The number of rational terms in the expansion | vedclass

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

Question

(D) The general term $T_{r+1}$ in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is given by: $T_{r+1} = {}^{144}C_r (3^{1/4})^{144-r} (7^{1/6})^r$ $T_{

Vedclass · Accepted Answer

$13$

Vedclass · Answer

(D) The general term $T_{r+1}$ in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is given by: $T_{r+1} = {}^{144}C_r (3^{1/4})^{144-r} (7^{1/6})^r$ $T_{r+1} = {}^{144}C_r (3)^{\frac{144-r}{4}} (7)^{\frac{r}{6}}$ For the term to be rational,the exponents of $3$ and $7$ must be integers. Thus,$\frac{144-r}{4} = 36 - \frac{r}{4}$ must be an integer,which implies $r$ must be a multiple of $4$. Also,$\frac{r}{6}$ must be an integer,which implies $r$ must be a multiple of $6$. Therefore,$r$ must be a multiple of $	ext{lcm}(4, 6) = 12$. Since $0 \le r \le 144$,the possible values for $r$ are $0, 12, 24, \dots, 144$. This is an arithmetic progression where $a = 0$,$d = 12$,and the last term $l = 144$. Using the formula $l = a + (n-1)d$: $144 = 0 + (n-1)12$ $12 = n - 1$ $n = 13$. Thus,there are $13$ rational terms.

The number of rational terms in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is

Similar Questions

In a transistor circuit,when the base current is increased by $50 \mu A$ keeping the collector voltage fixed at $2 \ V$,the collector current increases by $1 \ mA$. The current gain of the transistor is

For the reaction $C_{(s)} + CO_{2_{(g)}} \rightleftharpoons 2CO_{(g)}$,the partial pressures of $CO_2$ and $CO$ are $2.0 \ atm$ and $4.0 \ atm$ respectively at equilibrium. The $K_p$ for the reaction is

What are $X$ and $Y$ in the following reaction?
$CF_2Cl_2 \xrightarrow{UV} X + Y$

If the amplitude of $z-2-3i$ is $\frac{\pi}{4}$,then the locus of $z=x+iy$ is:

If $0 < y < 2^{1/3}$ and $x(y^3 - 1) = 1$,then $\frac{2}{x} + \frac{2}{3x^3} + \frac{2}{5x^5} + \dots = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module