Given data

• Mass of monkey: $m = 50\,\text{kg}$
• Maximum tension rope can withstand: $T_{\max} = 350\,\text{N}$
• Acceleration while moving downward: $a_1 = 4\,\text{m/s}^2$
• Acceleration while moving upward: $a_2 = 5\,\text{m/s}^2$
• Acceleration due to gravity: $g = 10\,\text{m/s}^2$

Tension while climbing downward

When the monkey accelerates downward with acceleration $4\,\text{m/s}^2$, forces on monkey are:

• Weight downward: $mg$
• Tension upward: $T$

Taking downward as positive: $mg - T = ma$

$50 \cdot 10 - T = 50 \cdot 4$

$500 - T = 200$

$T = 300\,\text{N}$

Since $300 < 350$, the rope will not break while going downward.

So:

• Option B: $T=350\,\text{N}$ while going downward $\Rightarrow$ False
• Option D: Rope will break while going downward $\Rightarrow$ False

Tension while climbing upward

When the monkey accelerates upward with acceleration $5\,\text{m/s}^2$, taking upward as positive: $T - mg = ma$

$T - 500 = 50 \cdot 5$

$T - 500 = 250$

$T = 750\,\text{N}$

Since $750 > 350$, the rope will break while climbing upward.

So:

• Option A: $T=700\,\text{N}$ while climbing upward $\Rightarrow$ False
• Option C: Rope will break while climbing upward $\Rightarrow$ True

Final answer The correct option is: $\boxed{\text{C}}$

Comparison with stored answer Stored correct answer: C

My derived answer matches the stored answer.