The correct options are
B The line joining the positions of the particle at any subsequent time makes a constant angle π2+(α+β2)with the horizontal.
C The magnitude of the relative velocity of the first particle with respect to the second is 2usin(α+β2)
For the first particle x1=ut cos α,y1=ut sin α−12gt2v1x=u cosα;viy=usinα−gt;For the second particle, x2=ut sin β,y2=ut sin β−12gt2v2x=ucosβ;v2y=usin β−gt
Angle made by the line joining the positions of two particles istan θ=y2−y1x2−x1=u(sin α−sinβ)u(cos α−cosβ)=−cot(α+β2)⇒θ=π2+(α+β2)v12x=u(cosα−cosβ) and v12y=u(sinα−sinβ). Hence v12=u√(cosα−cosβ)2+(sinα−sinβ)2 =√2−2cosαcosβ+2sinαsinβ=2usin(α+β2)