JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Binomial Theorem

More JEE Advanced questions on Binomial Theorem

1. Let $\mathbb{R}$ denote the set of all real numbers. Let $a_{\mathrm{i}},b_{\mathrm{i}}\in \mathbb{R}$ for $\mathrm{i}\in \{1,2,3\}$.
   Define the functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$, and …
   JEE Advanced 2025 Copy linkView solution →
2. Let $a_0,a_1,\dots .,a_{23}$ be real numbers such that
   

   $${\left(1+\frac{2}{5}x\right)}^{23}=\sum _{i=0}^{23}{a}_i{x}^i$$

   
   for every real number $x$. let $a_r$ be the largest among the numbers $a_j$ for $0\le j\le 23$.
   The the value of $r$ is ____.
   JEE Advanced 2025 Copy linkView solution →
3. For non-negative integers $s$ and $r$, let
   

   $$\left(\genfrac{}{}{0ex}{}{s}{r}\right)=\{\begin{array}{ll}\frac{s!}{r!(s-r)!}& \text{ if }r\le s,\\ 0& \text{ if }r>s.\end{array}$$

   
   For positive integers $m$ and $n$, let
   

   $$g(m,n)=\sum _{p=0}^{m+n}\frac{f(m,n,p)}{\left(\genfrac{}{}{0ex}{}{n+p}{p}\right)}$$

   
   where for any non …
   JEE Advanced 2020 Copy linkView solution →
4. Suppose $\det \left[\begin{array}{cc}{\sum }_{k=0}^{n}k& {\sum }_{k=0}^n{}^n{C}_k{k}^2\\ {\sum }_{k=0}^n{}^n{C}_{k}k& {\sum }_{k=0}^n{}^n{C}_k{3}^k\end{array}\right]=0$, holds for some positive integer $n$. Then …
   JEE Advanced 2019 Copy linkView solution →
5. Let
   

   $$X={\left({}^{10}{C}_1\right)}^2+2{\left({}^{10}{C}_2\right)}^2+3{\left({}^{10}{C}_3\right)}^2+\cdots +10{\left({}^{10}{C}_{10}\right)}^2,$$

   
   where ${}^{10}{C}_r,r\in \{1,2,\cdots ,10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is ___ …
   JEE Advanced 2018 Copy linkView solution →
6. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x{)}^2+(1+x{)}^3+\cdots +(1+x{)}^{49}+(1+mx{)}^{50}$ is $(3n+1{)}^{51}{C}_3$ for some positive integer $n$. Then the value of $n$ is
   JEE Advanced 2016 Copy linkView solution →