Q87 / 160 · WBJEE 2011 · Mathematics

If \(\mathrm{n} > 1\) is an integer and \(\mathrm{x} \neq 0\), then \((1+\mathrm{x})^{\mathrm{n}}-\mathrm{nx}-1\) is divisible by

  • A
    \(\mathrm{nx}^3\)
  • B
    \(\mathrm{n}^3 \mathrm{x}\)
  • Correct answer
    C
    \(\mathrm{x}\)
  • D
    \(\mathrm{nx}\)

Solution

\(\begin{array}{r}\text {Hints: } (1+\mathrm{x})^n={ }^n C_0+{ }^n C_1 x+{ }^n C_2 x^2+{ }^n C_3 \mathrm{x} 3+\ldots \ldots . \\ =1+\mathrm{nx}+\mathrm{x}^2\left({ }^n C_2+{ }^n C_3 \mathrm{x}+\ldots \ldots . .\right) \\ (1+\mathrm{x})^{\mathrm{n}}-\mathrm{nx}-1=\mathrm{x}^2\left({ }^n C_2+{ }^n C_3 \mathrm{x}+\ldots \ldots .\right) \end{array}\)

Hence, the correct answer is option: 'C'.

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