Amit M Agarwal Solutions for Chapter: Binomial Theorem, Exercise 1: Work Book Exercise 8.1

If the coefficients of $r^{\text{th}}$ and ${\left(r+1\right)}^{\text{th}}$ term in the expansion of $(3+7x{)}^{29}$ are equal, then $r$ equals

If ${\left(1+2x+3x^2\right)}^{10}=a_0+a_{1}x+a_2{x}^2+\dots +a_{20}{x}^{20},$ then $a_1$ equals

The number of terms which are free from radical signs in the expansion of ${\left(y^{\frac{1}{5}}+x^{\frac{1}{10}}\right)}^{55}$ is

The coefficient of $x^6$ in the expansion of ${\left(1+x^2-x^3\right)}^8$ is

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+\dots +(1+x{)}^{30}$ is

If the $9^{\text{th}}$ term in the expansion of ${\left[3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{\frac{-1}{8{\log }_3\left(5^{x-1}+1\right)}}\right]}^{10}$ is equal to $180$ and $x>1$, then $x$ equals

The expression

$\frac{1}{\sqrt{4x+1}}\left\{{\left(\frac{1+\sqrt{4x+1}}{2}\right)}^7-{\left(\frac{1-\sqrt{4x+1}}{2}\right)}^7\right\}$ is a polynomial in $x$ of degree

If$\frac{1}{\sqrt{4x+1}}\left\{{\left(\frac{1+\sqrt{4x+1}}{2}\right)}^n-{\left(\frac{1-\sqrt{4x+1}}{2}\right)}^n\right\}=a_0+a_{1}x+\dots +a_5{x}^5$, then $n$;is equal to