If the sum of the $10$ terms of an $A.P.$ is $4$ times to the sum of its $5$ terms, then the ratio of first term and common difference is

If the first, second and last terms of an $A.P.$ be $a,\;b,\;2a$ respectively, then its sum will be

If the sum of the series $2 + 5 + 8 + 11............$ is $60100$, then the number of terms are

The $8^{\text {th }}$ common term of the series $S _1=3+7+11+15+19+\ldots . .$ ; $S _2=1+6+11+16+21+\ldots .$ is $.......$.

If the angles of a quadrilateral are in $A.P.$ whose common difference is ${10^o}$, then the angles of the quadrilateral are

If $1, \log _{10}\left(4^{x}-2\right)$ and $\log _{10}\left(4^{x}+\frac{18}{5}\right)$ are in
arithmetic progression for a real number $x$ then the value of the determinant $\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|$ is equal to ...... .