grandes-ecoles 2010 QIII.A.3

grandes-ecoles · France · centrale-maths1__mp Complex Numbers Argand & Loci Similarity, Rotation, and Geometric Transformations in the Complex Plane

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\mathbf{C}$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$.
a) Show that $f\left([0,1]\cap\mathbf{Z}\left[\frac{1}{2}\right]\right)\subset\tau$.
b) Show that $f([0,1])\subset\tau$.

We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1])\subset\mathbf{C}$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$.

a) Show that $f\left([0,1]\cap\mathbf{Z}\left[\frac{1}{2}\right]\right)\subset\tau$.

b) Show that $f([0,1])\subset\tau$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.1 QII.2 QII.3 QII.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.A.6 QIII.B.1 QIII.B.2