Two persons $A$ and $B$ perform same amount of work in moving a body through a certain distance $d$ with application of forces acting at angles $45^{\circ}$ and $60^{\circ}$ with the direction of displacement respectively. The ratio of force applied by person $A$ to the force applied by person $B$ is $\frac{1}{\sqrt{x}}$. The value of $x$ is $\_\_\_\_$.

A particle experiences a variable force $\overrightarrow { \mathrm { F } } = \left( 4 x \hat { i } + 3 y ^ { 2 } \hat { j } \right)$ in a horizontal $x - y$ plane. Assume distance in meters and force is newton. If the particle moves from point $( 1,2 )$ to point $( 2,3 )$ in the $x - y$ plane, then Kinetic Energy changes by :
(1) 25 J
(2) 50 J
(3) 12.5 J
(4) 0 J

A block of mass 10 kg is moving along $x$-axis under the action of force $F = 5 x \mathrm {~N}$. The work done by the force in moving the block from $x = 2 \mathrm {~m}$ to 4 m will be $\_\_\_\_$ J.

A force $F = \left( 5 + 3 y ^ { 2 } \right)$ acts on a particle in the $y$-direction, where $F$ is newton and $y$ is in meter. The work done by the force during a displacement from $y = 2 \mathrm {~m}$ to $y = 5 \mathrm {~m}$ is $\_\_\_\_$ J.

A force $\vec{F} = (2 + 3x)\hat{i}$ acts on a particle in the $x$ direction where $F$ is in Newton and $x$ is in meter. The work done by this force during a displacement from $x = 0$ to $x = 4$ m is $\_\_\_\_$ J.