I'm not sure what you mean by half of these questions.

i) Assuming you meant 50e^(t^2+1)

Using the chain rule: 100t*e^(t^2+1)

ii) Independence is not a facet of the Methods course.

iii) ksinx+1=0

=> sinx = -1/k (Q3 and Q4)

=> x = Arcsin(-1/k) [By definition, Arcsin(A) is defined for -pi/2 ≤ Arcsin(A) ≤ pi/2]

This means Arcsin(-1/k) will return an answer from the interval [-pi/2, 0] (Q4). We need to add 2pi in order to move this into [0,2pi], then we should also construct the corresponding Q3 angle.

=> x = Arcsin(-1/k) + 2pi and x = 2pi - Arcsin(-1/k)

iv) tan(θ) = x,

=> tan(π/2 - θ) = 1 / tan(θ) = 1 / x

This can be seen if you draw a right-angled triangle, with angles: θ and π/2 - θ, then use the definition: tangent = opp/adj

v) What do you mean? What sort of question? If it tells you to describe the transformations, yes you can. I think the preferred terminology is "translation -5 units parallel to y-axis (down)"