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For part (c), you have to find the condition for the circular motion to exist. We set T2 > 0 and solve because T1 is always > 0 since r, w, g and m > 0 and also since 0 < θ < 90 (i.e. acute angle) -> implying sinθ and cosθ > 0. As for (d), think about when P is fixed (Hint: Consider the tensions (T1 & T2))