Recent questions tagged tifrmaths2025

There are three boxes. One box contains ten red socks, another contains ten blue socks, and the third contains a mix ve red and ve blue socks. The three boxes are labelle...

What is the number of functions $f: \mathbb{R} \backslash\{0,1\} \rightarrow \mathbb{R} \backslash\{0,1\}$ such that $|f(x)-f(y)|=|x-y|$ for all $x, y \in \mathbb{R} \bac...

Let $x_{n}=\left(1-\frac{1}{\sqrt{n}}\right)^{n} \exp \left(n^{\frac{1}{4}}\right)$. Which of the following statements is correct?$\displaystyle \lim _{n \rightarrow \inf...

Let $f:[0,1] \rightarrow \mathbb{R}$ be such that\[f(x)=\left\{\begin{array}{ll}\frac{1}{2^{n}}, & \text { if } x=\frac{p}{2^{n}}, \text { where } p \text { is an odd int...

Consider the linear map $T: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ defined by $T(p(x))=$ $\dfrac{d}{d x}(x p(x))$. Which of the following statements is correct?$T$ is i...

Let\[A=\left(\begin{array}{lll}1 & 2 & 1 \\2 & 1 & 2 \\1 & 2 & 1\end{array}\right)\]Consider the following statements:$A$ has a positive eigenvalue.$A$ has a negative eig...

Let $A$ be a nonzero $n \times n$ matrix with real entries, where $n>1$. Consider the following statements.If $A^{2}=0$ then $\operatorname{rank}(A) \leqslant\left\lfloor...

Consider the following subset of $\mathbb{R}^{2}$ :\[S=\left\{\left.\left(x, \sin \frac{1}{x}\right) \right\rvert\, x>0\right\} \cup\left\{\left.\left(0, \pm \frac{1}{n}\...

Consider the following statements:For any linear map $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ with a two-dimensional kernel, the trace of $T$ is an eigenvalue of $T...

For a positive integer $n$, define the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ by $f_{n}(x)=$ $\sin \left(\frac{x}{n}\right)$. Consider the sequences $\left\{...

Let $n>1$ be a positive integer. Let $A=\left[a_{i j}\right]_{1 \leqslant i, j \leqslant n}$ be an $n \times n$ matrix, such that $a_{i, i+1}=c_{i}$ for $i=1, \ldots,(n-1...

Let $S O_{4}=\left\{A \in \mathrm{M}_{4}(\mathbb{R}) \mid A A^{t}=\mathrm{Id}\right.$, and $\left.\operatorname{det}(A)>0\right\}$. Then every matrix in $\mathrm{SO}_{4}$...