Let the mean and the variance of seven observations $2, 4, \alpha, 8, \beta, 12, 14$ (where $\alpha < \beta$) be $8$ and $16$ respectively. Then the quadratic equation whose roots are $3\alpha + 2$ and $2\beta + 1$ is:

The mean of $10$ terms is $3$. If the first term is increased by $1$,the second by $2$,and so on,then the new mean is:

The mean and variance of the data $4, 5, 6, 6, 7, 8, x, y$ where $x < y$ are $6$ and $\frac{9}{4}$ respectively. Then $x^{4} + y^{2}$ is equal to

An online exam is attempted by $50$ candidates,out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance of $2$. The variance of marks obtained by $30$ girls is also $2$. The average marks of all $50$ candidates is $15$. If $\mu$ is the average marks of girls and $\sigma^{2}$ is the variance of marks of $50$ candidates,then $\mu+\sigma^{2}$ is equal to ...... .

The mean of $n$ observations is $\bar{x}$. If three observations $n+1, n-1, 2n-1$ are added such that the mean remains the same,then the value of $n$ is

If $\alpha$ and $\beta$ are respectively the mean deviation about the mean and the variance of the first five prime numbers,then the ordered pair $(\alpha, \beta)$ is: