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Maths-

General

Easy

Question

The term independent of x in expansion of $blank open parentheses fraction numerator x plus 1 over denominator x to the power of 2 divided by 3 end exponent minus x to the power of 1 divided by 3 end exponent plus 1 end fraction minus fraction numerator x minus 1 over denominator x minus x to the power of 1 divided by 2 end exponent end fraction close parentheses to the power of 10 end exponent$ is

120
210
310
4

The correct answer is: 210

$left parenthesis square root of 3 plus 1 right parenthesis to the power of 2 n end exponent minus left parenthesis square root of 3 minus 1 right parenthesis to the power of 2 n end exponent equals open square brackets left parenthesis square root of 3 plus 1 right parenthesis to the power of 2 end exponent close square brackets to the power of n end exponent minus open square brackets left parenthesis square root of 3 minus 1 right parenthesis to the power of 2 end exponent close square brackets to the power of n end exponent$
$equals left parenthesis 4 plus 2 square root of 3 right parenthesis to the power of n end exponent minus left parenthesis 4 minus 2 square root of 3 right parenthesis to the power of n end exponent$
$table row cell equals 2 to the power of n end exponent open square brackets left parenthesis 2 plus square root of 3 right parenthesis to the power of n end exponent minus left parenthesis 2 minus square root of 3 right parenthesis to the power of n end exponent close square brackets end cell row cell equals 2 to the power of n end exponent open curly brackets open square brackets blank to the power of n end exponent C subscript 0 end subscript 2 to the power of n end exponent plus blank to the power of n end exponent C subscript 1 end subscript 2 to the power of n minus 1 end exponent square root of 3 plus blank to the power of n end exponent C subscript 2 end subscript 2 to the power of n minus 2 end exponent 3 plus midline horizontal ellipsis close square brackets minus open square brackets blank to the power of n end exponent C subscript 0 end subscript 2 to the power of n end exponent close close end cell row cell open open negative blank to the power of n end exponent C subscript 1 end subscript 2 to the power of n minus 1 end exponent square root of 3 plus blank to the power of n end exponent C subscript 2 end subscript 2 to the power of n minus 2 end exponent 3 minus midline horizontal ellipsis close square brackets close curly brackets end cell row cell equals 2 to the power of n plus 1 end exponent open square brackets blank to the power of n end exponent C subscript 1 end subscript 2 to the power of n minus 1 end exponent square root of 3 plus blank to the power of n end exponent C subscript 3 end subscript 2 to the power of n minus 3 end exponent 3 square root of 3 plus midline horizontal ellipsis close square brackets equals 2 to the power of n plus 1 end exponent square root of 3 cross times left parenthesis text end text text s end text text o end text text m end text text e end text text end text text i end text text n end text text t end text text e end text text g end text text e end text text r end text text end text right parenthesis end cell end table$
$open parentheses fraction numerator x plus 1 over denominator x to the power of 2 divided by 3 end exponent minus x to the power of 1 divided by 3 end exponent plus 1 end fraction minus fraction numerator x minus 1 over denominator x minus x to the power of 1 divided by 2 end exponent end fraction close parentheses to the power of 10 end exponent$
$table row cell equals open parentheses fraction numerator open parentheses x to the power of 1 divided by 3 end exponent plus 1 close parentheses open parentheses x to the power of 2 divided by 3 end exponent minus x to the power of 1 divided by 3 end exponent plus 1 close parentheses over denominator open parentheses x to the power of 2 divided by 3 end exponent minus x to the power of 1 divided by 3 end exponent plus 1 close parentheses end fraction minus fraction numerator 1 over denominator square root of x end fraction times fraction numerator left parenthesis square root of x plus 1 right parenthesis left parenthesis square root of x minus 1 end root right parenthesis over denominator left parenthesis square root of x minus 1 right parenthesis end fraction close parentheses to the power of 2 end exponent end cell row cell equals open parentheses x to the power of 1 divided by 3 end exponent plus 1 equals fraction numerator x to the power of 1 divided by 2 end exponent plus 1 over denominator x to the power of 1 divided by 2 end exponent end fraction close parentheses equals open parentheses x to the power of 1 divided by 3 end exponent plus 1 equals 1 minus x to the power of negative 1 divided by 2 end exponent close parentheses equals open parentheses x to the power of 1 divided by 3 end exponent minus x to the power of negative 1 divided by 2 end exponent close parentheses to the power of 10 end exponent end cell end table$
Therefore,
$T subscript r plus 1 end subscript equals blank to the power of 10 end exponent C subscript r end subscript open parentheses x to the power of 1 divided by 3 end exponent close parentheses to the power of 10 minus r end exponent open parentheses negative x to the power of negative 1 divided by 2 end exponent close parentheses to the power of r end exponent rightwards double arrow left parenthesis negative 1 right parenthesis to the power of r blank to the power of 10 end exponent end exponent C subscript r end subscript x to the power of left parenthesis 20 minus 5 r right parenthesis divided by 6 end exponent$
$text For end text T subscript t plus 1 end subscript text to be independent of end text x text , end text$
20 - = 5 0 r r Þ = 4
$text Therefore, end text T subscript 5 end subscript equals T subscript 4 plus 1 end subscript equals left parenthesis negative 1 right parenthesis to the power of 410 end exponent C subscript 4 end subscript equals 210$

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