If $\int e^x(1+x) \cdot \sec ^2(x e^x) \, dx = f(x) + \text{constant}$, then $f(x)$ is equal to

Consider the following statements $A$ and $B$ and identify the correct answer:
$A$. When a person walks on a rough surface, the direction of the frictional force exerted by the surface on the person is in the direction of his motion.
$B$. When a cycle is in motion, the force of friction exerted by the ground on the front wheel is in the backward direction.

$A$ water barrel having water up to a depth $d$ is placed on a table of height $h$. $A$ small hole is made on the wall of the barrel at its bottom. If the stream of water coming out of the hole falls on the ground at a horizontal distance $R$ from the barrel,then the value of $d$ is

The centre of mass of three particles of masses $1 ~kg, 2 ~kg$ and $3 ~kg$ is at $(2,2,2)$. The position of the fourth mass of $4 ~kg$ to be placed in the system so that the new centre of mass is at $(0,0,0)$ is:

The function $f: R \rightarrow R$ is defined by $f(x) = \cos^2 x + \sin^4 x$ for $x \in R$. Then the range $f(R)$ is equal to

$A$ sodium atom is in one of the states labeled 'Lowest excited levels'. It remains in that state for an average time of $10^{-8} \ s$,before it makes a transition back to a ground state. What is the uncertainty in energy of that excited state?