Binomial Theorem and Its Simple Applications

If $\sum_{r=0}^{10} \left(\frac{10^{r+1} - 1}{10^r}\right) \cdot {}^{11}C_{r+1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$, then $\alpha$ is equal to:

The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is:

For an integer n \ge 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)^{2n-3} is 16, then the distance of the point P(2n-1, n²-4n) from the line x + y = 8 is:

The remainder when $(64^{64})^{64}$ is divided by 7 is equal to:

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

For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th 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:

The remainder, when $7^{103}$ is divided by 23, is equal to:

Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2n - 1}$. If $2A = 5B$, then $n$ is equal to:

If in the expansion of $(1 + x)^p (1 - x)^q$, the coefficients of $x$ and $x^2$ are 1 and -2, respectively, then $p^2 + q^2$ is equal to:

If $(1-x^3)^{10}=\displaystyle\sum_{r=0}^{10}a_rx^r(1-x)^{30-2r}$, then $\dfrac{9a_9}{a_{10}}$ is equal to ______.