Du’a for Exams: Authentic?
Question

What is the ruling on reciting some du`as before exams and afterwards, such as the du`a said when starting to revise, “O Allah I ask You for the understanding of the Prophets and the memory of the Messengers and angels who are close to You, make my tongue fresh with Your remembrance and my heart humble to You and obedient to You, for You are sufficient for me and You are the best disposer of affairs.” And when going out of the house and heading towards the exam, can I say, “O Allah, in You I have put my trust and I submit my affairs to You, and there is no refuge from You except with You.” And at the end of the exam: “Praise be to Allah Who has guided us and we could not have been guided had not Allah guided us.” And when stumbling over an answer: “There is no god but You, glory be to You, verily I was one of the wrongdoers. O Ever-Living, O Sustainer, by your mercy I seek help,” and many others.
Praise be to Allah.Du’a for exams: Authentic?
There is no report in the Sunnah which suggests any du`a for exams. The du`as that are widespread among students and are said during revision or when receiving the exam papers or when stumbling over an answer or when handing in the paper, etc., all have no basis in the Sunnah, either in authentic or weak reports. Rather they are all fabricated and are falsely attributed to the Prophet (peace and blessings of Allah be upon him). 
Prohibition of telling lies against the Prophet
Attributing any such thing to the Sunnah when one knows that it is not part of the Sunnah makes a person one of those who tell lies against the Prophet (peace and blessings of Allah be upon him). 
Al-Mughirah ibn Shu’bah (may Allah be pleased with him) said: I heard the Prophet (peace and blessings of Allah be upon him) say: “Telling a lie against me is not like telling a lie against anyone else. Whoever tells a lie against me deliberately, let him take his place in Hell.” (Narrated by al-Bukhari, 1229; Muslim, 4)
Whoever spreads these false hadiths and others like them, knowing that they cannot be soundly attributed to the Prophet (peace and blessings of Allah be upon him) is telling lies against the Prophet (peace and blessings of Allah be upon him) and his sin is the same as the sin of the one who fabricated them. 
It was narrated from al-Mughirah ibn Shu’bah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever narrated from me a hadith which he thinks is false is one of the liars.” (Narrated by Muslim in the Introduction to his Sahih)
Du’a at during times of hardship and difficulty
There is a proven du`a for leaving the house , but it is not as described by the questioner. The phrase is “Ya Hayyu ya Qayyum bi rahmatika astaghith (O Ever Living, O Sustainer, by Your mercy I seek help).” 
There are hadiths which indicate that this is to be said at times of hardship and difficulty. All of that includes exams and other difficulties . There follow the hadiths which have been narrated concerning that:  

Anas ibn Malik (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever says – when leaving his house – Bismillah, tawakkaltu ‘ala Allah, la hawla wa la quwwata illa Billah (In the name of Allah, I put my trust in Allah, there is no power and no strength except with Allah), it will be said to him: You are taken care of and you are protected, and the Shaytan will keep away from him.” (Narrated by Abu Dawud, 5095; al-Tirmidhi, 3426; classed as sahih by al-Albani in Sahih Abi Dawud)
Anas ibn Malik (may Allah be pleased with him) said: If something upset him, the Prophet (peace and blessings of Allah be upon him) would say: “Ya Hayyu ya Qayyum bi rahmatika astaghith (O Ever Living, O Sustainer, by Your mercy I seek help).” (Narrated by al-Tirmidhi, 3524; classed as hasan by al-Albani in al-Silsilah al-Sahihah, 3182) 
Anas (may Allah be pleased with him) reported that the Prophet (peace and blessings of Allah be upon him) said: “O Allah, nothing is easy but that which You make easy and You can make hardship easy if You will.” Narrated by Ibn Hibban, 3/255; classed as sahih by al-Albani in al-Silsilah al-Sahihah, 2886. 
Sa’d ibn Abi Waqqas (may Allah be pleased with him) said: The Prophet (peace and blessings of Allah be upon him) said: “The prayer of Dhu’l-Nun when he was in the belly of the fish: ‘La ilaha illa Anta, subhanaka inni kuntu min al-zalimin (There is no god but You, glory to You, verily I was one of the wrongdoers).’ A Muslim never calls upon his Lord with these words concerning any matter, but his prayer is answered.” (Narrated by al-Tirmidhi, 3505; classed as sahih by al-Albani in al-Silsilah al-Sahihah, 1644)

To sum up, it is not permissible to invent du`as and attribute them to Islam. What has been narrated in the hadiths that may be said during times of hardship and difficulty is sufficient.
For more, please see this category: Supplications prescribed in Shari’ah
And Allah knows best.