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JEE Main202522 Jan 2025Morning ShiftMathematicsBinomial TheoremActual

If _ r=0 ^5 11 2r+1 2 r+2 = m n , gcd ( m , n )=1 , then m - n is equal to _______

Correct answer

0

Step-by-step solution

aligned & (1+x)¹¹= ¹¹ C₀+ ¹¹ C₁ x+ ¹¹ C₂ x^2+ + 11 11 x¹¹ & ₀^1(1+x)¹¹ d x= ₀^1 ( ¹¹ C₀+ ¹¹ C₁ x+ ¹¹ C₂ x^2+ + 11 11 x x¹¹ ) d x & . . (1-x)¹² 12 |₀ ^1= 11 0 x + ¹¹ C₁ x^2 2 + ¹¹ C₂ x 3 + + 11 11 x¹² 12 ]₀^1 & 2¹²-1 12 =C₀+ C₁ 2 + C₂ 3 + + C₁₁ 12 (1) aligned Now, aligned & _ -1 ^0(1+x)¹¹ d x= _ -1 ^0 ( ¹¹ C₀+ ¹¹ C₁ x+ ¹¹ C₂ x^2+ + 11 11 x¹¹ ) d x & . . (1+x)¹² 12 ]_ -1 ^0= ¹¹ C₀ x+ ¹¹ C₁ x^2 2 + ¹¹ C₂ x^3 3 + + 11 11 x¹² 12 ]_ -1 ^0 aligned 1 12 =C₀- C₁ 2 + C₂ 3 aligned & (1)-(2) & = 2¹²-2 12 =2 [ C₁ 2 + C₃ 4 + ] &

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