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MHT CET202525 Apr 2025Evening ShiftMathematicsTrigonometric Ratios & IdentitiesActual

^4 8 + ^4 3 8 + ^4 5 8 + ^4 7 8 =

Options

  1. A1 2
  2. B3 2
  3. C1 4
  4. D3 4

Correct answer

B. 3 2

Step-by-step solution

Let S = ^4 8 + ^4 3 8 + ^4 5 8 + ^4 7 8 . Observe that 5 8 = - 3 8 and 7 8 = - 8 , but the fourth powers eliminate the negative signs: ^4 5 8 = ^4 3 8 and ^4 7 8 = ^4 8 . Thus, S = 2 ( ^4 8 + ^4 3 8 ) . Since 3 8 = 8 , we have S = 2 ( ^4 8 + ^4 8 ) . Using the identity a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 with a = 8 , b = 8 , and noting that ^2 x + ^2 x = 1 : S = 2 (1 - 2 ^2 8 ^2 8 ) . From the double-angle identity, 2 ^2 x ^2 x = 1 2 ^2 2x , so: S = 2 (1 - 1 2 ^2 4 ) . Since 4 = 1 2 , we have ^2 4 = 1 2 , and thus:

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