A) Remember that velocity is the derivative of the particle with relation to time. Since we're doing a parametric equation, that will be (dx/dt, dy/dt).
dx/dt = 1 + cos(t)
dy/dt = 1 + sin(t)
So the velocity of the particle is (1+cos(t), 1+sin(t)).
B) Speed is just the positive part of velocity. So we need to determine where, if ever, the velocity is negative. Since the sin and cos functions alternate between -1 and 1, the minimum velocity in the (x, y) directions is at minimum 0 for each variable. This means the speed of the particle is the same as the velocity (1+cos(t), 1+sin(t)).
C) For the particle to come to a complete stop, both dx/dt and dy/dt must be equal to zero. This can be shown to never be the case because dx/dt=0 when t=π+2nπ and dy/dt=0 when t=3π/2+2nπ. Since these are never equal, there is no point in time when the particles have stopped altogether.
