List of top Mathematics Questions on Binomial theorem asked in JEE Main

The term independent of $ x $ in the expansion of $$ \left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10} $$ for $ x>1 $ is:

Let $ (1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + ... + a_{20} x^{20} $. If $ (a_1 + a_3 + a_5 + ... + a_{19}) - 11a_2 = 121k $, then k is equal to _______

If $ \sum_{r=1}^{9} \left( \frac{r+3}{2^r} \right) \cdot {^9C_r} = \alpha \left( \frac{3}{2} \right)^9 - \beta $, $ \alpha, \beta \in \mathbb{N} $, then $ (\alpha + \beta)^2 $ is equal to
The sum of all rational terms in the expansion of $ \left( 2 + \sqrt{3} \right)^8 $ is

In the expansion of \[ \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n , \, n \in \mathbb{N}, \] if the ratio of the 15th term from the beginning to the 15th term from the end is \[ \frac{1}{6}, \] then the value of \[ {}^nC_3 \] is:

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $P(2n-1, n^2-4n)$ from the line $x + y = 8$ is:

If
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $

If \( \sum_{r=0}^5 \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n} \), gcd(m, n) = 1, then \( m - n \) is equal to:
The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
If the sum, \( \sum_{r=1}^9 \frac{r+3}{2r} \cdot \binom{9}{r} \) is equal to \( \alpha \cdot \left( \frac{3}{2} \right)^9 - \beta \), then the value of \( (\alpha + \beta)^2 \) is equal to:
The number of rational terms in the binomial expansion of $\left( \frac{1}{2} + \frac{1}{8} \right)^{1016}$ is
The sum of all rational terms in the expansion of $$ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} $$ is equal to: 
Let $\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}$ and $\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$, then the value of $n$ is _____.

If \(\frac{{^{11}C_1}}{2} + \frac{{^{11}C_2}}{3} + \ldots + \frac{{^{11}C_9}}{10} = \frac{n}{m}\) with \(\gcd(n, m) = 1\), then \(n + m\) is equal to:

Let \( 0 \le r \le n \). If \( ^{n+1}C_{r+1} : ^nC_r : ^{n-1}C_{r-1} = 55 : 35 : 21 \), then \( 2n + 5r \) is equal to:

The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of $$ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 $$ is: 
The coefficient of \( x^{70} \) in \( x^2 (1+x)^{98} + x^3 (1+x)^{97} + x^4 (1+x)^{96} + \ldots + x^{54} (1+x)^{46} \) is \( ^{99}C_p - ^{46}C_q \).
Then a possible value to \( p + q \) is:
If \( A \) denotes the sum of all the coefficients in the expansion of \( (1 - 3x + 10x^2)^n \) and \( B \) denotes the sum of all the coefficients in the expansion of \( (1 + x^2)^n \), then:
\(^{n-1}C_r = (k^2 - 8)  ^{n}C_{r+1}\) if and only if:
Let the coefficient of \( x^r \) in the expansion of \((x + 3)^{n-1} + (x + 3)^{n-2} (x + 2) + (x + 3)^{n-3} (x + 2)^2 + \ldots + (x + 2)^{n-1}\)
be \( \alpha_r \). If \( \sum_{r=0}^n \alpha_r = \beta^n - \gamma^n \), \( \beta, \gamma \in \mathbb{N} \), then the value of \( \beta^2 + \gamma^2 \) equals \(\_\_\_\_\_\).
If the coefficient of \(x^{30}\) in the expansion of \(\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8, \, x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.
Suppose \( 2 - p \), \( p \), \( 2 - \alpha \), \( \alpha \) are the coefficients of four consecutive terms in the expansion of \( (1 + x)^n \). Then the value of \( p^2 - \alpha^2 + 6\alpha + 2p \) equals
In the expansion of \[ (1 + x)(1 - x^2) \left( 1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3} \right)^5, \quad x \neq 0, \]the sum of the coefficients of \( x^3 \) and \( x^{-13} \) is equal to ____
The number of elements in the set S = {(x, y, z) : x, y, z ∈ Z, x + 2y + 3z = 42, x, y, z ≥ 0} equals ____
Let \( a \) be the sum of all coefficients in the expansion of \( (1 - 2x + 2x^2)^{2023} (3 - 4x^2 + 2x^3)^{2024} \). and \( b = \lim_{x \to 0} \frac{\int_0^x \frac{\log(1 + t)}{t^{2024} + 1} \, dt}{x^2} \).If the equations \( cx^2 + dx + e = 0 \) and \( 2bx^2 + ax + 4 = 0 \) have a common root, where \( c, d, e \in \mathbb{R} \), then \( d : c : e \) equals