If $f: R \rightarrow R$ is defined by $f(x)=x-[x]-\frac{1}{2}$ for $x \in R$,where $[x]$ is the greatest integer not exceeding $x$,then $\{x \in R: f(x)=\frac{1}{2}\}$ is equal to :

The equation of the circle of radius $3$ that lies in the fourth quadrant and touches the lines $x=0$ and $y=0$ is

Acetone on addition to methyl magnesium bromide forms a complex,which on decomposition with acid gives $X$ and $Mg(OH)Br$. Which one of the following is $X$?

$A$ line segment $l$ of length $a \ cm$ is rotated about a vertical line $L$ keeping the line $l$ in one of the following three positions: $(I)$ $l$ is parallel to $L$ and is at a distance of $r \ cm$ from $L$,$(II)$ $l$ is perpendicular to $L$ and its mid-point is at a distance $r \ cm$ from $L$,$(III)$ $l$ and $L$ are in the same plane and $l$ is inclined to $L$ at an angle $30^{\circ}$ with its mid-point at a distance $r \ cm$ from $L$. Let $A_1, A_2, A_3$ be the areas so generated. If $r > (a / 2)$,then

$A$ particle $P$ is moving in a circle of radius $a$ with a uniform speed $v$. $C$ is the centre of the circle and $AB$ is a diameter. When passing through $B$,the angular velocity of $P$ about $A$ and $C$ are in the ratio:

If $x$ and $y$ are acute angles such that $\cos x + \cos y = \frac{3}{2}$ and $\sin x + \sin y = \frac{3}{4}$,then $\sin(x+y)$ is equal to