Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to

  • [JEE MAIN 2022]
  • A

    $3$

  • B

    $4$

  • C

    $5$

  • D

    $6$

Similar Questions

Let $S=\{a+b \sqrt{2}: a, b \in Z \}, T_1=\left\{(-1+\sqrt{2})^n: n \in N \right\}$ and $T_2=\left\{(1+\sqrt{2})^n: n \in N \right\}$. Then which of the following statements is (are) $TRUE$?

$(A)$ $Z \cup T_1 \cup T_2 \subset S$

$(B)$ $T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set

$(C)$ $T_2 \cap(2024, \infty) \neq \phi$

$(D)$ For any given $a, b \in Z , \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$, where $i=\sqrt{-1}$

  • [IIT 2024]

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Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to $............$.

  • [JEE MAIN 2023]

If the coefficients of ${r^{th}}$ term and ${(r + 4)^{th}}$ term are equal in the expansion of ${(1 + x)^{20}}$, then the value of r will be