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Recent questions tagged tifrmaths2019

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1

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 $f\left ( \mathbb{R-Q} \right )\subseteq \mathbb{Q}.$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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2

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.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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3

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.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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4

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, then $A$ is the $2\times2$ identity matrix.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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5

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 $(ABC)=1$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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6

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 $\mathbb{R}$-vector space $M_{n}\left ( \mathbb{R} \right )$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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7

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 similar.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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8

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 $A$ is a compact subset of $M_{3}\left ( \mathbb{R} \right )\cong \mathbb{R}^{9}$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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9

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 $C\left ( \mathbb{R} \right )$ denotes the ring of all continuous functions $\mathbb{R}\rightarrow \mathbb{R}$ under pointwise addition and multiplication.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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10

TIFR-2019-Maths-B: 10
True/False Question : $\mathbb{R}$ and $\mathbb{R}\oplus \mathbb{R}$ are isomorphic as vector spaces over $\mathbb{Q}$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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11

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 as a sum of (not necessarily distinct) elements of $A$ is dense in $\left ( 0,\infty \right )$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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12

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 $x$ such that $x^{2}=x$).

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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13

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 ${x}',{y}',{z}'\in A$ such that $a^{50}{x}'+b^{20}{y}'+c^{15}{z}'=1$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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14

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.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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15

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 $n$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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16

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 necessarily contained in $N$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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17

TIFR-2019-Maths-B: 17
True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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18

TIFR-2019-Maths-B: 18
True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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19

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 $\sum_{n=1 }^{\infty }\frac{1}{f\left ( n \right )}$ converges.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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20

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 $t_{0}\in \left ( 0,1 \right )$ such that $v\left ( 1 \right )-v\left ( 0 \right )$ is a scalar multiple of $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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21

TIFR-2019-Maths-A: 1
The following sum of numbers (expressed in decimal notation) $1+11+111+\cdots +\underset{n}{\underbrace{11\dots1}}$ is equal to $\left ( 10^{n+1}-10-9n \right )/81$ $\left ( 10^{n+1}-10+9n \right )/81$ $\left ( 10^{n+1}-10-n \right )/81$ $\left ( 10^{n+1}-10+n \right )/81$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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22

TIFR-2019-Maths-A: 2
For $n\geq 1$, the sequence $\left \{ x_{n} \right \}^{\infty }_{n=1},$ where: $x_{n}=1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}-2\sqrt{n}$ is decreasing increasing constant oscillating

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

166 views

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23

TIFR-2019-Maths-A: 3
Define a function: $f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$ Consider the following statements: ${f}'\left ( 0 \right )$ exists and is equal to $1$ ... $f$ is increasing on $\mathbb{R}.$ How many of the above statements is/are true? $0$ $1$ $2$ $3$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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24

TIFR-2019-Maths-A: 4
Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have: $f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$ Then which one of the following ... $a,b \in \mathbb{R}$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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25

TIFR-2019-Maths-A: 5
Let $V$ be an n-dimensional vector space and let $T:V\rightarrow V$ be a linear transformation such that $Rank\:T \leq Rank\:T^{3}$. Then which one of the following statements is necessarily true? Null space$(T)$ = Range$(T)$ Null space$(T)$ $\cap$ Range$(T)$={ ... nonzero subspace $W$ of $V$ such that Null space$(T)$ $\cap$ Range$(T)$=$W$ Null space$(T)$ $\subseteq$ Range$(T)$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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26

TIFR-2019-Maths-A: 6
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$

soujanyareddy13 asked in Calculus Aug 29, 2020

by soujanyareddy13

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27

TIFR-2019-Maths-A: 7
Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements: If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$ If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$. Then (i) and (ii) are true (i) and (ii) are false (i) is true and (ii) is false (i) is false and (ii) is true

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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28

TIFR-2019-Maths-A: 8
Consider the following two statements: $(E)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=p\left ( x \right )$). $(O)$ ... false $(E)$ and $(O)$ are both true $(E)$ is true but $(O)$ is false $(E)$ is false but $(O)$ is true

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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29

TIFR-2019-Maths-A: 9
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ is bounded and uniformly continuous bounded but not uniformly continuous not bounded but uniformly continuous not bounded and not uniformly continuous

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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30

TIFR-2019-Maths-A: 10
Let $S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$ Then which of the following describes $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$

soujanyareddy13 asked in TIFR Aug 29, 2020

by soujanyareddy13

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Recent questions tagged tifrmaths2019