If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $\left(1+a x+b x^2\right) (1-3 x)^{15}$ in powers of $x$,...

If the fourth term in the Binomial expansion of $\left(\frac{2}{x}+x^{\log _8 x}\right)^6$ (x > 0) is 20 $\times 8^7$ , then a value of x...

The term independent of $x$ in the expansion of $\left(\frac{1}{60}-\frac{x^8}{81}\right) \cdot\left(2 x^2-\frac{3}{x^2}\right)^6$ is equal to :

For a positive integer $n,\left(1+\frac{1}{x}\right)^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $...

If ${ }^{20} \mathrm{C}_1+\left(2^2\right)^{20} \mathrm{C}_2+\left(3^2\right)^{20} \mathrm{C}_3+\ldots \ldots . .+\left(20^2\right)^{20} \mathrm{C}_{20}=\mathrm{A}\left(2^\beta\right)$, then the ordered pair $(\mathrm{A}, \beta)$ is equal to:

Let $\alpha>0, \beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)^{10}$...

A ratio of the 5 th term from the beginning to the $5^{\text {th }}$ term from the end in the binomial expansion of $\left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}$...

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is