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JEE Main 20244 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2024 · 4 April · Morning Shift · Mathematics · Binomial Theorem

The sum of all rational terms in the expansion of (2^ (1)/(5) +5^...

The sum of all rational terms in the expansion of (215+513)15\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15} is equal to :

Options

  1. A3133
  2. B931
  3. C6131
  4. D633

Correct answer

A. 3133

Step-by-step solution

Tr+1=15Cr(513)r(125)15−r=15Cr5r3⋅215−r5R=3λ,15μ⇒r=0,15\begin{aligned} & \mathrm{T}_{\mathrm{r}+1}={ }^{15} \mathrm{C}_{\mathrm{r}}\left(5^{\frac{1}{3}}\right)^{\mathrm{r}}\left(\frac{1}{2^5}\right)^{15-\mathrm{r}} \\ & ={ }^{15} \mathrm{C}_{\mathrm{r}} 5^{\frac{\mathrm{r}}{3}} \cdot 2^{\frac{15-\mathrm{r}}{5}} \\ & \mathrm{R}=3 \lambda, 15 \mu \\ & \Rightarrow \mathrm{r}=0,15 \end{aligned} 2 rational terms ⇒15C023+15C15(5)5=8+3125=3133\begin{aligned} & \Rightarrow{ }^{15} \mathrm{C}_0 2^3+{ }^{15} \mathrm{C}_{15}(5)^5 \\ & =8+3125=3133 \end{aligned}
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