UFM Additional Further Pure

Load questions...

Load questions...

Load questions...

View all 328 questions →

grandes-ecoles 2023 Q20 Ring and Field Structure View
Let $A$ be an algebraic $\mathbb{R}$-algebra without zero divisors. a) Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. b) Show that if $x \in A \setminus \mathbb{R}$, then $\mathbb{R} + \mathbb{R}x$ is an $\mathbb{R}$-algebra isomorphic to $\mathbb{C}$.
grandes-ecoles 2023 Q21 Ring and Field Structure View
We assume that $A$ is not isomorphic to one of the algebras $\mathbb{R}$ or $\mathbb{C}$. Show that there exists $i_A \in A$ such that $i_A^2 = -1$.
grandes-ecoles 2023 Q22 Algebra and Subalgebra Proofs View
We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map $$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$ We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$. a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$. b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.
grandes-ecoles 2023 Q23 Algebra and Subalgebra Proofs View
Show that $\ker(T + \mathrm{id}) = U$ and deduce that $\ker(T - \mathrm{id}) \neq \{0\}$.
grandes-ecoles 2023 Q24 Algebra and Subalgebra Proofs View
We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$. a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$. b) Show that $\beta^2 \in ]-\infty, 0[$. c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.
grandes-ecoles 2023 Q25 Algebra and Subalgebra Proofs View
Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Let $x, y \in A$ such that $xy = yx$ and such that $V = \mathbb{R}x + \mathbb{R}y$ is of dimension 2 over $\mathbb{R}$. Show that $$\forall u, v \in V \quad \|u+v\|^2 + \|u-v\|^2 \geq 4\|u\| \cdot \|v\|$$ and that the restriction of $\|\cdot\|$ to $V$ comes from an inner product on $V$.
grandes-ecoles 2023 Q26 Algebra and Subalgebra Proofs View
Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.
grandes-ecoles 2023 Q27 Ring and Field Structure View
Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Conclude that $A$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (Theorem C).
grandes-ecoles 2024 Q5 Binary Operation Properties View
For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
  • [(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
  • [(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
  • [(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.
grandes-ecoles 2024 Q8 Subgroup and Normal Subgroup Properties View
Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that $\mathscr{V}(A)$ is nonempty.
grandes-ecoles 2024 Q8a Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Justify that $(\mathbb{C}[A])^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.
grandes-ecoles 2024 Q8b Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.
grandes-ecoles 2024 Q8a Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Justify that $\mathbb{C}[A]^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.
grandes-ecoles 2024 Q8b Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.
grandes-ecoles 2024 Q8 Binary Operation Properties View
For which values of $d$ do we have $gg^{\prime} = g^{\prime}g$ for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$?
grandes-ecoles 2024 Q9 Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.
grandes-ecoles 2024 Q9 Subgroup and Normal Subgroup Properties View
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.
grandes-ecoles 2024 Q9 Group Actions and Surjectivity/Injectivity of Maps View
We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ the vector space of families of $n$ points in $\mathbb{R}^{d}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $g \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ we denote $g \cdot \boldsymbol{z} = (\phi_{g}(\boldsymbol{z}_{i}))_{1 \leqslant i \leqslant n}$.
  • [(a)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we have $g \cdot (g^{\prime} \cdot \boldsymbol{z}) = (gg^{\prime}) \cdot \boldsymbol{z}$.
  • [(b)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, if $\boldsymbol{x} = g \cdot \boldsymbol{y}$ then $\boldsymbol{y} = g^{-1} \cdot \boldsymbol{x}$.
grandes-ecoles 2024 Q11 Group Actions and Surjectivity/Injectivity of Maps View
We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$. For all $\boldsymbol{x} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $c(\boldsymbol{x}) = \{ \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R}) \mid \exists g \in \operatorname{Dep}(\mathbb{R}^{d}), g \cdot \boldsymbol{x} = \boldsymbol{y} \}$.
  • [(a)] Show that if $c(\boldsymbol{x}) \cap c(\boldsymbol{y}) \neq \emptyset$ then $c(\boldsymbol{x}) = c(\boldsymbol{y})$.
  • [(b)] Show that if $c(\boldsymbol{x}) = c(\boldsymbol{y})$ then $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.
isi-entrance 2013 Q73 4 marks Ring and Field Structure View
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a, b \in W$, the numbers $a - b$ and $ab$ are also in $W$. Let $S = \left\{ \left. \frac{m}{2^n} \right\rvert\, m, n \text{ integers} \right\}$ and $T = \left\{ \left. \frac{p}{q} \right\rvert\, p, q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings
isi-entrance 2015 Q28 4 marks Ring and Field Structure View
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:
(a) neither $S$ nor $T$ is a ring
(b) $S$ is a ring, $T$ is not a ring.
(c) $T$ is a ring, $S$ is not a ring.
(d) both $S$ and $T$ are rings.
isi-entrance 2015 Q28 4 marks Ring and Field Structure View
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:
(a) neither $S$ nor $T$ is a ring
(b) $S$ is a ring, $T$ is not a ring.
(c) $T$ is a ring, $S$ is not a ring.
(d) both $S$ and $T$ are rings.
isi-entrance 2016 Q73 4 marks Ring and Field Structure View
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a, b \in W$, the numbers $a - b$ and $ab$ are also in $W$. Let $S = \left\{ \left. \frac{m}{2^n} \right\rvert\, m, n \text{ integers} \right\}$ and $T = \left\{ \left. \frac{p}{q} \right\rvert\, p, q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings
isi-entrance 2016 Q73 4 marks Ring and Field Structure View
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings
jee-main 2025 Q17 Group Order and Structure Theorems View
The number of non-empty equivalence relations on the set $\{ 1,2,3 \}$ is:
(1) 6
(2) 5
(3) 7
(4) 4

Load questions...