If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}, \alpha \neq 0$, are equal, then the value of $\alpha$ is:

A data consists of 20 observations $x_1, x_2, \ldots, x_{20}$. If $\sum\limits_{i=1}^{20}\left(x_i+5\right)^2=2500$ and $\sum\limits_{i=1}^{20}\left(x_i-5\right)^2=100$, then the ratio of mean to standard deviation of this data is :

A bag contains $(\mathrm{N}+1)$ coins -N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$, then N is equal to:

If the eccentricity $e$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, passing through $(6,4 \sqrt{3})$, satisfies $15\left(e^2+1\right)=34 e$, then the length of the latus rectum of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{2\left(a^2+1\right)}=1$ is: