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Binomial Theorem — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsBinomial Theorem5 Marks
Question
Evaluate the following:
$\Big\{\text{a}^2+\sqrt{\text{a}^2-1}\Big\}^4+\Big\{\text{a}^2-\sqrt{\text{a}^2-1}\Big\}^4$
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Answer

$\Big\{\text{a}^2+\sqrt{\text{a}^2-1}\Big\}^4+\Big\{\text{a}^2-\sqrt{\text{a}^2-1}\Big\}^4$
$\text{Let}\ \text{a}^2=\text{A},\ \sqrt{\text{a}^2-1}=\text{B}$
$(\text{A}+\text{B})^4+(\text{A}-\text{B})^4$
$=\text{B}^4+{^4\text{C}}_1\text{AB}^3+{^4\text{C}}_2\text{A}^2\text{B}^2+{^4\text{C}}_3\text{A}^3\text{B}+\text{A}^2+\text{B}^4\\-{^4\text{C}}_1\text{AB}^3+{^4\text{C}}_2\text{A}^2\text{B}^2-{^4\text{C}}_3\text{A}^3\text{B}+\text{A}^4$
$=2\big(\text{A}^4+{^4\text{C}}_2\text{A}^2\text{B}^2+\text{B}^4\big)$
$=2\big(\text{A}^2+6\text{A}^2\text{B}^2+\text{B}^4\big)$
$=2\Big(\text{a}^8+6\text{a}^4(\text{a}^2-1)+(\text{a}^2-1)^2\Big)$
$=2\Big[\text{a}^8+6\text{a}^6-6\text{a}^4+\text{a}^4+1-2\text{a}^2\Big]$
$\Big\{\text{a}^2+\sqrt{\text{a}^2-1}\Big\}+\Big\{\text{a}^2-\sqrt{\text{a}^2-1}\Big\}$
$=2\text{a}^8+12\text{a}^6-10\text{a}^4-4\text{a}^4+2$

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