The monkey $\text{B}$ shown in the figure is holding onto the tail of the monkey $\text{A}$ which is climbing up a rope. The masses of monkeys $\text{A}$ and $\text{B}$ are $5\text{kg}$ and $2\text{kg}$ respectively. If $\text{A}$ can tolerate a tension of $30\text{N}$ in its tail, what force should it apply on the rope in order to carry the monkey $\text{B}$ with it ? (take $g=10{\text{m/s}}^2$)

The correct option is D $T\ge 70\text{N}andT\le 105\text{N}$
Suppose $\text{A}$ moves upward with acceleration $a$ such that maximum tension $30\text{N}$ is produced in its tail.


$T-5g-T_1=5a$
$\Rightarrow T-5g-30=5a....\left(i\right)$
$T_1-2g=2a$
$\Rightarrow 30-2g=2a--\left(ii\right)$

From equation $\left(i\right)$ and $\left(ii\right)$, we have
$a=5{\text{m/s}}^2$ and $T=105\text{N}$ (max)
So, $\text{A}$ can apply a maximum force of $105\text{N}$ on the rope to carry the monkey $\text{B}$ with it.

For minimum force there is no acceleration of monkey $\text{A}$ and $\text{B}$.
$\Rightarrow a=0$

From equation $\left(i\right)$,
$T_1-2g=0$
$\Rightarrow T_1=20\text{N}$ (weight of monkey $\text{B}$)

From equation $\left(ii\right)$,
$T-5g-20=0$ as $T_1=20\text{N}$
$\Rightarrow T=70\text{N}$

$\therefore$ The monkey $\text{A}$ should apply force between $70\text{N}$ and $105\text{N}$ to carry monkey $\text{B}$ with it.