## Roots of $f(f(..f(x)..))$, where $f(x) = ax^2 + bx + c$, are symmetric about $\frac{-b}{2a}$

Posted on Thu 30 April 2020 in Mathematics

Define $f(x) = ax^2 + bx + c , a,b,c \in \mathbb{R}$ and $f^n(x) = f(f^{n-1}(x)), n>1$. Prove that the real roots of $f^n(x)$ are symmetric about the vertical line passing through vertex i.e. $x = \frac{-b}{2a}$

This seems like …

## An integral involving ζ(2) (And Euler's first proof for the Basel Problem)

Posted on Wed 29 April 2020 in Mathematics

Evaluate the integral $$\int_0^1 \frac{\log(x)}{x-1}dx$$

There are a few methods of doing this: the first one uses the taylor series expansion of $$\log(1-x)$$: $$I = -\int_0^1 \frac{\log(1-x)}{x}dx$$ I = \int_0^1 \frac 1x \left( x + \frac{x^2}{2} + \frac{x …

## ICSE Mathematics - Last 23 years analysis and 2018 forecast

Posted on Tue 09 January 2018 in Mathematics

A speculative format for the 2018 ICSE Mathematics paper, created by analysis of the mathematics question papers of the past 23 years.