If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively,then the value of the expression $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A$ is

Two friends $A$ and $B$ are $30 \, km$ apart and they start simultaneously on motorcycles to meet each other. The speed of $A$ is $3$ times that of $B$. The distance between them decreases at the rate of $2 \, km$ per minute. Ten minutes after they start,$A$'s vehicle breaks down and $A$ stops and waits for $B$ to arrive. After how much time (in minutes) since $A$ started riding,does $B$ meet $A$?

If the curves $y^2=6x$ and $9x^2+by^2=16$ intersect each other at right angles,then the value of $b$ is

If $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are unit vectors,then $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$ does not exceed:

The number of values of $x$ in the interval $[0, 5\pi]$ satisfying the equation $3\sin^2 x - 7\sin x + 2 = 0$ is

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have a head on both sides,whereas the remaining $n+1$ coins are fair. $A$ coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$,then $n$ is equal to