grandes-ecoles 2015 QII.B.2

grandes-ecoles · France · centrale-maths2__psi Groups Group Homomorphisms and Isomorphisms

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ By which geometric transformation(s) is the set $\Omega_{x_0, y_0, \lambda}$ the image of $\Omega$? Justify that $\Omega_{x_0, y_0, \lambda}$ is an open set of $\mathbb{R}^2$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define:
$$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$
By which geometric transformation(s) is the set $\Omega_{x_0, y_0, \lambda}$ the image of $\Omega$? Justify that $\Omega_{x_0, y_0, \lambda}$ is an open set of $\mathbb{R}^2$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.D.1 QII.D.2 QII.D.3 QII.D.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2