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JEE Main201815 Apr 2018Morning ShiftMathematicsBinomial TheoremActual

If n is the degree of the polynomial, [ 1 5 x^3+1 - 5 x^3-1 ]^8+ [ 1 5 x^3+1 + 5 x^3-1 ]^8 and m is the coefficient of x ^ n in it, then the ordered pair ( n , m ) is equal to

Options

  1. A(12,(20)^4 )
  2. B(8,5(10)^4 )
  3. C(24,(10)^8 )
  4. D(12,8(10)^4 )

Correct answer

D. (12,8(10)^4 )

Step-by-step solution

[ 1 5 x^3+1 - 5 x^3-1 ]^8+ [ 1 5 x^3+1 + 5 x^3-1 ]^8 After rationalise the polynomial we get aligned & [ 1 5 x^3+1 - 5 x^3-1 5 x^3+1 + 5 x^3-1 5 x^3+1 + 5 x^3-1 ]^8 &+ [ 1 5 x^3+1 + 5 x^3-1 5 x^3+1 - 5 x^3-1 5 x^3+1 - 5 x^3-1 ]^8 aligned [ 5 x^3+1 + 5 x^3-1 (5 x^3+1 )- (5 x^3-1 ) ]^8+ [ 5 x^3+1 - 5 x^3-1 (5 x^3+1 )- (5 x^3-1 ) ]^8 aligned &= 1 2^8 [ ( 5 x^3+1 + 5 x^3-1 )^8+ ( 5 x^3+1 - 5 x^3-1 )^8 ] & .= 1 2^8 [ array l . ^8 C₀ ( 5 x^3+1 )^8+ ^8 C₄ ( 5 x^3+1 )^6 ( 5 x^3-1 )^2 )^4 ( 5 x^3-1 )^4+ ^8 C₆ ( 5 x ^ 3 + 1

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