How can cosmic jets exist? Cosmic jets are jets of matter ejected from super massive black holes.
The gravitational pull of a black hole is so high that even light can't escape it.
Yet particles with mass traveling at speeds slower than the speed of light are ejected out.
Why does this paradox exist?
Is the laws of physics violated?
(A humble doubt from a layman)
 A: The jets are not ejected from inside the black hole. They are ejected from the accretion disk outside the event horizon. There is no paradox, and no violation of the laws of physics. 
A: The models for astrophysics jets of particles depend on the topology of black holes and other heavy objects like neutron stars. These have what is called  an accretion disk :

An accretion disk is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a massive central body. The central body is typically a star. Friction causes orbiting material in the disk to spiral inward towards the central body. Gravitational and frictional forces compress and raise the temperature of the material, causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object's mass. Accretion disks of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the X-ray part of the spectrum. The study of oscillation modes in accretion disks is referred to as diskoseismology.

The existence of accretion disks also is used to model astrophysical jets of matter:

The formation and powering of astrophysical jets are highly complex phenomena that are associated with many types of high-energy astronomical sources. They likely arise from dynamic interactions within accretion disks, whose active processes are commonly connected with compact central objects such as black holes, neutron stars or pulsars. One explanation is that tangled magnetic fields are organised to aim two diametrically opposing beams away from the central source by angles only several degrees wide (c. > 1%). Jets may also be influenced by a general relativity effect known as frame-dragging

The accretion disk is kinetically outside the no return horizon of the black hole, and  matter composing it can gain  enough energy to form into jets, given a specific topology. Mathematical models exist for this.
