If the ratio of the coefficients of the third and fourth terms in the expansion of $(x - \frac{1}{2x})^n$ is $1 : 2$,then the value of $n$ is:

Let $x \in \mathbb{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$,if $(1-x)^3(2+x)^6$ is approximated by $a+bx+cx^2$,then $a+b+c=$

The value of $\sum\limits_{r = 1}^{19} {\frac{{{}^{20}{C_{r + 1}}{(-1)}^r}}{{{2^{2r + 1}}}}}$ is

If $(1+2x+3x^2)^{10} = a_0+a_1x+a_2x^2+\ldots+a_{20}x^{20}$,then $\frac{a_2}{a_1}$ is equal to

Let $S=\{a+b \sqrt{2}: a, b \in Z \}$,$T_1=\{(-1+\sqrt{2})^n: n \in N \}$ and $T_2=\{(1+\sqrt{2})^n: n \in N \}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $Z \cup T_1 \cup T_2 \subset S$
$(B)$ $T_1 \cap (0, \frac{1}{2024}) = \phi$,where $\phi$ denotes the empty set
$(C)$ $T_2 \cap (2024, \infty) \neq \phi$
$(D)$ For any given $a, b \in Z$,$\cos(\pi(a+b \sqrt{2})) + i \sin(\pi(a+b \sqrt{2})) \in Z$ if and only if $b=0$,where $i=\sqrt{-1}$

By neglecting $x^4$ and higher powers of $x$, find the approximate value of $\sqrt[3]{x^2+64}-\sqrt[3]{x^2+27}$.