Find the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}$
Find the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}$
We have ${T_{r + 1}} = {\,^6}{C_r}{\left( {\frac{3}{2}{x^2}} \right)^{6 - r}}\left( { - \frac{1}{{3x}}} \right)$
$ = {\,^6}{C_r}{\left( {\frac{3}{2}} \right)^{6 - r}}{\left( {{x^2}} \right)^{6 - r}}{( - 1)^r}{\left( {\frac{1}{x}} \right)^r}\left( {\frac{1}{{{3^r}}}} \right)$
$ = {( - 1)^r}{\quad ^6}{C_r}\quad \frac{{{{(3)}^{6 - 2r}}}}{{{{(2)}^{6 - r}}}}\quad {x^{12 - 3r}}$
The term will be independent of $x$ if the index of $x$ is zero, i.e., $12-3 r=0 .$ Thus, $r=4$
Hence $5^{\text {th }}$ term is independent of $x$ and is given by ${( - 1)^4}{\,^6}{C_4}\frac{{{{(3)}^{6 - 8}}}}{{{{(2)}^{6 - 4}}}} = \frac{5}{{12}}$
Similar Questions
Let $S=\{a+b \sqrt{2}: a, b \in Z \}, T_1=\left\{(-1+\sqrt{2})^n: n \in N \right\}$ and $T_2=\left\{(1+\sqrt{2})^n: n \in N \right\}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $Z \cup T_1 \cup T_2 \subset S$
$(B)$ $T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set
$(C)$ $T_2 \cap(2024, \infty) \neq \phi$
$(D)$ For any given $a, b \in Z , \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$, where $i=\sqrt{-1}$
Let $S=\{a+b \sqrt{2}: a, b \in Z \}, T_1=\left\{(-1+\sqrt{2})^n: n \in N \right\}$ and $T_2=\left\{(1+\sqrt{2})^n: n \in N \right\}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $Z \cup T_1 \cup T_2 \subset S$
$(B)$ $T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set
$(C)$ $T_2 \cap(2024, \infty) \neq \phi$
$(D)$ For any given $a, b \in Z , \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$, where $i=\sqrt{-1}$
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If the co-efficient of $x^9$ in $\left(\alpha x^3+\frac{1}{\beta x}\right)^{11}$ and the co-efficient of $x^{-9}$ in $\left(\alpha x-\frac{1}{\beta x^3}\right)^{11}$ are equal, then $(\alpha \beta)^2$ is equal to $.............$.
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