Two good syllabus questions

1

Quadratic Equations

If α, β, γ, δ are the four positive roots of the equation x⁴ -12x³+ax²+bx +81=0, then find “a” and “b”.

2

Permutations and Combinations

Find all possible arrangements of the word "COMBINATORICS" such that all the vowels are in alphabetical order and consonants are also in alphabetical order.

Note: Consonants and vowels to be considered separately.

Note: These are not puzzles but JEE-level questions

What's probable & what's not?

!!Please comment on the correctness of this question!!

In a lucky draw competition, a contestant had to draw 3 tickets from a pile of well-shuffled tickets. If the numbers on these tickets were in arithmetic progression, he would be adjudged the winner of the contest. If the numbers were not in arithmetic progression, he would return them to the pile. The tickets were numbered from 1 to 201. What is the minimum number of contestants to ensure that at least one of them wins the contest? If 100 people took part in the lucky draw, do you think we could take for granted that at-least one contestant would win the prize?

The Consecutive Numbers

A Math teacher, in order to keep two of her students occupied thought of two consecutive numbers in the range 1 to 10, and told one of the students Alex one of the numbers and the other student, Sam the other.

Sam and Alex then had the following conversation:

Alex: I don't know your number.
Sam: I don't know your number, either.
Alex: Now I know!

How many such sets of numbers can you find? Can you find all the sets which could have been thought up by the Math teacher?