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Binomial Theorem

Short Answer Type

81. Find the term independent of x in the expansion of:

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82. Find the term independent of x in the expansion of:
$open parentheses 2 straight x squared minus 1 over straight x close parentheses to the power of 12$

In the expansion of $open parentheses 2 straight x squared minus 1 over straight x close parentheses to the power of 12$
Suppose that $straight T subscript straight r plus 1 end subscript$ is the term independent of x.
∴   $straight T subscript straight r plus 1 end subscript$ contains $straight x to the power of 0$

Now, $straight T subscript straight r plus 1 end subscript space equals space straight C presuperscript 12 subscript straight r space left parenthesis 2 straight x squared right parenthesis to the power of 12 minus straight r end exponent times open parentheses negative 1 over straight x close parentheses to the power of straight r$
Power of x = (24 - 2r) - (r)
= 24 - 3r
Power of x required = 0
∴ 24 - 3r = 0 or r = 8
∴    $straight T subscript straight r plus 1 end subscript space equals space straight T subscript 9$ is the term independent of x.
$straight T subscript 9 space equals space straight C presuperscript 12 subscript 8 space left parenthesis 2 straight x squared right parenthesis to the power of 4 space open parentheses negative 1 over straight x close parentheses to the power of 8$
   $equals space fraction numerator 12 factorial over denominator 8 factorial space 4 factorial end fraction cross times 16 straight x to the power of 8 cross times 1 over straight x to the power of 8 equals fraction numerator 12 cross times 11 cross times 10 cross times 9 over denominator 4 cross times 3 cross times 2 cross times 1 end fraction cross times 16 equals 7920$

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83. Find the term independent of x in the expansion of:

open parentheses 2 straight x minus straight x squared over 4 close parentheses to the power of 9

110 Views

Long Answer Type

84.

The absolute term in the expansion of: $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ is 405. Find value of k.

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85. Show that the co-efficient of the middle term in the expansion of (1 + x)²ⁿ is equal to the sum of the co-efficients of two middle terms in the expansion of (1 + x)^2n-1

94 Views

86.

Find a, b and n in the expansion of $left parenthesis straight a plus straight b right parenthesis to the power of straight n$ if the first three terms of the expansion
are 729, 7290 and 30375, respectively.