Register

Recent questions tagged true-false

1 1 vote
1 1 answer
361
361 views
TIFR-2018-Maths-A: 25
True/False Question :Let $f(x)$ and $g(x)$ be uniformly continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Then their pointwise product $f(x)g(x)$ is uniformly conti...
User Avatar
1 1 vote
0 0 answers
377
377 views
TIFR-2019-Maths-B: 1
True/False Question :There exists a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f\left ( \mathbb{Q} \right )\subseteq \mathbb{R}-\mathbb{Q}$ and $...
User Avatar
0 0 votes
1 1 answer
621
621 views
TIFR-2019-Maths-B: 2
True/False Question :If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
User Avatar
0 0 votes
0 0 answers
359
359 views
TIFR-2019-Maths-B: 3
True/False Question :Let $X\subseteq \mathbb{Q}^{2}$. Suppose each continuous function $f:X\rightarrow \mathbb{R}^{2}$ is bounded. Then $X$ is necessarily finite.
User Avatar
0 0 votes
0 0 answers
344
344 views
TIFR-2019-Maths-B: 4
True/False Question :If $A$ is a $2\times2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^{2}$ have the same characteristic polynomial, t...
User Avatar
0 0 votes
1 1 answer
639
639 views
TIFR-2019-Maths-B: 5
True/False Question :Suppose $A,B,C$ are $3\times3$ real matrices with Rank $A =2$, Rank $B=1$, Rank $C=2$. Then Rank $(ABC)=1$.
User Avatar
0 0 votes
0 0 answers
330
330 views
TIFR-2019-Maths-B: 6
True/False Question :For any $n\geq 2$, there exists an $n\times n$ real matrix $A$ such that the set $\left \{ A^{p} \mid p\geq 1 \right \}$ spans the $\mathbb{R}$-vecto...
User Avatar
0 0 votes
0 0 answers
319
319 views
TIFR-2019-Maths-B: 7
True/False Question :The matrices$$\begin{pmatrix} 0 & i & 0\\ 0& 0& 1\\ 0& 0 & 0 \end{pmatrix} and \begin{pmatrix} 0 & 0 & 0\\ -i& 0& 0\\ 0& 1 & 0 \end{pmatrix}$$are si...
User Avatar
0 0 votes
0 0 answers
385
385 views
TIFR-2019-Maths-B: 8
True/False Question :Consider the set $A\subset M_{3}\left ( \mathbb{R} \right )$ of $3\times 3$ real matrices with characteristic polynomial. $x^{3}-3x^{2}+2x-1$. Then $...
User Avatar
0 0 votes
0 0 answers
335
335 views
TIFR-2019-Maths-B: 9
True/False Question :There exists an injective ring homomorphism from the product ring $\mathbb{R}\times \mathbb{R}$ into $C\left ( \mathbb{R} \right )$, where $C\left ( ...
User Avatar
0 0 votes
0 0 answers
365
365 views
TIFR-2019-Maths-B: 10
True/False Question :$\mathbb{R}$ and $\mathbb{R}\oplus \mathbb{R}$ are isomorphic as vector spaces over $\mathbb{Q}$.
User Avatar
0 0 votes
0 0 answers
340
340 views
TIFR-2019-Maths-B: 11
True/False Question :If $0$ is a limit point of a set $A\subseteq \left ( 0,\infty \right )$, then the set of all $x\in\left ( 0,\infty \right )$ that can be expressed a...
User Avatar
0 0 votes
0 0 answers
315
315 views
TIFR-2019-Maths-B: 12
True/False Question :The only idempotents in the ring $\mathbb{Z}_{51} \left ( i.e.,\mathbb{Z}/51\mathbb{Z} \right )$ are $0$ and $1$. (An idempotent is an element $x$ su...
User Avatar
0 0 votes
0 0 answers
395
395 views
TIFR-2019-Maths-B: 13
True/False Question :Let $A$ be a commutative ring with $1$, and let $a,b,c\in A$. Suppose there exist $x,y,z\in A$ such that $ax+by+cz=1.$ Then there exist ${x}',{y}',{z...
User Avatar
0 0 votes
0 0 answers
308
308 views
TIFR-2019-Maths-B: 14
True/False Question :The ring $\mathbb{R}\left [ x \right ]/\left ( x^{5} +x-3\right )$ is an integral domain.
User Avatar
0 0 votes
0 0 answers
310
310 views
TIFR-2019-Maths-B: 15
True/False Question :Given any group $G$ of order $12$, and any $n$ that divides $12$, there exists a subgroup $H$ of $G$ of order $n$.
User Avatar
0 0 votes
0 0 answers
357
357 views
TIFR-2019-Maths-B: 16
True/False Question : Let $H,N$ be subgroups of a finite group $G$, with $N$ a normal subgroup of $G$. If the orders of $G/N$ and $H$ are relatively prime, then $H$ is ne...
User Avatar
0 0 votes
0 0 answers
271
271 views
TIFR-2019-Maths-B: 17
True/False Question :If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
User Avatar
0 0 votes
0 0 answers
287
287 views
TIFR-2019-Maths-B: 18
True/False Question :Each solution of the differential equation $${y}''+e^{x}y=0$$remains bounded as $x\rightarrow \infty$.
User Avatar
0 0 votes
0 0 answers
271
271 views
TIFR-2019-Maths-B: 19
True/False Question :There exists a uniformly continuous function $f:\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$ such that $$\sum_{n=1 }^{\infty }\fra...
User Avatar
0 0 votes
0 0 answers
300
300 views
TIFR-2019-Maths-B: 20
True/False Question :Let $v:\mathbb{R}\rightarrow \mathbb{R}^{2}$ be $C^{\infty }$ (i.e., has derivatives of all orders). Then there exists $t_{0}\in \left ( 0,1 \right )...
User Avatar
0 0 votes
0 0 answers
521
521 views
TIFR-2020-Maths-B: 1
True/False Question :There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.
User Avatar
0 0 votes
0 0 answers
416
416 views
TIFR-2020-Maths-B: 2
True/False Question :Let $C\left ( \left [ 0,1 \right ] \right )$ denote the set of continuous real valued functions on $\left [ 0,1 \right ]$, and $\mathbb{R}^{\mathbb{N...
User Avatar
0 0 votes
0 0 answers
356
356 views
TIFR-2020-Maths-B: 3
True/False Question :Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a bounded sequence of positive real numbers. Then: $$\underset{n\rightarrow\infty }{lim sup}\:\frac...
User Avatar
0 0 votes
0 0 answers
473
473 views
TIFR-2020-Maths-B: 4
True/False Question :Let $C\left ( \left [ 0,1 \right ] \right )$ denote the metric space of continuous real valued functions on $\left [ 0,1 \right ]$ under the supremum...
User Avatar
0 0 votes
0 0 answers
346
346 views
TIFR-2020-Maths-B: 5
True/False Question :If $X$ is a metric space such that every continuous function $f:X\rightarrow \mathbb{R}$ is uniformly continuous, then $X$ is compact.
User Avatar
0 0 votes
0 0 answers
287
287 views
TIFR-2020-Maths-B: 6
True/False Question :Let $X$ be a metric space, and let $C\left ( X \right )$ denote the $\mathbb{R}-$vector space of continuous real valued functions on $X$. Then $X$ is...
User Avatar
0 0 votes
0 0 answers
340
340 views
TIFR-2020-Maths-B: 7
True/False Question :Let $A$ be a countable union of lines in $\mathbb{R}^{3}$. Then $\mathbb{R}^{3} \setminus A$ is connected.
User Avatar
0 0 votes
0 0 answers
429
429 views
TIFR-2020-Maths-B: 8
True/False Question :An invertible linear map from $\mathbb{R}^{2}$ to itself takes parallel lines to parallel lines.
User Avatar
0 0 votes
1 1 answer
305
305 views
TIFR-2020-Maths-B: 9
True/False Question :For any matrix $C$ with entries in $\mathbb{C}$, let $m\left ( C \right )$ denote the minimal polynomial of $C$, and $p\left ( C \right )$ its charac...
User Avatar
Page: