Step 1: The force on a moving charge.
A charge in a magnetic field feels \[ \vec F=q(\vec v\times\vec B). \] The direction of this force decides what happens to the motion.
Step 2: Note the directions given.
Velocity $\vec v=3\hat i+4\hat j$ lies in the $x$-$y$ plane, while $\vec B=5\hat k$ points along $z$. They are at right angles, so the force is non-zero.
Step 3: A useful general fact.
The magnetic force is always perpendicular to $\vec v$ (since $\vec v\times\vec B$ is perpendicular to $\vec v$). A force perpendicular to velocity does no work.
Step 4: Effect on speed.
No work means no change in kinetic energy, so the speed stays exactly the same.
Step 5: Effect on direction.
Even though speed is constant, the sideways force keeps bending the velocity. So the direction, and hence the path, keeps changing.
Step 6: Combine the two results.
Speed unchanged, but path curved. The correct statement is that only the path changes.
\[ \boxed{\text{Its path will change (speed stays the same)}} \]