denominator 2 end fraction straight i

    Here, 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 z space equals space 1 half minus fraction numerator square root of 3 over denominator 3 end fraction straight i

    Let    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 z space equals 1 half minus fraction numerator square root of 3 over denominator 2 end fraction straight i equals space straight r space left parenthesis cos space straight theta space plus space straight i space sin space straight theta right parenthesis                             ...(i)

    Equating real and imaginary parts, 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 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 cos space straight theta space equals space 1 half                                          ...(ii)

                               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 space sin space straight theta space equals space minus fraction numerator square root of 3 over denominator 2 end fraction space space space space space space space space space space space                                  ...(iii)

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


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    rightwards double arrow                      r2 =1             rightwards double arrow     r = 1

    From (ii)            space space space space space space space space space space open parentheses straight I close parentheses space cosθ space equals space 1 half               space space rightwards double arrow     <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>


    From (iii)          (I) space space space space space space space space space sin space straight theta space equals negative space fraction numerator square root of 3 over denominator 2 end fraction            rightwards double arrow       space space space space space space space space sin space straight theta space equals space minus fraction numerator square root of 3 over denominator 2 end fraction

    Since,space space space space space space space space space sin space straight theta space is space minus ve space and space cos space straight theta space is space plus ve

     

    ∴     <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>  lies in IVth quadrant

    ∴          space space space space space space space space space space space space space space space space space space sin space straight theta space equals space fraction numerator negative square root of 3 over denominator 2 end fraction equals space minus space sin 60 degree space equals space sin left parenthesis negative 60 right parenthesis degree space equals space sin open parentheses fraction numerator negative straight pi over denominator 3 end fraction close parentheses


    rightwards double arrow                  space space space space space space space space space straight theta space equals space fraction numerator negative straight pi over denominator 3 end fraction

    Putting in (i), we get

                       <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>

    Which  is required polar  form  of z.

    Hence,       space space space space space space space space space open vertical bar straight z close vertical bar space space equals space straight r equals space 1 comma space space space arg left parenthesis straight z right parenthesis space equals space straight theta space equals space fraction numerator negative straight pi over denominator 3 end fraction

                                  
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    26.

    If z1- -3x + 2iy and z2 = 6 -5i are equal, find value of x and y.

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    27.

    If Z1 = (x - 3y) - i (2x + y), and Z2 = -1 - 5i are equal,   find values of x a nd y.

    300 Views
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    28.

    Evaluate :

    1 + i2 + i4 + i6 + .....+ i20

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    29.

    Evaluate :

    space space space space space space space space 1 over straight i to the power of 4 straight n plus 2 end exponent

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    30.

    Simplify

    (1 -i)4

    195 Views
    Answer
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