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TIFR CSE 2023 | Part B | Question: 2
Which of the following is true about the set of regular languages and the set of recursively enumerable languages over a finite alphabet $\Sigma$ ? The set of regular languages is countable while the set of recursively enumerable languages ... whether the set of recursively enumerable languages is countable or not is not known and is a longstanding open problem.
Kabir5454
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Mar 14
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Kabir5454
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tifr2023
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2
NIELIT 2022 April Scientist B | Section A | Question: 21
Choose the word which is least like the other words in the group : Shimmer Simmer Glimmer Glint
Sagnick p
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Mar 14
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Sagnick p
70
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NIELIT 2022 April Scientist B | Section A | Question: 27
The first republic day of India was celebrated on January $26, 1950.$ What day of the week was it ? Wednesday Friday Thursday Tuesday
Sagnick p
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Mar 14
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Sagnick p
57
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nielit2022apr-scientistb
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4
TIFR Mathematics 2023 | Part B | Question: 1
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 $|f(x)-f(y)| \leq$ $|x-y|^{\alpha}$ for all $x, y \in(0,1)$. Then $f$ can be extended to a continuous function $[0,1] \rightarrow \mathbb{R}$.
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Mar 14
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admin
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tifrmaths2023
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5
TIFR Mathematics 2023 | Part B | Question: 2
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 continuous. Then at least one of the two functions $f$ and $g$ is uniformly continuous.
admin
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Mar 14
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admin
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tifrmaths2023
true-false
1
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0
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6
TIFR Mathematics 2023 | Part B | Question: 3
Answer whether the following statements are True or False. Let $\left\{f_{n}\right\}_{n}$ be a sequence of (not necessarily continuous) functions from $[0,1]$ to $\mathbb{R}$. Let $f:[0,1] \rightarrow \mathbb{R}$ be such that for any $x \in[0,1]$ ... $\lim _{n \rightarrow \infty} f_{n}\left(x_{n}\right)=f(x)$. Then $f$ is continuous.
admin
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Mar 14
by
admin
16
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tifrmaths2023
true-false
1
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7
TIFR Mathematics 2023 | Part B | Question: 4
Answer whether the following statements are True or False. Let $A, B \in \mathrm{M}_{2}(\mathbb{Z} / 2 \mathbb{Z})$ be such that $\operatorname{tr}(A)=\operatorname{tr}(B)$ and $\operatorname{tr}\left(A^{2}\right)=\operatorname{tr}\left(B^{2}\right)$. Then $A$ and $B$ have the same eigenvalues.
admin
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Mar 14
by
admin
25
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tifrmaths2023
true-false
1
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8
TIFR Mathematics 2023 | Part B | Question: 5
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 matrix $A$ such that $A v_{1}=v_{2}$ and $A w_{1}=w_{2}$.
admin
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Mar 14
by
admin
11
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tifrmaths2023
true-false
1
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0
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9
TIFR Mathematics 2023 | Part B | Question: 6
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 $1 \leq i, j \leq n$. Assume that $\lim _{m \rightarrow \infty} A^{m}$ exists, and denote it by $B=\left(b_{i j}\right)$. Then, for all $1 \leq i, j \leq n$, we have $b_{i j} \in\{0,1\}$.
admin
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Mar 14
by
admin
11
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tifrmaths2023
true-false
1
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0
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10
TIFR Mathematics 2023 | Part B | Question: 7
Answer whether the following statements are True or False. Given any monic polynomial $f(x) \in \mathbb{R}[x]$ of degree $n$, there exists a matrix $A \in \mathrm{M}_{n}(\mathbb{R})$ such that its characteristic polynomial equals $f$.
admin
asked
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Mar 14
by
admin
9
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tifrmaths2023
true-false
1
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0
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11
TIFR Mathematics 2023 | Part B | Question: 8
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 diagonalizable in $\mathrm{M}_{4}(\mathbb{C})$.
admin
asked
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Mar 14
by
admin
14
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tifrmaths2023
true-false
1
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0
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12
TIFR Mathematics 2023 | Part B | Question: 9
Answer whether the following statements are True or False. If $A \in \mathrm{M}_{n}(\mathbb{R})$ is such that $A B=B A$ for all invertible matrices $B \in \mathrm{M}_{n}(\mathbb{R})$, then $A=\lambda \cdot$ Id for some $\lambda \in \mathbb{R}$.
admin
asked
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Mar 14
by
admin
18
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tifrmaths2023
true-false
1
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0
answers
13
TIFR Mathematics 2023 | Part B | Question: 10
Answer whether the following statements are True or False. There exists a homeomorphism $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(2 x)=3 f(x)$ for all $x \in \mathbb{R}$.
admin
asked
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Mar 14
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admin
15
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tifrmaths2023
true-false
1
vote
0
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14
TIFR Mathematics 2023 | Part B | Question: 11
Answer whether the following statements are True or False. There exists a continuous bijection from $[0,1] \times[0,1]$ to $\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\}$, which is not a homeomorphism.
admin
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Mar 14
by
admin
16
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tifrmaths2023
true-false
1
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0
answers
15
TIFR Mathematics 2023 | Part B | Question: 12
Answer whether the following statements are True or False. Let $f \in \mathbb{C}\left[z_{1}, \ldots, z_{n}\right]$ be a nonzero polynomial $(n \geq 1)$, and let \[ X=\left\{z \in \mathbb{C}^{n} \mid f(z)=0\right\} . \] Then $\mathbb{C}^{n} \backslash X$ is path connected.
admin
asked
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Mar 14
by
admin
14
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tifrmaths2023
true-false
1
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0
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16
TIFR Mathematics 2023 | Part B | Question: 13
Answer whether the following statements are True or False. A connected metric space with at least two points is uncountable.
admin
asked
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Mar 14
by
admin
11
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tifrmaths2023
true-false
1
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17
TIFR Mathematics 2023 | Part B | Question: 14
Answer whether the following statements are True or False. If $A$ and $B$ are disjoint subsets of a metric space $(X, d)$, then \[ \inf \{d(x, y) \mid x \in A, y \in B\} \neq 0. \]
admin
asked
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Mar 14
by
admin
12
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tifrmaths2023
true-false
1
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18
TIFR Mathematics 2023 | Part B | Question: 15
Answer whether the following statements are True or False. A countably infinite complete metric space has infinitely many isolated points (an element $x$ of a metric space $X$ is said to be an isolated point if $\{x\}$ is an open subset of $X$ ).
admin
asked
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Mar 14
by
admin
13
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tifrmaths2023
true-false
1
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19
TIFR Mathematics 2023 | Part B | Question: 16
Answer whether the following statements are True or False. Suppose $G$ and $H$ are two countably infinite abelian groups such that every nontrivial element of $G \times H$ has order $7$ . Then $G$ is isomorphic to $H$.
admin
asked
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Mar 14
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admin
15
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tifrmaths2023
true-false
1
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0
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20
TIFR Mathematics 2023 | Part B | Question: 17
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.
admin
asked
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Mar 14
by
admin
10
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tifrmaths2023
true-false
1
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0
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21
TIFR Mathematics 2023 | Part B | Question: 18
Answer whether the following statements are True or False. Let $G$ be a group generated by two elements $x$ and $y$, each of order $2$ . Then $G$ is finite.
admin
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Mar 14
by
admin
7
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tifrmaths2023
true-false
1
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0
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22
TIFR Mathematics 2023 | Part B | Question: 19
Answer whether the following statements are True or False. $\mathbb{R}[x] /\left(x^{4}+x^{2}+2023\right)$ is an integral domain.
admin
asked
in
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Mar 14
by
admin
10
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tifrmaths2023
true-false
1
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0
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23
TIFR Mathematics 2023 | Part B | Question: 20
Answer whether the following statements are True or False. Every finite group is isomorphic to a subgroup of a finite group generated by two elements.
admin
asked
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Mar 14
by
admin
8
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tifrmaths2023
true-false
1
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0
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24
TIFR Mathematics 2023 | Part A | Question: 1
Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=\left(3 x^{2}+1\right) /\left(x^{2}+3\right)$. Let $f^{\circ 1}=f$, and let $f^{\circ n}=f^{\circ(n-1)} \circ f$ for all integers $n \geq 2$. Which of the following statements is ... $\lim _{n \rightarrow \infty} f^{\circ n}(1 / 2)$ nor $\lim _{n \rightarrow \infty} f^{\circ n}(2)$ exists.
admin
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Mar 14
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admin
16
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tifrmaths2023
1
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25
TIFR Mathematics 2023 | Part A | Question: 2
Consider the following properties of a sequence $\left\{a_{n}\right\}_{n}$ of real numbers. $\text{(I)} \lim _{n \rightarrow \infty} a_{n}=0$. $\text{(II)}$ There exists a sequence $\left\{i_{n}\right\}_{n}$ ... $\text{(I)}$ does not imply $\text{(II)}$, and $\text{(II)}$ does not imply $\text{(I)}$.
admin
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Mar 14
by
admin
9
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tifrmaths2023
1
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0
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26
TIFR Mathematics 2023 | Part A | Question: 3
Consider sequences $\left\{x_{n}\right\}_{n}$ of real numbers such that \[ \lim _{n \rightarrow \infty}\left(x_{2 n-1}+x_{2 n}\right)=2 \quad \text { and } \quad \lim _{n \rightarrow \infty}\left(x_{2 n}+x_{2 n+1}\right)=3 . \] Which of the ... $\left\{x_{n}\right\}_{n}$, for which $\lim _{n \rightarrow \infty} \frac{x_{2 n+1}}{x_{2 n}}$ does not exist.
admin
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Mar 14
by
admin
10
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tifrmaths2023
1
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0
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27
TIFR Mathematics 2023 | Part A | Question: 4
Consider the function $f:(0, \infty) \rightarrow(0, \infty)$ given by $f(x)=x e^{x}$. Let $L:(0, \infty) \rightarrow(0, \infty)$ ... $\lim _{x \rightarrow \infty} \frac{L(x)}{\sqrt{\log x}}=1$. None of the remaining three options is correct.
admin
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Mar 14
by
admin
15
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tifrmaths2023
1
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0
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28
TIFR Mathematics 2023 | Part A | Question: 5
Let $\left\{b_{n}\right\}_{n}$ be a monotonically increasing sequence of positive real numbers such that $\lim _{n \rightarrow \infty} b_{n}=$ $\infty$. Which of the following statements is true about \[ \lim _{n \rightarrow \ ... $0$. The limit exists for all such sequences, and its value is always $1$. None of the remaining three options is correct.
admin
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Mar 14
by
admin
7
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tifrmaths2023
1
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29
TIFR Mathematics 2023 | Part A | Question: 6
For every positive integer $n$, define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by $f_{n}(x)=\frac{\sin \left(n^{2} x\right)+\cos \left(e^{n} x\right)}{1+n^{2} x^{2}}$. Then \[ \lim _{n \rightarrow \infty} \int_{0}^{1-\sin (1 / n)} f_{n}(x) d x \] equals $1$. $0.$ $\infty$. $1 / 2$.
admin
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Mar 14
by
admin
12
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tifrmaths2023
1
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0
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30
TIFR Mathematics 2023 | Part A | Question: 7
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 following statements is correct? $f_{1}$ and $f_{2}$ ... $f_{2}$ is not. $f_{2}$ is uniformly continuous, but $f_{1}$ is not. Neither $f_{1}$ nor $f_{2}$ is uniformly continuous.
admin
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Mar 14
by
admin
7
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tifrmaths2023
1
vote
0
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31
TIFR Mathematics 2023 | Part A | Question: 8
Let $x_{1} \in \mathbb{R}^{2} \backslash\{0\}$ be fixed, and inductively define $x_{n+1}=A x_{n}$ for $n \geq 1$, where $A$ is the $2 \times 2$ real matrix given by \[ A:=\left(\begin{array}{cc} \frac{\sqrt{3}}{ ... , but it has a convergent subsequence. $\lim _{n \rightarrow \infty}\left\|x_{n}\right\|=0$. None of the remaining three options is correct.
admin
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Mar 14
by
admin
15
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tifrmaths2023
1
vote
0
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32
TIFR Mathematics 2023 | Part A | Question: 9
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 dimension of the kernel of $T$ equals $2$. $8$. $1$. None of the remaining three options.
admin
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Mar 14
by
admin
8
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tifrmaths2023
1
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0
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33
TIFR Mathematics 2023 | Part A | Question: 10
Let $V=\{f(x) \in \mathbb{R}[x] \mid f(0)=0\}$, viewed as a real vector space. Consider the following assertions: $\text{(I)}$ $V$ contains three linearly independent polynomials of degree $2$ . $\text{(II)}$ $V$ contains two linearly independent ... $\text{(II)}$ is true. Neither $\text{(I)}$ nor $\text{(II)}$ is true.
admin
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Mar 14
by
admin
6
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tifrmaths2023
0
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1
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34
NIELIT 2022 April Scientist B | Section B | Question: 119
In a binary data transmission $\text{DPSK}$ is preferred to $\text{PSK}$ because : a coherent carrier is not required to be generated at the receiver for a given energy per bit, the probability of error is less the $1800$ phase shifts of the carrier are unimportant more protection is provided against impulse noise
Sagnick p
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Others
Mar 14
by
Sagnick p
63
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nielit2022apr-scientistb
1
vote
1
answer
35
TIFR Mathematics 2022 | Part B | Question: 17
Answer whether the following statements are True or False. There exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ \lim _{x \rightarrow \infty} f(x)=2 \quad \text { and } \quad \lim _{x \rightarrow \infty} f^{\prime}(x)=1 .\]
KG
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Mar 12
by
KG
64
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tifrmaths2022
true-false
0
votes
1
answer
36
NIELIT 2022 April Scientist B | Section A | Question: 6
The following question is based on the information given below: Data of $450$ candidates, who took part in an examination in social sciences, Mathematics and Science is given as below : Number of Students who : passed in all subjects are $167$ ... in Social sciences only $= 62.$ How many failed in two subjects only? $162$ $152$ $100$ $52$
abhisham62
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Mar 7
by
abhisham62
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nielit2022apr-scientistb
0
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2
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37
NIELIT 2021 Dec Scientist A - Section B: 93
Which of the following step is not a part of the requirement engineering process? Feasibility Study Programming Language Requirement Specification Software Requirement Specification Requirement Gathering & Validation
Bharat Bhushan
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Mar 5
by
Bharat Bhushan
67
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nielit2021dec-scientista
0
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38
NIELIT 2021 Dec Scientist A - Section B: 87
The postfix equivalent of the infix expression $(a+b)^{*}(c^{*}d-e)^{*}f/g$ is : $ab + cd^{*}e – fg^{*}/^{*}$ $ab + cd^{*}e - fg/^{**}$ $ab + cde^{*} – fg/^{**}$ $abcd + e^{*} fg – /^{**}$
khushbook
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Mar 4
by
khushbook
68
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nielit2021dec-scientista
0
votes
1
answer
39
NIELIT 2022 April Scientist B | Section A | Question: 31
In a family of six persons $\text{A, B, C, D, E}$ and $\text{F},$ there are two married couples. The family has equal number of male and female. $\text{D}$ is the grandmother of $\text{A}$ and mother of $\text{B}$ $\text{C}$ is ... $ $\text{A}$ is the sister of $\text{F}.$ $\text{D}$ has two grandsons$ None of these
DMeher
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Feb 15
by
DMeher
66
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nielit2022apr-scientistb
0
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1
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40
NIELIT 2022 April Scientist B | Section A | Question: 12
Read following information and answer the question. There is a group of five persons, $\text{P, Q, R, S}$ and $\text{T.}$ One of them is a Horticulturist, one is Physicist, one is Journalist, one is Industrialist and one is an Advocate. Three of the ... $\text{R's}$ brother. Who is industrialist ? $\text{P}$ $\text{Q}$ $\text{R}$ $\text{S}$
DMeher
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Feb 15
by
DMeher
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