If n is a positive integer- then -3-12n-3-12n is -

(√(3)+1)^2n = ^2nC_0(√(3))^2n+ ^2nC_1(√(3))^2n 1+ ^2nC_2(√(3))^2n 2+⋯ Also we have, (√(3) 1)^2n = ^2nC_0(√(3))^2n ^2nC_1(√(3))^2n 1+ ^2nC_2(√(3))^2n 2 ...

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Standard XII

Mathematics

Sum of Binomial Coefficients of Odd Numbered Terms

If n is a pos...

Question

If $n$ is a positive integer, then $(\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n}$ is :

an irrational number

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an odd positive integer

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an even positive integer

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a rational number other than positive integer

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Solution

The correct option is A an irrational number
$(\sqrt{3} + 1)^{2 n} =^{2 n} C_{0} (\sqrt{3})^{2 n} +^{2 n} C_{1} (\sqrt{3})^{2 n - 1} +^{2 n} C_{2} (\sqrt{3})^{2 n - 2} + \dots$

Also we have,
$(\sqrt{3} - 1)^{2 n} =^{2 n} C_{0} (\sqrt{3})^{2 n} -^{2 n} C_{1} (\sqrt{3})^{2 n - 1} +^{2 n} C_{2} (\sqrt{3})^{2 n - 2} + \dots$

$∴ (\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n} = 2 [^{2 n} C_{1} (\sqrt{3})^{2 n - 1} +^{2 n} C_{3} (\sqrt{3})^{2 n - 3} + \dots]$ which is an irrational number

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Similar questions

Q. If

n

is a positive integer, then

(\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n}

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If n is a positive integer, then (√3+1)2n−(√3−1)2n is :

The correct option is A an irrational number(√3+1)2n= 2nC0(√3)2n+ 2nC1(√3)2n−1+ 2nC2(√3)2n−2+⋯ Also we have, (√3−1)2n= 2nC0(√3)2n− 2nC1(√3)2n−1+ 2nC2(√3)2n−2+⋯ ∴(√3+1)2n−(√3−1)2n=2[ 2nC1(√3)2n−1+ 2nC3(√3)2n−3+⋯] which is an irrational number

If $n$ is a positive integer, then $(\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n}$ is :