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NDA2024MathematicsBinomial TheoremActual

What is the coefficient of x¹⁰ in the expansion of (1-x^2 )²⁰ (2-x^2- 1 x^2 )⁻⁵ ?

Options

  1. A-1
  2. B1
  3. C10
  4. DCoefficient of x¹⁰ does not exist

Correct answer

A. -1

Step-by-step solution

(1-x^2 )²⁰ [- ( 1 x^2 +x^2-2 ) ]⁻⁵ aligned & =- (1-x^2 )²⁰ (x- 1 x )⁻¹⁰ & =- (1-x^2 )²⁰ (x^2-1 )⁻¹⁰ x¹⁰ & =- (1-x^2 )¹⁰ x¹⁰ & ~T _ r+1 =- ¹⁰ C _r(1)^ 10-r (-x^2 )^ 10-r x¹⁰ & =(-1)^ 11-r ¹⁰ C _r x^ 30-2 r & 30-2 r=10 & r=10 & Coefficient of x¹⁰=(-1)^1 ¹⁰ C ₁₀=-1 1-1 . aligned

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