When a charged body is placed near the conductor, the opposite charge is appeared on the one side of the uncharged conductor body near the charged body under the effect of induction.
The opposite charge on the another body is also known as the induced charge.
The perpex rod in the given question is charged.
let the charge on the perpex is positive as shown in the figure.
Then, the polystrene sphers (insulator uncharged sphere) will not have any charge induced as the charges are not free to move inside the insulator.
Thus, when a charged perspex rod is placed close to a suspended polystyrene sphere then the induced charge on the sphere is not observed under the phenomenon of induction process and sphere remain uncharged.