Sum of Binomial Coefficients of Odd Numbered Terms

Q. If $n$ is a positive integer, then $(\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}$ is :

1. an irrational number
2. an odd positive interger
3. an even positive interger
4. a rational number other than positive integers

Q. Find the value of ${}^n{C}_1{+}^{n}C5{+}^{n}C9{+}^n{C}_{13}.....$

1. $2^{n-2}(1+sin\frac{n\pi }{4})$
2. $2^{\frac{n-2}{2}}sin\frac{n\pi }{4}+2^{n-2}$
3. $2^{\frac{n-1}{2}}sin\frac{n\pi }{4}+2^{n-2}$
4. $2^{\frac{n}{2}}sin\frac{n\pi }{4}+2^{n-1}$

Q. Let the coefficients of powers of $x$ in the second, third and fourth terms in the binomial expansion of $(1+x{)}^n,$ where $n$ is a positive integer, be in arithmetic progression. The sum of the coefficients of odd powers of $x$ in the expansion is

1. $32$
2. $64$
3. $128$
4. $256$

Q. The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

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

Q. The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

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

Q. Let $P=\sum _{r=1}^{50}\frac{{}^{50+r}{C}_r(2r-1)}{{}^{50}{C}_r(50+r)},Q=\sum _{r=0}^{50}{(}^{50}{C}_r{)}^2$ and $R=\sum _{r=0}^{100}(-1{)}^r{(}^{100}{C}_r{)}^2$. Then

1. $Q=R$
2. $P-Q=-1$
3. $Q+R=2P+1$
4. $P-R=1$