Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Binomial Theorem

Short Answer Type

71.

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

227 Views

72.

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

181 Views

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

<pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/17/af/d635b2ae437cbfac35c5d2761edb.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/17/af/d635b2ae437cbfac35c5d2761edb.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...') #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...') #4 /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...') #5 /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) #6 /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) #7 {main}</pre>

term from the beginning in the expression of

left parenthesis straight a plus straight b right parenthesis to the power of straight n

space space straight T subscript straight r plus 1 end subscript equals straight C presuperscript straight n subscript straight r space straight a to the power of straight n minus straight r end exponent space straight b to the power of straight r

∴ Co-efficient of (r + 1)^th term from the beginning =

straight C presuperscript straight n subscript straight r

...(i)

<pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/17/af/d635b2ae437cbfac35c5d2761edb.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/17/af/d635b2ae437cbfac35c5d2761edb.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...') #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...') #4 /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...') #5 /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) #6 /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) #7 {main}</pre>

term from the end is

straight T subscript straight n minus left parenthesis straight r plus 1 right parenthesis plus 2 end subscript

i.e.,

straight T subscript straight n minus straight r plus 1 end subscript

But,

straight T subscript straight n minus straight r plus 1 end subscript equals straight C presuperscript straight n subscript straight n minus straight r end subscript space straight a to the power of straight n minus left parenthesis straight n minus straight r right parenthesis end exponent space straight b to the power of straight n minus straight r end exponent

equals straight C presuperscript straight n subscript straight n minus straight r end subscript space straight a to the power of straight r space straight b to the power of straight n minus straight r end exponent

∴ Co-efficient of (r + 1)^th term from end

equals straight C presuperscript straight n subscript straight n minus straight r end subscript space equals space straight C presuperscript straight n subscript straight n minus left parenthesis straight n minus straight r right parenthesis end subscript

(∵

straight C presuperscript straight n subscript straight r equals straight C presuperscript straight n subscript straight n minus straight r end subscript

)

space space equals straight C presuperscript straight n subscript straight r

...(ii)
From (i) and (ii), we have co-efficient of (r + 1)^th term from the beginning is equal to the coefficient of (r + 1)^th from the end.

107 Views

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 $square root of 6 colon 1.$

251 Views

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

120 Views

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

118 Views

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

115 Views

78. In the expansion of (1 + x)ⁿ, the co-efficients of x^p-1 and of x^q-1 are equal. Show that p + q = n + 2, p ≠ q.

115 Views

79. In the expansion of (1 + a)^m+n , prove that co-efficient of a^m and aⁿ are equal.

112 Views

80. Find a, if the co-efficients of x² and x³ in the expansion of (3 + ax)⁹ are equal.

115 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

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

Long Answer Type

Short Answer Type

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