Which you may already know,there are 3 sub exams in total whlie taking the license exam.
They are writen exam,practice exam I and practice exam II.
If you score over 50 points in total,you pass the exam.However,if you score enough high,you can fight against il and if you win,you can fight against sophie.If you beat sophie,the achievement "Master vs Apprentice!" is unlocked.You can continue your story even you failed to beat il or sophie,however you can not unlock the achievement mentioned,obviously.

An easy way to score that high points is to forget about writen exam and practice exam II and to synthesise a 999 quality item in the practice exam I.

The result is shown in the screenshot below:
(200pts is far more enough to have the opptunity to fight against sophie.)

So how to synthesise a item of 999 quality in the Practice Exam I?

1. The following materials are required:
   
   • 2 hakurei stone
   and you need at least following number of materials have certain traits:
   
   • 1 quality up
     
   • 3 quality up+
     
   • 2 quality up++
     
   • optional:another quality up++
   !NOTE: The aim of the need of the traits listed above is to compose "well made","pro perfection"and "super quality".If you already have one of more of those,the actual amout required of the listed traits would decrease.
   
2. How to compose traits:For instance,while you synthesise something,if some materials have the trait of "quality up" and "quality up+"(does not matter whether in the same material),you can neither transfer "quality up" nor "quality up+" but "well made"to the synthesis.
   
3. So,compose "well made"using"quality up" and "quality up+";"pro perfection"using"quality up+" & "quality up++";"super quality"using "well made" & "pro perfection".Use your way to synthesise some "distilled water"with the traits of "super quality" , "pro perfection" and "quality up++"(optional).
   
4. Use the "distilled water" with those traits as material to synthesise "distilled water".The other quality of other material may be low but it does not matter at all.Transfer those 3(or2) traits to the new-made "distilled water" you would found that the quality is significantly higher.(Hint:You may use "reactive catalyst" as catalyst to increase 2 transferable trait slots.The "reactive catalyst" can be leart in the village of "North Auoro Snowfield",from the local licesed alchemist,Kirsche.)
   
5. Repeat step 4 until you have 999 quality distilled water.
   
6. Use some of the 999 distilled water to synthesise some neutralizer,any color is ok,however blue is recommanded.(You must keep at least 2 999 distilled water,they are used in the exam.)
   
7. Sleep to the exam day.Synthesise a 999 quality "Ice Bomb" in the exam.
   (Note:Step 7 is NOT redundant because synthesise 999 quality neutralizer in the exam will only score 20 points.)

About Battle Against Sophie
Really do not have much to say.Just farm some levels and bring some bombs and medicine.And you may switch difficuly to "easy" before the exam starts.