$A$ particle of mass $1 \ kg$ is moving in $SHM$ with a path length of $0.01 \ m$ and frequency of $50 \ Hz$. The maximum force in newton,acting on the particle is (in $pi^2$)

The number of natural numbers less than $1000$,in which no two digits are repeated,is:

If $S_n = 1^3 + 2^3 + \ldots + n^3$ and $T_n = 1 + 2 + \ldots + n$,then

The vectors $b$ and $c$ are in the direction of north-east and north-west respectively and $|b|=|c|= 4$. The magnitude and direction of the vector $d = c - b$ are:

If $u=\log \left(x^3+y^3+z^3-3 x y z\right)$,then $(x+y+z)(u_x+u_y+u_z)$ is equal to

Three samples of the same gas $A, B$ and $C$ $(\gamma = 3/2)$ initially have equal volume. Now the volume of each sample is doubled. The process is adiabatic for $A$,isobaric for $B$ and isothermal for $C$. If the final pressures are equal for all three samples,the ratio of their initial pressures is: