Find the $12^{\text{th}}$ term of a $G.P.$ whose $8^{\text{th}}$ term is $192$ and the common ratio is $2$.

Let $a$ and $b$ be roots of $x^2 - 3x + p = 0$ and let $c$ and $d$ be the roots of $x^2 - 12x + q = 0$,where $a, b, c, d$ form an increasing $G$.$P$. Then the ratio of $(q + p) : (q - p)$ is equal to

In a certain test, there are $n$ questions. In this test, $2^{n-i}$ students gave wrong answers to at least $i$ questions, where $i = 1, 2, \ldots, n$. If the total number of wrong answers given is $2047$, then $n$ is equal to:

If $1+\sin x+\sin ^{2} x+\ldots$ up to $\infty = 4+2 \sqrt{3}$,where $0 < x < \pi$ and $x \neq \frac{\pi}{2}$,then $x$ is equal to

The value of ${4^{1/3}} \cdot {4^{1/9}} \cdot {4^{1/27}} \cdots \infty$ is

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = 2^{n}$.