Sum of Binomial Coefficients of Even Numbered Terms

Q. For any positive integer $m,n$ with $n\ge m,$ let $\left(\begin{array}{c}n\\ m\end{array}\right)={}^n{C}_m.$ Then
$\left(\begin{array}{c}n\\ m\end{array}\right)+\left(\begin{array}{c}n-1\\ m\end{array}\right)+\left(\begin{array}{c}n-2\\ m\end{array}\right)+\cdots +\left(\begin{array}{c}m\\ m\end{array}\right)$ is equal to

1. ${}^n{C}_m$
2. ${}^n{C}_{m+1}$
3. ${}^{n+1}{C}_{m+1}$
4. ${}^{n+1}{C}_m$

Q.

The coefficient of ${\mathrm{t}}^{24}$ in the expansion of ${(1+{\mathrm{t}}^2)}^{12}(1+{\mathrm{t}}^{12})(1+{\mathrm{t}}^{24})$ is

1. ${}^{12}C_6+2$

2. ${}^{12}C_5$

3. ${}^{12}C_6$

4. ${}^{12}C_7$

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. If the sum of the series ${}^{40}{C}_0+{}^{40}{C}_4+{}^{40}{C}_8+\cdots +{}^{40}{C}_{40}$ is $2^a(2^a+1)$, then the value of $a$ is

Q. If $A\ne A^2=I,$ then $|I+A|=$