x2 + (2m - 1)x + (m - 6) = 0

The nice thing about these type of equation is that there is a formulation for the sum and the product of roots of a specific equation.

The sum of the two roots of a quadratic equation (regardless of whether the roots are real or not) is

“Negative of the coefficient of x divided by the coefficient of x2, or -b/a for a quadratic equation ax2 + bx + c = 0”.

The product of two roots of a quadratic equation (regardless of whether the roots are real or not) is

“The constant term divided by the coefficient of x2, or c/a for a quadratic equation ax2 + bx + c = 0”.

x^2 + (2m - 1)x + (m - 6) = 0

Here, the sum of roots based on the equation is 1 - 2m.

The product of roots based on the equation is m - 6.

Since the roots are defined be x1 and x2,

x1 + x2 = 1 - 2m
x1x2 = m - 6

Since x1 <= -1 and x2 >= 1, x1x2 <= -1. In contrast, there is no limit for x1 + x2.

We need to find numbers such that m - 6 <= -1

So m <= 5

Greatest m is 5.

thank you for your answers! does anyone know the answers to the rest of the questions?

Q29 seems to be a binomial theorem question by setting x as x - 1 + 1 so that

x^5 = [(x - 1) + 1]^5

Will look at it again when I am more free

The binomial coefficients for the expansion are 1, 5, 10, 10, 5 and 1 so I believe that a1 is 5

Will look again at a later time

okay thank you!!

As for the other three questions, I am stuck in the (x + 2)^2017 question for some time already so I Guess I can’t help much for now. If I am suddenly able to come up with the working, I will post them here.

However, if I never post any answers by this week, chances are, I may have forgotten them already.

As for the other question involving marbles, I will look into it again also another time. It falls under the topic “permutations and combinations” in JC2 statistics.