The coefficient of x−5 in the binomial expansion of 

\small \left ( \frac{x + 1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1} -\frac{x-1}{x-x^{\frac{1}{2}}}\right )^{10},

where \small x\neq 0,1, is :

  • Option 1)

    1

  • Option 2)

    4

  • Option 3)

    -4

  • Option 4)

    1

 

Answers (1)

As we learnt in

General Term in the expansion of (x+a)^n -

T_{r+1}= ^{n}c_{r}\cdot x^{n-r}\cdot a^{r}
 

- wherein

Where r\geqslant 0 \, and \, r\leqslant n

r= 0,1,2,----n

 

 We will use general term in the expansion of (x+a)^{n} offer simplyfing question. 

\left(\frac{(x^{1/3})^{3}+1^{3}}{x_{3}^{3}-x^{1/3}+1}-\frac{(x^{1/2})^{2}-1}{x^{1/2}(x^{1/2}-1)} \right )^{10}

= \left(\frac{(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)}{(x^{2/3}-x^{1/3}+1)}-\frac{(x^{1/2}-1)(x^{1/2}+1)}{x^{1/2}(x^{1/2}-1)} \right )^{10}

= \left(x^{1/3}+1-(1+x^{-1/2}) \right )^{10}

= \left(x^{1/3}-x^{-1/2} \right )^{10}

General term of this expansion =\ ^{10}C_{r}(x^{1/3})^{10-r}(x^{-1/2})^{r}

=\ ^{10}C_{r}x^{\frac{10}{3}\frac{-r}{3}\frac{-r}{3}}

=\ ^{10}C_{r}x^{\frac{20-5r}{6}}

Coefficient of x-5 means \frac{20-5r}{6}=-6\ \Rightarrow\ \; r = 10

Coefficient  =\ ^{10}C_{10}

Correct option is 1.

 


Option 1)

1

This is the correct option.

Option 2)

4

This is an incorrect option.

Option 3)

-4

This is an incorrect option.

Option 4)

1

This is an incorrect option.

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