How to use Serre spectral sequence to compute cup product structures? 

Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are zero. Can we obtain that for any field $k$,
$$
H^*(E;k)\cong H^*(F;k)\otimes H^*(B,k)
$$
as rings?

For example, how to solve the following problem?

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ixANp.jpg