Widal test is used for the diagnosis of

The velocity at the mean position of a particle executing $S.H.M.$ is $v$. What is the velocity of the particle at a distance equal to half of the amplitude?

The equivalent resistance of a series combination of two resistors is $s$. When they are connected in parallel,the equivalent resistance is $p$. If $s = np$,then the minimum value for $n$ is (Round off to the nearest integer).

$A$ plane mirror is placed at the bottom of a tank containing a liquid of refractive index $\mu$. $P$ is a small object at a height $h$ above the mirror. An observer $O$,vertically above $P$ outside the liquid,sees $P$ and its image in the mirror. The apparent distance between these two will be

In the $HCl$ molecule,the separation between the nuclei of the two atoms is about $1.27 \ \mathring{A}$. The approximate location of the centre of mass of the molecule from the hydrogen atom is ....... $\ \mathring{A}$. (Assuming the chlorine atom to be about $35.5$ times as massive as the hydrogen atom.)

The value of the integral $\int_0^4 \frac{dx}{1+x^2}$ obtained by using the Trapezoidal rule with $h=1$ is