Binomial Theorem — JEE Advanced PYQs

11 previous year questions on Binomial Theorem (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $R$ denote the set of all real numbers. Let $a_{i}, b_{i} \in R$ for $i \in {1, 2, 3}$ .
Define the functions $f : R \to R, g : R \to R$ , and …
JEE Advanced 2025View solution →
Let $a_{0}, a_{1}, \dots ., a_{23}$ be real numbers such that
$(1 + \frac{2}{5} x)^{23} = i = 0 \sum 23 a_{i} x^{i}$

for every real number $x$ . let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ .
The the value of $r$ is ____.
JEE Advanced 2025View solution →
Let a and b be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of $(ax^{2} + \frac{70}{27 bx})^{4}$ is equal to the coefficient of $x^{- 5}$ is equal to the coefficient of $(a x - \frac{1}{b x ^{2}})^{7}$ , then t …
JEE Advanced 2023View solution →
For non-negative integers $s$ and $r$ , let
$(r s) = {\frac{s !}{r ! ( s - r )!} 0 if r \leq s, if r > s .$

For positive integers $m$ and $n$ , let
$g (m, n) = p = 0 \sum m + n \frac{f ( m , n , p )}{( p n + p )}$

where for any non …
JEE Advanced 2020View solution →
Suppose $det [\sum_{k = 0}^{n} k \sum_{k = 0}^{n}^{n} C_{k} k \sum_{k = 0}^{n}^{n} C_{k} k^{2} \sum_{k = 0}^{n}^{n} C_{k} 3^{k}] = 0$ , holds for some positive integer $n$ . Then …
JEE Advanced 2019View solution →
Let
$X = (^{10} C_{1})^{2} + 2 (^{10} C_{2})^{2} + 3 (^{10} C_{3})^{2} + \dots + 10 (^{10} C_{10})^{2},$

where $^{10} C_{r}, r \in {1, 2, \dots, 10}$ denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is ___ …
JEE Advanced 2018View solution →
Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(1 + x)^{2} + (1 + x)^{3} + \dots + (1 + x)^{49} + (1 + m x)^{50}$ is $(3 n + 1)^{51} C_{3}$ for some positive integer $n$ . Then the value of $n$ is
JEE Advanced 2016View solution →
The coefficient of $x^{9}$ in the expansion of $(1 + x) (1 + x^{2}) (1 + x^{3}) \dots (1 + x^{100})$ is
JEE Advanced 2015View solution →
Coefficient of $x^{11}$ in the expansion of $(1 + x^{2})^{4} (1 + x^{3})^{7} (1 + x^{4})^{12}$ is
JEE Advanced 2014View solution →
The coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ . Then $n =$
JEE Advanced 2013View solution →
For $r = 0, 1, \dots, 10$ , let $A_{r}, B_{r}$ and $C_{r}$ denote, respectively, the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and …
JEE Advanced 2010View solution →