$A$ light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (vertically). If the acceleration of the system is $\frac{g}{8}$,then the ratio of masses $m_2 : m_1$ is:

There are four forces acting at a point $P$ produced by strings as shown in the figure. Point $P$ is at rest. The forces $F_1$ and $F_2$ are respectively:

$A$ presbyopic patient has a near point of $30 \, cm$ and a far point of $40 \, cm$. The dioptric power of the corrective lens required for seeing distant objects is: (in $D$)

$A$ wire $50 \ cm$ long and $1 \ mm^2$ in cross-section carries a current of $4 \ A$ when connected to a $2 \ V$ battery. The resistivity of the wire is:

The half-life of $Ra^{226}$ is $1620$ years. Calculate the number of atoms that decay in one second in $1 \ g$ of radium (Avogadro number $= 6.023 \times 10^{23}$).

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \text{ m}$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \text{ m}$ respectively. If the velocity of the first particle is $0.4 \hat{i} \text{ ms}^{-1}$,what is the velocity of the second particle in $\text{ms}^{-1}$ if they collide after $10 \text{ s}$?