The sum of all rational terms in the expansio | vedclass

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

Question

(B) The general term in the expansion of $(1+2^{1/3}+3^{1/2})^6$ is given by $\frac{6!}{r_1! r_2! r_3!} (1)^{r_1} (2^{1/3})^{r_2} (3^{1/2})^{r_3}$,whe

Vedclass · Accepted Answer

$612$

Vedclass · Answer

(B) The general term in the expansion of $(1+2^{1/3}+3^{1/2})^6$ is given by $\frac{6!}{r_1! r_2! r_3!} (1)^{r_1} (2^{1/3})^{r_2} (3^{1/2})^{r_3}$,where $r_1+r_2+r_3=6$.For the term to be rational,$r_2$ must be a multiple of $3$ and $r_3$ must be a multiple of $2$.Possible values for $(r_1, r_2, r_3)$ are:$1$. $(6, 0, 0) \implies \frac{6!}{6!0!0!} (1)^6 (2)^0 (3)^0 = 1$$2$. $(4, 0, 2) \implies \frac{6!}{4!0!2!} (1)^4 (2)^0 (3)^1 = 15 	imes 3 = 45$$3$. $(2, 0, 4) \implies \frac{6!}{2!0!4!} (1)^2 (2)^0 (3)^2 = 15 	imes 9 = 135$$4$. $(0, 0, 6) \implies \frac{6!}{0!0!6!} (1)^0 (2)^0 (3)^3 = 1 	imes 27 = 27$$5$. $(3, 3, 0) \implies \frac{6!}{3!3!0!} (1)^3 (2)^1 (3)^0 = 20 	imes 2 = 40$$6$. $(1, 3, 2) \implies \frac{6!}{1!3!2!} (1)^1 (2)^1 (3)^1 = 60 	imes 6 = 360$$7$. $(0, 6, 0) \implies \frac{6!}{0!6!0!} (1)^0 (2)^2 (3)^0 = 1 	imes 4 = 4$Sum $= 1 + 45 + 135 + 27 + 40 + 360 + 4 = 612$.

The sum of all rational terms in the expansion of $(1+2^{1/3}+3^{1/2})^6$ is equal to . . . . . . .

Similar Questions

If ${a_1}, {a_2}, {a_3}, {a_4}$ are the coefficients of any four consecutive terms in the expansion of ${(1 + x)^n}$,then $\frac{{{a_1}}}{{{a_1} + {a_2}}} + \frac{{{a_3}}}{{{a_3} + {a_4}}}$ =

If $a, b, c, d$ are any four consecutive coefficients of any expanded binomial,then $\frac{a + b}{a}, \frac{b + c}{b}, \frac{c + d}{c}$ are in

If the coefficients of the second,third,and fourth terms in the expansion of $(1 + x)^n$ are in $A.P.$,then $n$ is equal to

The coefficient of $xy^2z^3$ in the expansion of $(x-2y+3z)^6$ is

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+2}$,which are in the ratio $1:3:5$,is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module