Hi all,

There are now solutions available for the mixed STEP I Pure questions.

in the hints and partial solutions for question 3iia. it is stated that there is only one solution to this question, however when I completed the question I ended up with x=2 and x=142. I plugged both into the original equation and x=2 works and when plugging in x=142 if the negative square root of 144 is taken instead of the positive, this solution also works. Have I made a mistake?

Okay, I've come across a method which may or may not be valid and doesn't involve the adding of fractions (I don't think it does, anyway).

Essentially, I label my two n's n1 and n2.

n1 can be in m + 1 positions, since it cannot be at the start but can be anywhere else. So n1 goes with m + 1

n2 cannot go right next to n1, so its range of positions is m - 1.

So I basically said that the total range of spaces for the n's is m + 1, making that the denominator. I said that since it can only work with m - 1 spaces, so this is the numerator.

Hello,

I have worked through the warm up question in assignment 2 but am slightly unsure about part (iv).

I took the difference of squares and got x^4 + x^2 - 2x + 1. However I am not sure how to use this to "hence" derive the solutions for x^4 + 1 = 0.

Any guidance is appreciated!

For assignment 9, although I see where all the values come from, I am struggling to see how (27b^2) + (4a^3b) < 0 implies that there are 3 distinct real roots.

If I use the cubic x^3 - 4x + 4 then

(27b^2)+(4a^3)(b) = -592, which satisfies the above inequality, yet there exists only one distinct real root?

I'm struggling to understand the difference between "if", "if and only if" and "only if"

to my understanding :

"A if B" means B⟹A.
"A if and only if B" means A⟺B so A implies B and B implies A.
Both of these make sense.

A only if B means "only if B is true, A is true", so B⟹A. But if A is true, as B being true is the only statement that makes A true, B must be true?...so A⟹B as well? which is the same as "if and only if"?

At the end, they have the STEP question on probability but I was unable to actually find a solution which uses the factorial method instead of the 'n choose r' method, if anyone has a solution that works, please post it;
Question is as follows, I seat n boys and 3 girls in a line at random, so that each order of the n + 3 children is as
likely to occur as any other. Let K be the maximum number of consecutive girls in the line
so, for example, K = 1 if there is at least one boy between each pair of girls.
i) Find P(K=3)
(ii) Show that

If you don't know the question, you can find it online. The whole book is free to download. It involves sequences and proofs.

In part ii, we're given a few equations and it says that we should find and prove a general rule wherein the equations below are special cases.

However, the answers bit doesn't tell me anything about how I get to the equation. It tells me how to prove it, but I have no clue on how to get to the equation in the first place. Could anyone give some insight? I am completely lost.

I've got two things here. One about the assignment, the other about STEP preparation in general.

First, I've done a thing which to me seems to work, but I don't know if it qualifies as an answer. It's not one which is listed, but still.

So, the RHS of the answer is shown to be 5 - 4a. However, after going through the question and then checking what I did with the hints, I found the RHS to be 4 - 4a. Perhaps this is a mistake on my part, but I can't see where I've made any sort of mistake.