Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 1: JEE Main - 10th April 2015

The remainder on dividing $5^{99}$ by $11$ is _____ .

Let $A=\left[a_{\hat{i}\hat{j}}\right] \forall a_{ij}\in Z\cap \left[0,4\right], 1\le i,j\le 2$. The number of matrices $A$ such that the sum of all entries is $a$ prime number $p\in \left(2,13\right)$ is _____ .

The coefficient of $x^{-6}$, in the expansion of ${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$, is

If the constant term in the binomial expansion of ${\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)}^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha }\beta$ where $\beta <0$ is an odd number, then $\left|\alpha l-\beta \right|$ is equal to _____ .

The value of

$\frac{1}{1!50!}+\frac{1}{3!48!}+\frac{1}{5!46!}+\dots .+\frac{1}{49!2!}+\frac{1}{51!1!}$ is 

The remainder when ${19}^{200}+{23}^{200}$ is divided by $49$, is _____ .

Let the sixth term in the binomial expansion of ${\left(\sqrt{2^{{\log }_2\left(10-3^x\right)}}+\sqrt[5]{2^{(x-2){\log }_{2}3}}\right)}^{\mathrm{m}}$ powers of $2^{\left(x-2\right){\log }_{2}3}$, 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 A.P., then the sum of the squares of all possible values of $x$ is _____ .

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