Recent questions tagged tifrmaths2018

The set of real numbers in the open interval $(0,1)$ which have more than one decimal expansion is emptynon-empty but finitecountably infiniteuncountable

How many zeroes does the function $f\left ( x \right )=e^{x}-3x^{2}$ have in $\mathbb{R}$?$0$$1$$2$$3$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as follows:$$f\left ( x \right )=\left\{\begin{matrix} 1, & if &x=0 \\ 0,&if &x \in \mathbb{R}\setminus \mathbb{Q}, \:...

Let $A$ be the set of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following two properties:$f$ has derivatives of all orders, andfor all $x,y \in ...

Denote by the set all $n \times n$ complex matrices $A$ ($n\geq 2$ a natural number) having the property that $4$ is the only eigenvalue of $A$. Consider the following fo...

Consider the following four sets of maps $f:\mathbb{Z}\rightarrow \mathbb{Q}$:$\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is bijective and increasing$\}$,$\{f:\mathbb{Z...

Consider a cube $C$ centered at the origin in $\mathbb{R^{3}}$. The number of invertible linear transformation of $\mathbb{R^{3}}$ which map $C$ onto itself is$72$$48$$24...

The number of rings of order $4$, up to isomorphism, is:$1$$2$$3$$4.$

For a sequence $\left \{ a_{n} \right \}$ of real numbers, which of the following is a negation of the statement ‘$\underset{n\rightarrow \infty }{lim}\:a_{n}=0$’?There e...

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Then which of the following statements implies that $f\left ( 0 \right )=0$?$\underset{n\rightarrow \infty }{lim}\...

True/False Question :Let $A$ be a countable subset of $\mathbb{R}$ which is well-ordered with respect to the usual ordering on $\mathbb{R}$ (where ‘well-ordered’ means th...

True/False Question :$\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.

True/False Question :For any closed subset $A\subset \mathbb{R}$, there exists a continuous function $f$ on $\mathbb{R}$ which vanishes exactly on $A$.

True/False Question :Let $f$ be a nonnegative continuous function on $\mathbb{R}$ such that $\int_{0}^{\infty }f\left ( t \right )dt$ is finite. Then $\underset{x\rightar...

True/False Question ;Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\r...

True/False Question :Let $G,H$ be finite groups. Then any subgroup of $G \times H$ is equal to $A \times B$ for some subgroups $A<G$ and $B<H$.

True/False Question :Let $g$ be a continuous function on $\left [ 0,1 \right ]$ such that $g\left ( 1 \right )=0$. Then the sequence of functions $f_{n}\left ( x \right )...

True/False Question :Let $A,B,C \in M_{3} \left(\mathbb{R}\right)$ be such that $A$ commutes with $B$, $B$ commutes with $C$ and $B$ is not a scalar matrix. Then $A$ comm...