# Recent questions tagged tifrmaths2017 1
Show that the subset $GL_{n} \left ( \mathbb{R} \right)$ of $M_{n}\left ( \mathbb{R} \right )$ consisting of all invertible matrices is dense in $M_{n}\left ( \mathbb{R} \right )$.
2
Let $f$ be a continuous function on $\mathbb{R}$ satisfying the relation $f\left ( f\left ( f\left ( x \right ) \right ) \right )=x \:for\:all \:x \in \mathbb{R}.$ Prove or disprove that $f$ is the identity function.
3
Prove or disprove: the group of positive rationals under multiplication is isomorphic to its subgroup consisting of rationals which can be expressed as $p/q$, where both $p$ and $q$ are odd positive integers.
4
Show that the only elements in $M_{n}\left ( \mathbb{R} \right )$ commuting with every idempotent matrix are the scalar matrices. ($A$ matrix $P$ in $M_{n}\left ( \mathbb{R} \right )$ is said to be idempotent if $P^{2}=P$.)
5
Prove or disprove the following: let $f:X\rightarrow X$ be a continuous function from a complete metric space $\left ( X,d \right )$ into itself such that $d\left ( f\left ( x \right ),f\left ( y \right ) \right )< d\left ( x,y \right )$ whenever $x\neq y$. Then $f$ has a fixed point.
6
How many isomorphism classes of associative rings (with identity) are there with $35$ elements? Prove your answer.
7
Prove or disprove: If $G$ is a finite group and $g,h \in G$, then $g,h$ have the same order if and only if there exists a group $H$ containing $G$ such that $g$ and $h$ are conjugate in $H$.
8
Prove or disprove: there exists $A\subset \mathbb{N}$ with exactly five elements, such that sum of any three elements of $A$ is a prime number.
9
Show that there does not exist any continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ that takes every value exactly twice.
10
For which positive integers $n$ does there exist a $\mathbb{R}$-linear ring homomorphism $f:\mathbb{C}\rightarrow M_{n}\left (\mathbb{R} \right)$? Justify your answer.
11
True/False Question : Let $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ be a continuous function such that $f\left ( x \right )\geq x^{3}$ for all $x \in \left [ 0,1 \right ]$ with $\int_{0}^{1}f\left ( x \right )dx=\frac{1}{4}$. Then $f\left ( x \right )=x^{3}$ for all $x \in \mathbb{R}$.
12
True/False Question : Suppose $a, b, c$ are positive real numbers such that $\left ( 1+a+b+c \right )\left ( 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right )=16.$ Then $a+b+c=3.$
13
True/False Question : There exists a function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying, $f\left ( -1 \right )=-1, f(1)=1$ and $\left | f\left ( x \right )-f\left ( y \right )\left | \leq \right |x-y \right |^{\frac{3}{2}},$ for all $x,y \in \mathbb{R}.$
14
True/Flase Question : Over the real line, $\underset{x\rightarrow \infty }{lim}\: log\left ( 1+\sqrt{4+x} -\sqrt{1+x}\right )=log\left ( 2 \right ).$
15
True/False Question : Suppose $f$ is a continuously differentiable function on $\mathbb{R}$ such that $f\left ( x \right )\rightarrow 1$ and ${f}'\left ( x \right )\rightarrow b$ as $x\rightarrow \infty$. Then $b=1.$
16
True/False Question : If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and bijective, then $f^{-1}$ is also differentiable.
17
True/False Question : Let $H_{1},H_{2},H_{3},H_{4}$ be four hyperplanes in $\mathbb{R}^{3}$. The maximum possible number of connected components of $\mathbb{R}^{3}-\left (H_{1}\cup H_{2}\cup H_{3}\cup H_{4} \right )$ is $14.$
18
True/False Question : Let $n\geq 2$ be a natural number. Let $S$ be the set of all $n\times n$ real matrices whose entries are only $0,1$ or $2$. Then the average determinant of a matrix in $S$ is greater than or equal to $1$.
19
True/False Question : For any metric space $\left ( X,d \right )$ with $X$ finite, there exists an isometric embedding $f:X\rightarrow \mathbb{R}^{4}$.
20
True/False Question : There exists a non-negative continuous function $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ such that $\int_{0}^{1}f^{n}dx\rightarrow 2$ as $n\rightarrow \infty.$
21
True/False Question : There exists a subset $A$ of $\mathbb{N}$ with exactly five elements such that the sum of any three elements of $A$ is a prime number.
22
True/False Question : There exists a finite abelian group $G$ containing exactly $60$ elements of order $2$.
23
True/False Question : Let $\alpha ,\beta$ be complex numbers with non-positive real parts. Then $\left | e^{\alpha } -e^{\beta }\right |\leq \left | \alpha -\beta \right |.$
24
True/False Question : Every $2 \times 2$ matrix over $\mathbb{C}$ is a square of some matrix.
25
True/False Question : Under the projection map $\mathbb{R}^{2}\rightarrow \mathbb{R}$ sending $\left ( x,y \right )$ to $x$, the image of any closed set is closed.
26
True/False Question : The number of ways a $2\times 8$ rectangle can be tiled with rectangular tiles of size $2\times 1$ is $34$.
27
True/False Question : Over the real line, $\underset{x\rightarrow \infty }{lim}\left ( \frac{x+log\:9}{x-log\:9} \right )^{x}=81.$
True/False Question : Let $f:[0,\infty )\rightarrow \mathbb{R}$ be a continuous function with $lim_{x\rightarrow \infty }f\left ( x \right )= 0.$ Then $f$ has a maximum value in $[0,\infty )$
True/False Question : Given a continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$, there exists a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$ such that the restriction of $g$ to $\mathbb{Q}$ is $f$.
True/False Question :​​​​​​ For all positive integers $m$ and $n$, if $A$ is an $m \times n$ real matrix, and $B$ is an $n \times m$ real matrix such that $AB = I$, then $BA=I$.