Question

Match the elements of List I with List II

List I	List II
A) lf $\lambda$ be the number of terms which are integers, in the expansion of $(5^{\frac{1}{6}}+7^{\frac{1}{9}}{)}^{1824}$, then $\lambda$ is divisible by	P) 2
B) lf $\lambda$ be the number of terms which are rational in the expansion of $(5^{\frac{1}{6}}+2^{\frac{1}{8}}{)}^{100}$, then $\lambda$ is divisible by	Q) 3
C) lf $\lambda$ be the number of terms which are irrational in the expansion of $(3^{\frac{1}{4}}+4^{\frac{1}{3}}{)}^{99}$, then $\lambda$ is divisible by	R) 7
	S) 13
	T) 17

The correct option which matches all the elements correctly, is :

• A. (A) - P,Q,T (B) - P (C) - R,S
• B. (A) - P,T (B) - Q (C) - S,T
• C. (A) - P,S,T (B) - S,Q (C) - Q
• D. (A) - P,S (B) - P,A (C) - P,R

Answer 1

The correct option is A (A) - P,Q,T (B) - P (C) - R,S

(i) $(5^{\frac{1}{6}}+7^{\frac{1}{9}}{)}^{1824}$

$T_{r+1}{=}^{1824}{C}_r(5{)}^{\frac{1824-r}{6}}{7}^{\frac{r}{9}}$

For integer terms, $r$ should be multiple of $9$.

For $r=18,36,54,72,.....1818$, terms comes as integer.

This is an A.P.

$1818=18+(n-1)18$

$\Rightarrow n=101$

Also, for $r=0$ , we would get an integer

So, total number of terms which gives integer values are $101+1=102.$

So, $\lambda =102$

So, $\lambda$ is divisible by $2,3,17$

(ii) $(5^{\frac{1}{6}}+2^{\frac{1}{8}}{)}^{1824}$

$T_{r+1}{=}^{100}{C}_r(5{)}^{\frac{100-r}{6}}{2}^{\frac{r}{8}}$

For rational terms, $r$ should be multiple of $8$.

For $r=16,40,64,88$, terms comes as rational.

So, number of rational terms are $4$.

So, $\lambda =4$

which is divisble by $2$.

(iii) $(3^{\frac{1}{4}}+4^{\frac{1}{3}}{)}^{99}$

$T_{r+1}{=}^{99}{C}_r(3{)}^{\frac{99-r}{64}}{4}^{\frac{r}{3}}$

For rational terms, $r$ should be multiple of $3$.

For $r=3,15,27,.....97$, terms comes as rational.

This is an AP

$97=3+(n-1)12$

$\Rightarrow n=8$

For $r=99$ also, there is a rational value

So, number of rational terms are $8+1=9$

Now, number of irrational terms = total number of terms -rational number of terms

$=99+1-9=91$

So, $\lambda =91$

which is divisble by $7,13$.

Hence, option A is the correct answer.