Sum of Binomial Coefficients of Even Numbered Terms

Q.

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt{x}{)}^{50}$ is

1. $\frac{1}{2}(3^{50}+1)$

2. $\frac{1}{2}\left(3^{50}\right)$

3. $\frac{1}{2}(3^{50}-1)$

4. $\frac{1}{2}(2^{50}+1)$

Q. If $n$ is the degree of the polynomial, ${\left[\frac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]}^8+{\left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]}^8$ and $m$ is the coefficient of $x^n$ in it, then the ordered pair $(n,m)$ is equal to :

1. $(24,(10{)}^8)$
2. $(8,5(10{)}^4)$
3. $(12,(20{)}^4)$
4. $(12,8(10{)}^4)$

Q. If ${(1+x-2x^2)}^6=1+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_{12}{x}^{12},$ then the value of
$a_2+a_4+a_6+\cdots +a_{12}$ will be

1. $32$
2. $31$
3. $64$
4. $1024$

Q. Let $(x+10{)}^{50}+(x-10{)}^{50}=a_0+a_{1}x+a_2{x}^2+...+a_{50}{x}^{50}$, for all $x\in R$, then $\frac{a_2}{a_0}$ is equal to:

1. $12.75$
2. $12.50$
3. $12.25$
4. $12.00$