Ten trucks, numbered $1$ to $10$ , are carrying packets of sugar. Each packet weights either $999\,g$ or $1000\,g$ and each truck carries only the packets equal weights. The combined weight of $1$ packet selected from the first truck,$2$ packets from the second,$4$ packets from the third, and so on, and $2^9$ packet from the tenth truck is $1022870\,g$. The trucks that have the lighter bags are

The solution of the equation $1 + a + {a^2} + {a^3} + ....... + {a^x}$ $ = (1 + a)(1 + {a^2})(1 + {a^4})$ is given by $x$ is equal to

Consider an infinite $G.P. $ with first term a and common ratio $r$, its sum is $4$ and the second term is $3/4$, then

The first term of a $G.P.$ is $7$, the last term is $448$ and sum of all terms is $889$, then the common ratio is

If $G$ be the geometric mean of $x$ and $y$, then $\frac{1}{{{G^2} - {x^2}}} + \frac{1}{{{G^2} - {y^2}}} = $

Suppose that the sides $a,b, c$ of a triangle $A B C$ satisfy $b^2=a c$. Then the set of all possible values of $\frac{\sin A \cot C+\cos A}{\sin B \cot C+\cos B}$ is