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Senior two math problems. Urgent. High salary.
1, let BD∩AC=O,

∵ quadrilateral ABCD is a square,

∴AC⊥BD,

∵PD⊥ plane ABCD, AC∈ plane ABCD,

∴AC⊥PD,

∫PD∩BD = D,

∴AC⊥ aircraft PDB,

∫AC∈ plane ACE,

Aircraft, ACE⊥. Aircraft.

2. link EO,

To sum up, the AC(O)⊥ plane PDB,

∴ < AEO is the angle formed by AE and plane PDB,

AB = AD = PD = a,

∴BD=√2a,

PB=√(a^2+2a^2)=√3a,

BE=BP/3=√3a/3,

On the plane PBD, EF⊥BD, vertical foot h,

EH/PD=BE/PB= 1/3,

∴EF=a/3,

OB=BD/2=√2a/2,

BH/BD=BE/PB= 1/3,

∴BH=√2a/3,

OH=OB-BH=√2a/6,

OE=√(OF^2+EF^2)=√6a/6,

ae=√(ao^2+oe^2)=√(2a^2/4+6a^2/36)=√6a/3,

∴cos<; AEO = EO/AE =(√6a/6)/(√6a/3)= 1/2

The angle between AE and PDB plane is 60 degrees.

3.∫DF// plane ACE,

And plane DFB∩ plane ACE=OE,

∴DF//OE,

O is the midpoint of BD,

∴OE is the center line of △BDF,

∴EF=BE=PB/3=√3a/3,

∴pf=pd-ef-be=pd-2be=√3a-2√3a/3=√3a/3.