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Question

Value of $$(\sqrt{2} + 1)^6 + (\sqrt{2} -1)^6$$ is


A
90
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B
70
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C
142
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D
198
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Solution

The correct option is D 198
$$1^2 + ^6C_3 . x^3 . 1^3 + ^6C_4 . x^2$$
$$1^4 + ^6C_5. x.1^5 +^6C_6.x^0. 1^6$$
$$=x^6 + 6x^5 +15^4+ 20x^3 + 15^2 + 6x+1$$    ....... (i)
$$(x-1)^6 = x^6 + ^6C_1 . x^5 . (-1) + ^6C_2 x^4 . (-1)^2 +^6C_3 . x^3 . (-1)^3 + ^6C_4 . x^2 . (-1)^4 + ^6C_5 .x. (-1)^5 + ^6C_6.x.(-1)^6$$
$$=x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 -6x+1$$  .... (ii)
Adding (i) and (ii)
$$(x+1)^6 + (x-1)^6 = 2 [x^6 + 15x^4 + 15x^4 + 15x^2+1]$$
Putting x $$= \sqrt{2}$$
$$(\sqrt{2} + 1)^6 + (\sqrt{2 -1})^6 = 2 [(\sqrt{2})^6 + 15 (\sqrt{2})^4 + 15(\sqrt{2})^2 +1]$$
$$= 2 [8 +60 + 30 + 1] = 2 \times 99 = 198$$

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