Answer all the questions.

Q1. Consider a parallelogram with vertices A, B, C and D.

Show that the area of triangle ABD is equal to the area of the triangle ABC.

Q2. Show that if A(a₁, a₂ , a₃ ), B(b₁,b₂ ,b₃ ) and C(c₁, c₂ , c₃) are three fixed points on a plane and P(x, y, z) is a general point on the same plane, then

Q3. Two particles move in a plane, both starting from the origin, in such a way that, after time t seconds, their position vectors are 3(cos t - 1)i + (4sin t) j and (3sin t)i + 4(cos t - 1) j . Find the value of t when the particles are first moving in the opposite direction.

Q4. Two particles P and Q move in the positive direction on the x-axis, P with constant acceleration 2 m/s² and Q with constant acceleration of 1 m/s². At time t = 0, P is projected from the origin with speed 1 m/s and at time t = 4, Q is projected from O with speed 16 m/s. Find the times between which Q is ahead of P. Find also the distance from O at which Q overtakes P, and the distance from O at which P overtakes Q.

Q5. Particle A is located at the Origin, while particle B is located 10 units due north of A. B is moving with velocity vB = 2i - 5j.

(a) If A can move with speed 2 unit/s, in what direction should A move in order to intercept with B? How long does the interception take?

(b) If A move with velocity v_A = 4i + j, what is the shortest distance between A and B?