A positively charged particle placed on a frictionless table by attaching it to a string fixed at one

point of magnetic field is switched on in the vertical direction, the tension in the string will

remain unchanged as the particle is not in motion, hence no force acts on it.

Magnetic force exerts force on a moving particle only and not on stationary charge. By the situation given in your problem it is not very clear whether the particle is moving or not. If it is stationary, there will not be any force hence the tension will remain same.