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Binomial Theorem

Short Answer Type

71.

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

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72.

Find the 11^th term from the end in the expansion of $open parentheses 2 straight x minus 1 over straight x squared close parentheses to the power of 25$

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73. Prove that in the expansion of (a + b)ⁿ, the co-efficients of terms equidistant from the beginning and end are the same.

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Long Answer Type

74.

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $open parentheses fourth root of 2 plus fraction numerator 1 over denominator fourth root of 3 end fraction close parentheses to the power of straight n$ is $square root of 6 colon 1.$

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Short Answer Type

75. Find the co-efficient of

straight x to the power of 5 space in space left parenthesis straight x plus 3 right parenthesis to the power of 8

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76. Find the co-efficient of:

straight a to the power of 5 straight b to the power of 7 space in space left parenthesis straight a minus 2 straight b right parenthesis to the power of 12

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77. Find the positive value of m for which the co-efficient of x² in the expansion of (1 + x)^m is 6.

In the expansion of $left parenthesis 1 plus straight x right parenthesis to the power of straight m comma$ the co-efficient of $straight x squared equals straight C presuperscript straight m subscript 2$

∴ $straight C presuperscript straight m subscript 2 equals 6$

$rightwards double arrow$ $fraction numerator straight m factorial over denominator left parenthesis straight m minus 2 right parenthesis factorial space 2 factorial end fraction equals 6$ $rightwards double arrow$ $space space fraction numerator straight m left parenthesis straight m minus 1 right parenthesis over denominator 2 end fraction equals 6$

$rightwards double arrow$ $straight m squared minus straight m minus 12 equals 0$ $rightwards double arrow$ $left parenthesis straight m minus 4 right parenthesis space left parenthesis straight m plus 3 right parenthesis space equals 0$

$rightwards double arrow$ m = 4, m = -3

∴ The positive value of m = 4.