What is the ruling on using an automatic washing machine to wash clothes on which Quranic verses have been written for the purpose of cheating in an exam, knowing that the verses will be removed by washing, and is this considered a degradation of the Holy Quran?

Cheating on exams is forbidden and a reprehensible act, due to the Prophet's (peace and blessings be upon him) saying: "Whoever cheats us is not one of us." What your brother did, cheating and writing the Quran on trousers, is a grave reprehensible act that indicates a lack of reverence for the word of Allah. The scholars have disliked writing the Quran on clothes, walls, and anything that is exposed to disrespect.

As for how to deal with the trousers on which Quranic verses are written, they must be washed alone with pure water, away from impurities. The wash water should be poured into a pure place, as it is blessed water. Ibn Taymiyyah indeed stated the permissibility of washing the Quran and drinking it for healing.

Your disapproval of your brother's actions is correct, but you should advise him gently and wisely.