Recent questions tagged tifrmaths2022

Answer whether the following statements are True or False.If $X$ is a connected metric space, and $F$ is a subring of $C(X, \mathbb{R})$ that is a field, then every eleme...

Answer whether the following statements are True or False.Let $K \subseteq[0,1]$ be the Cantor set. Then there exists no injective ring homomorphism $C([0,1], \mathbb{R})...

Answer whether the following statements are True or False.There exists a metric space $(X, d)$ such that the group of isometries of $X$ is isomorphic to $\mathbb{Z}$.

Answer whether the following statements are True or False.Let $A \subset \mathbb{R}^{2}$ be a nonempty subset such that any continuous function $f: A \rightarrow \mathbb{...

Answer whether the following statements are True or False.Every subring of a unique factorization domain is a unique factorization domain.

Answer whether the following statements are True or False.Let $f_{1}, f_{2}, f_{3}, f_{4} \in \mathbb{R}[x]$ be monic polynomials each of degree exactly two. Then there e...

Answer whether the following statements are True or False.There exists a finite abelian group $G$ such that the group Aut$(G)$ of automorphisms of $G$ is isomorphic to $\...

Answer whether the following statements are True or False.There exists an integral domain $R$ and a surjective homomorphism $R \rightarrow R$ of rings that is not injecti...

Answer whether the following statements are True or False.There exists $f \in C([0,1], \mathbb{R})$ satisfying the following two conditions:$\displaystyle{}\int_{0}^{1} f...

Answer whether the following statements are True or False.Let $f:[0,1] \rightarrow[0, \infty)$ be continuous on $[0,1]$ and twice differentiable in $(0,1)$. If $f^{\prime...

Answer whether the following statements are True or False.The set $\{f(x) \in \mathbb{R}[x] \mid f(n) \in \mathbb{Z}$ for all $n \in \mathbb{Z}\}$ is uncountable.

Consider the following properties of a metric space $(X, d)$ :$(X, d)$ is complete as a metric space.For any sequence $\left\{Z_{n}\right\}_{n \in \mathbb{N}}$ of closed ...

Consider the following assertions:$\left\{(x, y) \in \mathbb{R}^{2} \mid x y=1\right\}$ is connected.$\left\{(x, y) \in \mathbb{C}^{2} \mid x y=1\right\}$ is connected.Wh...

What is the number of solutions of: \[ x=\frac{x^{2}}{50}-\cos \frac{x}{2}+2 \] in $[0,10]?$ $0$$1$$2$$\infty$

Let $A$ be an element of $\mathrm{M}_{4}(\mathbb{R})$ with characteristic polynomial $t^{4}-t$. What is the characteristic polynomial of $A^{2}$ ?$t^{4}-t$$t^{4}-2 t^{3}+...

Let $n$ be a positive integer, and let $V=\{f \in \mathbb{R}[x] \mid \operatorname{deg} f \leq n\}$ be the real vector space of real polynomials of degree at most $n$. Le...

Let $T$ be the linear transformation from the real vector space $\mathbb{R}[x]$ to itself, given by $T(f)=f^{\prime}$, where $f^{\prime}$ is the derivative of $f$. Consid...

What is the largest value of $n$ for which there exists a set $\left\{A_{1}, \ldots, A_{n}\right\}$ of (distinct) nonzero matrices in $\mathrm{M}_{2}(\mathbb{C})$ such th...

Let $p$ be a prime number. What is the number of elements in the group $\mathbb{Q} / \mathbb{Z}$ that have order exactly $p?$$0$$p-1$$p$$\infty$