If sum of coefficients of all positive even powers of x in binomial expansion of left(2 x^{3}+frac{3

English

Gujarati

Hindi

7.Binomial Theorem

hard

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

A

$36$

B

$75$

C

$89$

D

$83$

(JEE MAIN-2022)

Solution

$T_{r+1}={ }^{10} C_{r}\left(2 x^{3}\right)^{10-r}\left(\frac{3}{x}\right)^{r}$

$={ }^{10} C_{r} 2^{10-r} 3^{r} x^{30-4 r}$

Put $r=0,1,2, \ldots 7$ and we get $\beta=83$

Standard 11

Mathematics

Share

Similar Questions

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + …. + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n – 2}}{C_n}$ equals

medium

A possible value of $^{\prime}x^{\prime}$, for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$ , is:

hard

(JEE MAIN-2021)

The coefficient of $x^8$ in the expansion of $(x-1) (x- 2) (x-3)……………(x-10)$ is :

normal

The coefficient of $x^r (0 \le r \le n – 1)$ in the expression :

$(x + 2)^{n-1} + (x + 2)^{n-2}. (x + 1) + (x + 2)^{n-3} . (x + 1)^2; + …… + (x + 1)^{n-1}$ is :

normal

If $x + y = 1$, then $\sum\limits_{r = 0}^n {{r^2}{\,^n}{C_r}{x^r}{y^{n – r}}} $ equals

hard