If the fourth term in the expansion of (x+x^{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The general term $T_{r+1}$ in the expansion of $(a+b)^n$ is given by $T_{r+1} = {}^{n}C_{r} a^{n-r} b^{r}$.For the fourth term $(T_4)$,we set $r=3

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(A) The general term $T_{r+1}$ in the expansion of $(a+b)^n$ is given by $T_{r+1} = {}^{n}C_{r} a^{n-r} b^{r}$.For the fourth term $(T_4)$,we set $r=3$:$T_4 = {}^{7}C_{3} (x)^{7-3} (x^{\log_{2} x})^{3} = 4480$.Since ${}^{7}C_{3} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = 35$,we have:$35 \cdot x^{4} \cdot x^{3 \log_{2} x} = 4480$.Dividing by $35$:$x^{4 + 3 \log_{2} x} = \frac{4480}{35} = 128 = 2^{7}$.Let $t = \log_{2} x$,then $x = 2^t$. Substituting this into the equation:$(2^t)^{4 + 3t} = 2^{7} \Rightarrow t(4 + 3t) = 7$.$3t^{2} + 4t - 7 = 0$.$(3t + 7)(t - 1) = 0$.Thus,$t = 1$ or $t = -7/3$.If $t = 1$,then $\log_{2} x = 1 \Rightarrow x = 2^1 = 2$.Since $x \in N$,the value is $2$.

If the fourth term in the expansion of $(x+x^{\log _{2} x})^{7}$ is $4480$,then the value of $x$ where $x \in N$ is equal to

Similar Questions

The sum of the coefficients of the first $50$ terms in the binomial expansion of $(1-x)^{100}$ is equal to

If the sum of the coefficients in the expansion of $(x - 2y + 3z)^n$ is $128$,then the greatest coefficient in the expansion of $(1 + x)^n$ is

The numerically greatest term in the binomial expansion of $(2x - 3y)^5$ when $x = \frac{3}{2}$ and $y = \frac{2}{3}$ is

The constant term in the expansion of $\left(1+\frac{1}{x}\right)^{20}\left(30 x(1+x)^{29}+(1+x)^{30}\right)$ is

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module