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Find the coefficient of (i) x^{7} in \left(a x^{2}+\frac{1}{b x}\right)^{11}, (ii) and x^{-7}in \left(a x-\frac{1}{b x^{2}}\right)^{11}. Find the relation between a and b if these coefficients are equal.

 

 

Option: 1

ab=2


Option: 2

a^{2}=b


Option: 3

ab=1


Option: 4

a=b^2


Answers (1)

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The general term in \left(a x^{2}+\frac{1}{b x}\right)^{11}={ }^{11} C_{r}\left(a x^{2}\right)^{11-r}\left(\frac{1}{b x}\right)^{r}

={ }^{11} C_{r} \frac{a^{11-r}}{b^{r}} x^{22-3 r}

If in this term power of x is 7 , then 22-3 r=7 \Rightarrow r=5

\therefore coefficient of x^7={ }^{11} \mathrm{C}_{5} \frac{\mathrm{a}^{6}}{\mathrm{~b}^{5}}

The general term in \left(a x-\frac{1}{b x^{2}}\right)^{11}=(-1)^{r 11} C_{r}(a x)^{11-r}\left(\frac{1}{b x^{2}}\right)^{r}=(-1)^{r 11} C_{r} \frac{a^{11-r}}{b^{r}} x^{11-3 r}

If in this term power of x is -7 , then 11-3r=-7 \Rightarrow r=6

\therefore coefficient of x^{-7}=(-1)^{6}{ }^{11} C_{6} \frac{a^{11-6}}{b^{6}}={ }^{11} C_{5} \frac{a^{5}}{b^{6}}

If these two coefficient are equal, then { }^{11} \mathrm{C}_{5} \frac{\mathrm{a}^{6}}{\mathrm{~b}^{5}}={ }^{11} \mathrm{C}_{5} \frac{\mathrm{a}^{5}}{\mathrm{~b}^{6}}

\Rightarrow a^{6} b^{6}=a^{5} b^{5} \Rightarrow a^{5} b^{5}(a b-1)=0 \Rightarrow a b=1(a \neq 0, b \neq 0)

 

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Gunjita

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