If $a,\,b,\,c,\,d$ are positive real numbers such that $a + b + c + d$ $ = 2,$ then $M = (a + b)(c + d)$ satisfies the relation

Let $a$, $b \in R$  be such that $a$, $a + 2b$ , $2a + b$ are in $A.P$. and $(b + 1)^2$, $ab + 5$, $(a + 1)^2$ are in $G.P.$ then $(a + b)$ equals

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a $G.P.$ be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an $A.P.$ If the $5^{\text {th }}$ term of the $G.P.$ is the $\mathrm{n}^{\text {th }}$ term of the $A.P.$, then $\mathrm{n}$ is equal to

The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original term is

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ then the common ratio of the $G.P.$ is:

The minimum value of $2^{sin x}+2^{cos x}$ is