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Algebra: Binomial Expansion Fraction

XAT 2011 · Quantitative Ability & Data Interpretation · Medium · Algebra

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If x=(9+45)48=[x]+fx = (9 + 4\sqrt{5})^{48} = [x] + f, where [x][x] is defined as integral part of xx and ff is a fraction, then x(1−f)x(1 - f) equals

  1. A.

    1

  2. B.

    Less than 1

  3. C.

    More than 1

  4. D.

    Between 1 and 2

  5. E.

    None of the above

Answer

A

Explanation

Let y=(9−45)48y = (9 - 4\sqrt{5})^{48}. Since 0<9−45<10 < 9 - 4\sqrt{5} < 1, 0<y<10 < y < 1. Thus y=1−fy = 1 - f. Then x(1−f)=xy=((9+45)(9−45))48=(81−80)48=1x(1 - f) = x y = ((9 + 4\sqrt{5})(9 - 4\sqrt{5}))^{48} = (81 - 80)^{48} = 1.

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