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Find the value of (a2+a2-1)4+(a2-a2-1)4 - Mathematics

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Question

Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`

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Solution

Firstly, the expression (x + y)4 + (x – y)4 is simplified by using Binomial Theorem.

(x + y)4 = 4C0x4 + 4C1x3y + 4C2x2y2 + 4C3xy3 + 4C4y4

= x4 + 4x3y + 6x2y2 + 4xy3 + y4

(x - y)4 = 4C0x4 - 4C1x3y + 4C2 x2y2 - 4C3xy3 + 4C4y4

= x4 - 4x3y + 6x2y2 - 4xy3 + y4

∴ (x + y)4 + (x - y)4 = 2(x4 + 6x2 y2 + y4)

Putting x = a2 and y = `sqrt(a^2 - 1)`, we obtain

`a^2  + sqrt((a^2 - 1)^4)  +  a^2  - sqrt((a^2 - 1)^4)  = 2[(a^2)^4   + 6 (a^2)^2  sqrt((a^2 - 1)^2)  + sqrt(a^2  - 1)^4]`

= `2[a^8  + 6a^4  (a^2 - 1)  + (a^2  - 1)^2]`

= `2[a^8  +  6a^6  - 6a^4  + a^4  - 2a^2  + 1]`

= `2[a^8  + 6a^6 - 5a^4  - 2a^2 + 1]`

= `2a^8  + 12a^6 - 10a^4 - 4a^2  + 2`

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Chapter 8: Binomial Theorem - Miscellaneous Exercise [Page 175]

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NCERT Mathematics [English] Class 11
Chapter 8 Binomial Theorem
Miscellaneous Exercise | Q 6 | Page 175

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