List of top Mathematics Questions on binomial expansion formula asked in JEE Main

Let $ \alpha, \beta, \gamma $ and $ \delta $ be the coefficients of $ x^7, x^5, x^3, x $ respectively in the expansion of $ (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, \, x > 1 $. If $ \alpha u + \beta v = 18 $, $ \gamma u + \delta v = 20 $, then $ u + v $ equals:

For some $ n \neq 10 $, let the coefficients of the 5^th, 6^th, and 7^th terms in the binomial expansion of $ (1 + x)^{n+4} $ be in A.P. Then the largest coefficient in the expansion of $ (1 + x)^{n+4} $ is:

Let the coefficients of three consecutive terms $ T_r $, $ T_{r+1} $, and $ T_{r+2} $ in the binomial expansion of $ (a + b)^{12}$ be in a G.P. and let $ p $ be the number of all possible values of $ r $. Let $ q $ be the sum of all rational terms in the binomial expansion of $ \left( 4\sqrt{3} + 3\sqrt{4} \right)^{12} $. Then $ p + q $ is equal to:

Number of integral terms in the expansion of $ \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} $ is equal to ______.

In the expansion of $(1 + x)(1 - x^2) (1 + \frac 3x + \frac {3}{x^2}+ \frac {1}{x^3})^5$ the sum of coefficients of $x^3$ and $x^{-13}$ is

Coefficient of x²⁰¹² in (1-x)²⁰⁰⁸(1+x+x²)²⁰⁰⁷ is equal to ___

Let the sixth term in the binomial expansion of $({\sqrt{2}^{log_{2}}(10-3^{x})+\sqrt[5]{2^{(x-2)log_{2}{3}}}})^{m}$, in the increasing powers of $2^{(x-2)log_{2}3}$, be 21 If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an AP, then the sum of the squares of all possible values of x is

For the expression $(1-x)^{100}$. Then sum of coefficient of first 50 terms is:

The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of $\left(2x^2 + \frac{1}{2x}\right)^{11}$ is equal to:

The coefficient of x⁷ in (1 – 2x + x³)¹⁰ is?

Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)n be the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is _______.

If the $1011^{th}$ term from the end in the binominal expansion of $(\frac{4x}{5}-\frac{5}{2x})^{2022}$ is 1024 times $1011^{th}$ term from the beginning, then |x| is equal to

Let α be the constant term in the binomial expansion of $ (√x−\frac{6}{x^{\frac{3}{2}}} )^n ,n≤15.$ If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x^-n is λα, then λ is equal to _______.

The term independent of x in the expansion of
$(1−x^2+3x^3)(\frac{5}{2}x^3−\frac{1}{5x^2})11,x≠0 $
is:

The number of positive integers k such that the constant term in the binomial expansion of $( 2x^3 + \frac {3}{x^k} )^{12}, x ≠ 0$ is $2^8$ . l, where l is an odd integer, is______.

If the coefficient of x10 in the binomial expansion of
$\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}} + \frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$
is 5^k.l, where l, k∈N and l is co-prime to 5, then k is equal to ___________.

Let the coefficients of x^–1 and x^–3 in the expansion of
$(2x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}} )^{15} , x > 0$be m and n respectively. If r is a positive integer such that
$mn² = ^{15}C_r.2^r$
then the value of r is equal to ______.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ($\frac{\sqrt{24}+1}{\sqrt{34}}$)n, in the increasing powers of 134 be 64:1.If the sixth term from the beginning is $\frac{\alpha}{\sqrt{34}}$, then α is equal to ___________.

If the sum of the co-efficients of all the positive even powers of x in the binomial expansion of $(2x^3+\frac{3}{x})^{10} $ is 5¹⁰ – β·3⁹, the β is equal to ____.

The coefficient of $x^7$ in the expression $\left(1+x\right)^{10}+x\left(1+x\right)^{9}+x^{2}\left(1+x\right)^{8}+...+x^{10}$ is :

For a positive integer $n$, if the mean of the binomial coefficients in the expansion of $(a + b)^{2n - 3}$ is 16, then $n$ is equal to :

The term independent of $x$ in the binomial expansion of $\left(1-\frac{1}{x}+3x^{5}\right)\left(2x^{2}-\frac{1}{x}\right)^{8}$ is :

The coefficient of $x^{1012}$ in the expansion of $(1 + x^n + x^{253})^{10}$, (where n $\le$ 22 is any positive integer), is :-

The coefficient of $x^{50}$ in the binomial expansion of $(1 + x)^{1000} + x (1 + x)^{999} + x^2(1 + x)^{998} + .... + x^{1000}$ is:

Find the number of rational numbers in the expansion of $(2^{\frac{1}{5}} + 5^{\frac{1}{3}})^{15}$.