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(D) The general term is $T_{r+1} = {}^{10}C_r (2x^3)^{10-r} (3x^{-1})^r = {}^{10}C_r 2^{10-r} 3^r x^{30-4r}$.For even powers of $x$,$30-4r$ must be an

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$83$

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(D) The general term is $T_{r+1} = {}^{10}C_r (2x^3)^{10-r} (3x^{-1})^r = {}^{10}C_r 2^{10-r} 3^r x^{30-4r}$.For even powers of $x$,$30-4r$ must be an even integer,which is true for all $r \in \{0, 1, \dots, 10\}$.However,we require positive even powers,so $30-4r > 0 \implies 4r Let $f(x) = (2x^3 + \frac{3}{x})^{10} = \sum_{r=0}^{10} {}^{10}C_r 2^{10-r} 3^r x^{30-4r}$.Let $S_e$ be the sum of coefficients of even powers and $S_o$ be the sum of coefficients of odd powers.$f(1) = S_e + S_o = (2+3)^{10} = 5^{10}$.$f(-1) = S_e - S_o = (-2-3)^{10} = 5^{10}$.Thus $S_e = \frac{f(1) + f(-1)}{2} = 5^{10}$.This $S_e$ includes the constant term ($r=7.5$ is not possible,but $r=7$ gives $x^2$,$r=6$ gives $x^6$,$r=5$ gives $x^{10}$,$r=4$ gives $x^{14}$,$r=3$ gives $x^{18}$,$r=2$ gives $x^{22}$,$r=1$ gives $x^{26}$,$r=0$ gives $x^{30}$). The constant term occurs when $30-4r=0$,which is not possible for integer $r$.Sum of coefficients of positive even powers is $S_e = \sum_{r=0}^7 {}^{10}C_r 2^{10-r} 3^r = 5^{10} - \sum_{r=8}^{10} {}^{10}C_r 2^{10-r} 3^r$.For $r=8, 9, 10$,the powers are $x^{-2}, x^{-6}, x^{-10}$.Sum $= {}^{10}C_8 2^2 3^8 + {}^{10}C_9 2^1 3^9 + {}^{10}C_{10} 2^0 3^{10} = 45 \cdot 4 \cdot 3^8 + 10 \cdot 2 \cdot 3^9 + 3^{10} = 180 \cdot 3^8 + 20 \cdot 3^9 + 3^{10} = 60 \cdot 3^9 + 20 \cdot 3^9 + 3 \cdot 3^9 = 83 \cdot 3^9$.Thus $\beta = 83$.

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $(2x^3 + \frac{3}{x})^{10}$ is $5^{10} - \beta \cdot 3^9$,then $\beta$ is equal to

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