Recent questions tagged tifrmaths2023

Answer whether the following statements are True or False.Let $\alpha$ be a positive real number, and let $f:(0,1) \rightarrow \mathbb{R}$ be a function such that $|f(x)-...

Answer whether the following statements are True or False.Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions such that $f^{2}+g^{2}$ is uniformly ...

Answer whether the following statements are True or False.Let $A, B \in \mathrm{M}_{2}(\mathbb{Z} / 2 \mathbb{Z})$ be such that $\operatorname{tr}(A)=\operatorname{tr}(B)...

Answer whether the following statements are True or False.Let $v_{1}, v_{2}, w_{1}, w_{2}$ be nonzero vectors in $\mathbb{R}^{2}$. Then there exists a $2 \times 2$ real m...

Answer whether the following statements are True or False.Let $A=\left(a_{i j}\right) \in \mathrm{M}_{n}(\mathbb{R})$ be such that $a_{i j} \geq 0$ for all $1 \leq i, j \...

Answer whether the following statements are True or False.If $A \in \mathrm{M}_{4}(\mathbb{Q})$ is such that its characteristic polynomial equals $x^{4}+1,$ then $A$ is d...

Answer whether the following statements are True or False.There exists a nonabelian group $G$ of order $26$ such that every proper subgroup of $G$ is abelian.

Consider the following properties of a sequence $\left\{a_{n}\right\}_{n}$ of real numbers.$\text{(I)}\displaystyle{} \lim _{n \rightarrow \infty} a_{n}=0$.$\text{(II)}$ ...

Consider the functions $f_{1}, f_{2}:(0, \infty) \rightarrow \mathbb{R}$ defined by\[f_{1}(x)=\sqrt{x}, \quad \text { and } \quad f_{2}(x)=\sqrt{x} \sin x .\]Which of the...

Let $T: \mathrm{M}_{3}(\mathbb{R}) \rightarrow \mathbb{R}^{3}$ be the linear map defined by $T(A)=A\left(\begin{array}{c}1 \\ 0 \\ -1\end{array}\right)$. Then the dimensi...