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

If
$\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}$then L is equal to _____.

Let n ≥ 5 be an integer. If 9ⁿ – 8n – 1 = 64α and 6ⁿ – 5n – 1 = 25β, then α – β is equal to

Let $ƒ(x)$ be a polynomial function such that $ƒ(x) + ƒ^′(x) + ƒ^{′′}(x) = x^5 + 64$. Then, the value of $\lim\limits_{x \to 1}\frac {f(x)}{x−1}$ is equal to :

Let $C_r$ denote the binomial coefficient of $x^r $in the expansion of $(1 + x)^{10}$. If for $α, β∈R$, $C_1 + 3⋅2 C_2 + 5⋅3 C_3 + …$ upto 10 terms =$\frac {α×211}{2^β−1}(C_0+\frac {C_1}{2}+\frac {C_2}{3}…..$upto 10 terms) then the value of $α + β$ is equal to _______.

The remainder when (2021)²⁰²³ is divided by 7 is

If the coefficients of x and x² in the expansion of (1 + x)^p (1 – x)^q, p, q≤15, are – 3 and – 5 respectively, then coefficient of x³ is equal to ______.

If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots .+{ }^{n} C_{2}\right)$ is:

A possible value of 'x', for which the ninth term in the expansion of
$\left\{ 3^{\log_3\sqrt{25^{x-1}+7}} + 3^{(-\frac{1}{8})\log_3(5^{x-1}+1)} \right\}^{10}$ in the increasing powers of $3^{(-\frac{1}{8})\log_3(5^{x-1}+1)}$ is equal to 180, is :

If ${}^{20}C_r$ is the co-efficient of $x^r$ in the expansion of $(1+x)^{20$, then the value of $\sum_{r=0}^{20} r^2 \cdot {}^{20}C_r$ is equal to :}

Let $\binom{n}{k}$ denote $^nC_k$.
If $A_k = \sum_{i=0}^{9} \binom{9}{i} \binom{12}{12-k+i} + \sum_{i=0}^{8} \binom{8}{i} \binom{13}{13-k+i}$ and $A_4 - A_3 = 190 p$, then p is equal to ___________.

$3 \times 7^{22} + 2 \times 10^{22} - 44$ when divided by 18 leaves the remainder _________.

If $\left(\frac{3^6}{4^4}\right)k$ is the term independent of $x$ in the binomial expansion of $\left(\frac{x}{4} - \frac{12}{x^2}\right)^{12}$, then $k$ is equal to ___________

If the coefficient of $ a^7b^8 $ in the expansion of $ (a + 2b + 4ab)^{10} $ is $ K \cdot 2^{16} $, then K is equal to _________.

If the coefficients of $x^7$ in $\left(x^2 + \frac{1}{bx}\right)^{11}$ and $x^{-7}$ in $\left(x - \frac{1}{bx^2}\right)^{11}$, $b \neq 0$, are equal, then the value of b is equal to :

The value of $ 2^{15}C_1 + 2^{15}C_2 - 3^{15}C_3 + \cdots - 15^{15}C_{15} + \binom{14}{1} + \binom{14}{3} + \binom{14}{5} + \cdots + \binom{14}{11} $ is :

The term independent of x in the expansion of [ (x + 1)/(x^{2/3} - x^{1/3} + 1) - (x - 1)/(x - x^{1/2}) ]¹⁰, x ≠ 1, is equal to ________.

Let n C_r denote the binomial coefficient of x^r in the expansion of (1 + x)^n. If ∑_{k=0}^{10} (2² + 3k) n C_k = α 3^{10} + β 2^{10}, α, β ∈ R, then α + β is equal to ________.

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + ....... + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$, then the value of $\gamma$ is :

The term independent of '$x$' in the expansion of $\left( \frac{x+1}{x^{2/3}-x^{1/3}+1} - \frac{x-1}{x-x^{1/2}} \right)^{10}$, where $x \neq 0, 1$ is equal to __________.

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is __________.

Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + ⋯ + a₄₀x⁴⁰. Then, a₁ + a₃ + a₅ + ⋯ + a₃₇ is equal to :

If n ≥ 2 is a positive integer, then the sum of the series $^{n+1}C_2 + 2( ^2C_2 + ^3C_2 + ^4C_2 + ....... + ^nC_2 )$ is :

The maximum value of k for which the sum $\sum_{i=0}^k \binom{10}{i} \binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i} \binom{13}{k+1-i}$ exists, is equal to __________.