Which metal will spontaneously react with zn2+(aq), but will not spontaneously react with mg2+(aq)?
(1) mn(s)(2) cu(s)
(3) ni(s)(4) ba(s)

Correct option (1) Mn(s)

Reason: 
The spontaneity of electrochemical cell, depends on the it's Eo value. Electrochemical cells with positve Eo are spontanous and vice-versa.

In present case, the  Eo of half-cell of interest are as follows:
Eo Zn2+/Zn = -0.763v
Eo Mg2+/Mg = 2.37v
Eo Mn2+/Mn = -1.18v

Therefore, Eo cell (with Zn as one of the half-cell) =  Eo Zn2+/Zn - Eo Mn2+/Mn
                                                                                =  -0.763 - (-1.18)
                                                                                = 0.417v

On other hand, Eo cell (with Mg as one of the half-cell) =  Eo Mg2+/Zn - Eo Mn2+/Mn
                                                                                =  -2.37 - (-1.18)
                                                                                = -1.19v

Thus, Mn(s) metal will spontaneously react with Zn2+(aq), but will not spontaneously react with Mg2+(aq)