Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}$ be a $GP$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .$ If $c_{2}=12$ and $\mathrm{c}_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}$ is equal to ..... .

Let $a, b, c$ be the sides of a triangle. If $t$ denotes the expression $\frac{\left(a^2+b^2+c^2\right)}{(a b+b c+c a)}$, the set of all possible values of $t$ is

Three numbers are in an increasing geometric progression with common ratio $\mathrm{r}$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $\mathrm{d}$. If the fourth term of GP is $3 \mathrm{r}^{2}$, then $\mathrm{r}^{2}-\mathrm{d}$ is equal to:

If $x, y, z \in R^+$ such that $x + y + z = 4$, then maximum possible value of $xyz^2$ is -

If the arithmetic, geometric and harmonic means between two positive real numbers be $A,\;G$ and $H$, then

Suppose the sequence $a_1, a_2, a_3, \ldots$ is a n arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is