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

If the term independent of \(x\) in the expansion of \[ \left( \sqrt{ax^2} + \frac{1}{2x^3} \right)^{10} \] is 105, then \(a^2\) is equal to:

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:

If $\left({ }^{30} C _1\right)^2+2\left({ }^{30} C _2\right)^2+3\left({ }^{30} C _3\right)^2+\ldots+30\left({ }^{30} C _{30}\right)^2=\frac{\alpha 60 !}{(30 !)^2}$ then $\alpha$ is equal to :
The coefficient of x and x\(^2\) in (1 + x)\(^p\) (1 – x)\(^q\) are 4 and – 5, then 2p + 3q is
If the coefficient of $x^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1 / 3}-\frac{1}{b x^3}\right)^{15}$, where a and $b$ are positive real numbers, then for each such ordered pair $(a, b)$ :
The remainder, when $19^{200}+23^{200}$ is divided by $49$ , is_______

If the term without \(x\) in the expansion of \(\left(x^{\frac{2}{3}}+\frac{\alpha}{x^3}\right)^{22}\)is 7315 , then \(|\alpha|\) is equal to ___

The constant term in the expansion of $\left(2 x+\frac{1}{x^7}+3 x^2\right)^5$ is_____

Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0 n \in N$, be $376$. Then the coefficient of $x^4$ is ______

The coefficient of \(x^{18}\) in the expansion of \(\left(x^4 - \frac{1}{x^3}\right)^{15}\) is ________
In the expansion of \((2^\frac{1}{4}+3^{-\frac{1}{4}})^n\), the ratio of \(5^{th}\) term from start and \(5^{th}\) term form end is \(\sqrt 6 : 1\) , then find \(3^{rd}\) term
\((2x^3-\frac{1}{3}x^8)^5\) β†’ coefficient of \(x^4\)

If the coefficients of three consecutive terms in the expansion of (1+x)n are in the ratio 1:5:20, then the coefficient of the fourth term of the expansion is?

Let $\alpha>0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-a}, \beta \in N$.
Then \(Ξ±\) is equal to _________. 

If the coefficient of x7 in expansion of \((ax- \frac{1}{bx^2})^{13}\) is equal to the coefficient of x-5 in expansion of \((ax + \frac{1}{bx^2})13\), then a4b4 is ______
The mean of the coefficients of \( x^n, x^{n+1}, \dots, x^r \) in the binomial expansion of \( (2 + x)^r \) is:
The number of integral terms in the expansion of (31/2 + 51/4)680 is equal to:
If the coefficients of \( x \) and \( x^2 \) in \( (1 + x)^p(1 - x)^q \) are 4 and -5 respectively, then \( 2p + 3q \) is equal to:
The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)n+2, which are in the ratio 1 : 3 : 5, is equal to
Let 𝐾 be the sum of the coefficients of the odd powers of π‘₯ in the expansion of $(1+π‘₯)^{99}$. Let π‘Ž be the middle term in the expansion of \((2+\frac{1}{\sqrt 2} )^{200}\). If $\frac{⁑^{200}𝐢_{99} 𝐾}{π‘Ž} =\frac{2^𝑙 π‘š}{𝑛} $, where π‘š and 𝑛 are odd numbers, then the ordered pair (𝑙, 𝑛) is equal to
If coefficient of x15 in expansion of \((ax^3+\frac{1}{bx^{1/3}})^{15}\) is equal to coefficient of x –15 in expansion of \((ax^{1/3}+\frac{1}{bx^3})^{15}\) then |ab – 5| is equal to
The coefficient of x5 in the expansion of \((2x^3 - \frac{1}{3x^2})^5\) is
The remainder when 7103 is divided by 17, is
Let the coefficients of the middle terms in the expansion of\(\begin{array}{l} \left(\dfrac{1}{\sqrt 6}+ \beta x\right)^4, \left(1 – 3\beta x\right)^2 \text{and}\left(1 – \dfrac{\beta}{2}x\right )^6, \beta > 0,\end{array}\)respectively form the first three terms of an A.P. If d is the common difference of this A.P., then \(\begin{array}{l} 50-\frac{2d}{\beta^2} \end{array}\)is equal to _______.
If\(\begin{array}{l}1 + \left(2 + ^{49}C_1 + ^{49}C_2 + … ^{49}C_{49}\right) \left(^{50}C_2 + ^{50}C_4 + … ^{50}C_{50}\right) \end{array}\)is equal to 2n. m, where m is odd, then n + m is equal to ______.