Which three-digit number, made of consecutive digits, like 567, is 2 less than a cube and 2 more than a square?

During a secret mission, an agent gave the following code to the higher authorities
AIM DUE OAT TIE MOD

However, the information is in one word only and the rest are fake. To assist the authorities in understanding better, he also sent them a clue, If I tell you any one character of the code, you can easily find out the number of vowels in the codeword.

Can you find out the code word?

John was piloting a plane behind a car but was never able to overtake it. Why?

A person was found dead on a big green land with no trees or buildings around. When the police arrived, they found a piece of rock lying next to him and the man was wearing clothes comprising of red, blue and yellow colours.

Can you tell who the person is and what was the cause of death?

A man was found murdered on Sunday morning. His wife immediately called the police. The police questioned the wife and staff and got these alibis: The Wife said she was sleeping. The Cook was cooking breakfast. The Gardener was picking vegetables. The Maid was getting the mail. The Butler was cleaning the closet. The police instantly arrested the murderer. Who did it and how did they know?

What is the word or phrase?

What always comes into a house through the keyhole?

The following four words might seem abrupt to you. But you can make them meaningful by just adding vowels.
RDSH
GGPLNT
TMT
NN

Hint: All of them are vegetables.

Find the next number in the series?
2, 12, 36, 80, 150, 252, 392, _ ?

Alpha is Beta's sister.

Gamma is a Beta mother.

Delta is Gamma's father.

Tango is Delta's mother.

Then,

How is Alpha related to Delta?