$A$ 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:

There is a rectangular sheet of dimension $(2m-1) \times (2n-1)$,where $m > 0, n > 0$. It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

Consider the following statements for $X = \dot{C}H_3$ and $Y = \dot{C}F_3$:
$(I)$ When $X$ dimerises,the bond angle decreases.
$(II)$ When $X$ dimerises,the bond angle increases.
$(III)$ In the $X-Y$ molecule,the $C-C$ bond length is less than that in the $Y-Y$ molecule.
$(IV)$ The bond angle in $X$ is greater than that in $Y$.

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y = f(x)$,$x$-axis and the ordinates $x = \frac{\pi}{4}$ and $x = \beta > \frac{\pi}{4}$ is $\left( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta \right)$. Then $f\left( \frac{\pi}{2} \right)$ is

Three identical bodies of equal mass $M$ each are moving along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each body is

The electron affinity of sulfur is........