The money invested in a company is compounded continuously. If ₹ $200$ invested today becomes ₹ $400$ in $6$ years, then at the end of $33$ years it will become ₹ (in $\sqrt{2}$)

Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is:

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$,then

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a geometric progression are $a, b, c$ respectively,then $a^{q-r} \cdot b^{r-p} \cdot c^{p-q} = \dots\dots$

If $1 + \sin x + \sin^2 x + \dots \infty = 4 + 2\sqrt{3}$ for $0 < x < \pi$,then:

How many terms of the $G.P.$ $3, \frac{3}{2}, \frac{3}{4}, \ldots$ are needed to give the sum $\frac{3069}{512} ?$