Hi,
I have not been able to do the first part of the question. However I have been able to do i,ii & iii. How many marks would I get for this? Thank you.

Do you know good book(s) for preparing for MAT exam?

The last part of Q2 asks us to show that $y=3e^{2x}+14\ln\left(\frac{4}{3}-x\right)$ has a minimum between $x=\frac{1}{2}$ and $x=1$. Normally I would do this by differentiating and subbing in the two $x$ values and looking for a change of sign (from negative to positive). But it seems from the way the question is set out that we're supposed to do it by using the Maclaurin series expansion of $e^x$. Do we also need to use the Maclaurin expansion of $\ln (1+x)$?

I am on the second part of the step Question where I have to find and prove the maximum and minimum values when c>0. Any help would be appreciated thank you.

This is possibly one of the "worst" quadratics to try and solve:
\[
x^2 + ax + b = c \, .
\]

I'll leave it here for a week and then post the solutions.

Task: Factorise $6y^3 -y^2 -21y + 2x^2 + 12x -4xy +x^2y -5xy^2 + 10 $ into 3 linear factors

Workings: let $$ f(x,y) = 6y^3 -y^2 -21y + 2x^2 + 12x -4xy +x^2y -5xy^2 + 10 $$
$$ f(x,-2) = 0 \Rightarrow f(x,y)= (y+2)(ay+cx+d)(by+ex+f) $$

I need help with STEP 1 2003 - Question 3 iii.

I completed the square and found that cos(theta/2) = 1/2 - root3/2.
I have reached the stage where I have that half theta = arcos(1/2-root3/2)
From here I'm not sure how to reach end conclusion.

Thank you.

Hi again everyone!

I've been looking at the STEP specification, I find that it's pretty vague in describing what you actually need to know. I don't know if anyone shares my views in this but this is how I feel. Also, from looking at STEP questions specifically on Sequences & Series, it seems you need to know a lot more than what's on the syllabus.

I would like to go through a list of things and ask for peoples opinion on whether I need to need to know it or not.

Right, the specification says:

Hi - Assignment 20 is now available! The hints and partial solutions is quite long this time, and has undergone several edits. As a result there may well be some typos/stray bits still present which should not be. Please let us know if you find any mistakes!

4x^2 +16xy +y^2 +24x =0. Prove that x is less than or equal to zero and greater than or equal to 2/5, and find similar restrictions for y.

I started by rearranging the above equation to (2x+y)^2 +12x(y+2), and then
(2x+y)^2 =-12x(y+2), so since LHS is positive, then RHS must be positive, but that can only happen if x is less than or equal to zero, (to cancel with the negative 12), or if (y+2) is less than or equal to zero, or y is less than or equal to-2. So I've proved one of the inequalities given. But how do I prove the other one.