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 A and G are respectively the arithmetic and geometric means between two distinct positive numbers a and b then prove that A>G.

Q.

Simplify the expression
${(1+x)}^{1000}+x(1+x{)}^{999}+x^2(1+x{)}^{998}+\cdots +x^{1000}$ and find the coefficient of $x^{50}$

Q. By using binomial theorem, expand the following:
$(99{)}^5$

Q. Using binomial theorem, evaluate the following:
$(96{)}^3$

Q.

Find the value of x, if the ratio of 10th term to 11th term of the expansion $(2-3x^3{)}^{20}$ is 45 : 22.

Or

Find the value of a, so that the term independent of x in ${(\sqrt{x}+\frac{a}{x^2})}^{10}$ is 405.

Q.

If the thrid term in the binomial expansion of $(1+x{)}^m$ is - $\frac{1}{8}$$x^2$, then the rational value of m is

1. 2

2. 1/2

3. 4

4. 3