Questions by soujanyareddy13 / GATE Overflow for GATE CSE

0 votes

0 answers

981

TIFR-2020-Maths-A: 4
Let $\left \{ a_{n}\right \}_{n=1}^{\infty }$ be a strictly increasing bounded sequence of real numbers such that $\lim_{n\rightarrow \infty }a_{n}=A$. Let $f:\left [ a_{1},A \right ]\rightarrow \mathbb{R}$ be a continuous function such that ... $B$ is: necessarily $0$ at most $1$ possibly greater than $1$, but finite possibly infinite

Let $\left \{ a_{n}\right \}_{n=1}^{\infty }$ be a strictly increasing bounded sequence of real numbers such that $\lim_{n\rightarrow \infty }a_{n}=A$. Let $f:\left [ a_{...

235 views

asked Aug 28, 2020

0 votes

0 answers

982

TIFR-2020-Maths-A: 5
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function that satisfies: $|f\left ( x \right )-f\left ( y \right )|\leq \left | x-y \right |\left | sin\left ( x-y \right ) \right |$ ... be uniformly continuous. $f$ is uniformly continuous but not necessarily differentiable. $f$ is differentiable, but its derivative may not be continuous. $f$ is constant.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function that satisfies: $|f\left ( x \right )-f\left ( y \right )|\leq \left | x-y \right |\left | sin\le...

182 views

asked Aug 28, 2020

0 votes

0 answers

983

TIFR-2020-Maths-A: 6
Let $C = \{ f:\mathbb{R}\rightarrow \mathbb{R}| f$ is differentiable, and $\lim_{x\rightarrow \infty }\left ( 2f\left ( x \right ) +f{}'\left ( x \right )\right )=0\left \} \right.$. Which of the following statements is correct? For each $L$ ... $f \in C$ such that $\lim_{x\rightarrow \infty }f\left ( x \right )\frac{1}{2}$

Let $C = \{ f:\mathbb{R}\rightarrow \mathbb{R}| f$ is differentiable, and $\lim_{x\rightarrow \infty }\left ( 2f\left ( x \right ) +f{}’\left ( x \right )\rig...

214 views

asked Aug 28, 2020

0 votes

1 answer

984

TIFR-2020-Maths-A: 7
Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m$. Then the sequence $\{a_{m}\} \infty_{m=1}$ diverges ... limit point converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$ converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$

Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m...

268 views

asked Aug 28, 2020

0 votes

0 answers

985

TIFR-2020-Maths-A: 8
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that : $\left | f\left ( x \right ) -f\left ( y \right )\right |\geq log\left ( 1+\left | x-y \right | \right ),$ for all $x,y \in \mathbb{R}$. Then: $f$ is injective but not surjective $f$ is surjective but not injective $f$ is neither injective nor surjective $f$ is bijective

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that : $\left | f\left ( x \right ) -f\left ( y \right )\right |\geq log\left (...

139 views

asked Aug 28, 2020

0 votes

1 answer

986

TIFR-2020-Maths-A: 9
What is the greatest integer less than or equal to $\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$ $1332$ $1352$ $1372$ $1392$

What is the greatest integer less than or equal to$$\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$$$1332$$1352$$1372$$1392$

330 views

asked Aug 28, 2020

0 votes

0 answers

987

TIFR-2020-Maths-A: 10
Consider the following sentences: $\left ( I \right )$ For every connected subset $Y$ of a metric space $X$, its interior $Y^{\circ}$ is connected. $\left ( II \right )$ For every connected subset $Y$ of a metric space $X$, its boundary $\partial Y$ is connected. Which ... $\left ( II \right )$ are both true. $\left ( I \right )$and $\left ( II \right )$ are both true.

Consider the following sentences:$\left ( I \right )$ For every connected subset $Y$ of a metric space $X$, its interior $Y^{\circ}$ is connected.$\left ( II \right )$ Fo...

193 views

asked Aug 28, 2020

0 votes

0 answers

988

TIFR-2020-Maths-A: 11
Consider a set $\left \{ A_{1} ,\dots,A_{n}\right \}$ of events, $n> 1$. Suppose that one of the events in $\left \{ A_{1} ,\dots,A_{n}\right \}$ is certain to occur, but that no more than two of them can occur. Suppose that for each $1\leq r,s\leq n$ such that $r\neq s$, ... $q\leq 2/n$ $p\leq 1/n$ and $q\geq 2/n$ $p\geq 1/n$ and $q\leq 2/n$ $p\geq 1/n$ and $q\geq 2/n$

Consider a set $\left \{ A_{1} ,\dots,A_{n}\right \}$ of events, $n 1$. Suppose that one of the events in $\left \{ A_{1} ,\dots,A_{n}\right \}$ is certain to occur, but ...

129 views

asked Aug 28, 2020

0 votes

0 answers

989

TIFR-2020-Maths-A: 12
Let $\left \{ z_{1}, z_{2},\dots,z_{7}\right \}$ be a set of seven complex numbers with unit modulus. Assume that they form the vertices of a regular heptagon in the complex plane. Define $w=\sum_{i< j}^{}z_{i}z_{j}.$ Then: $w=0.$ $\left | w \right |=\sqrt{7}.$ $\left | w \right |=7.$ $\left | w \right |=1.$

Let $\left \{ z_{1}, z_{2},\dots,z_{7}\right \}$ be a set of seven complex numbers with unit modulus. Assume that they form the vertices of a regular heptagon in the comp...

200 views

asked Aug 28, 2020

0 votes

0 answers

990

TIFR-2020-Maths-A: 13
Consider $\mathbb{R}^{3}$ as the space of $3\times 1$ real matrices. The multiplicative group $GL_{3}\left ( \mathbb{R} \right )$ of invertible $3\times 3$ real matrices acts on this space by left multiplication. What is the number of orbits for this action? $1$. $2$. $4$. $\infty$.

Consider $\mathbb{R}^{3}$ as the space of $3\times 1$ real matrices. The multiplicative group $GL_{3}\left ( \mathbb{R} \right )$ of invertible $3\times 3$ real matrices ...

87 views

asked Aug 28, 2020

1 votes

0 answers

991

TIFR-2020-Maths-A: 14
Let $V$ be a finite dimensional vector space over $\mathbb{R}$ , and $W\subset V$ a subspace. Then $W\cap T\left ( W \right )\neq \left \{ 0 \right \}$ for every linear automorphism $T:V\rightarrow V$ if and only if: $W=V$. $dim W< \frac{1}{2}dimV$. $dim W= \frac{1}{2}dimV$. $dim W> \frac{1}{2}dimV$.

Let $V$ be a finite dimensional vector space over $\mathbb{R}$ , and $W\subset V$ a subspace. Then $W\cap T\left ( W \right )\neq \left \{ 0 \right \}$ for every linear a...

139 views

asked Aug 28, 2020

0 votes

0 answers

992

TIFR-2020-Maths-A: 15
Let $A \in M_{n}\left ( \mathbb{C} \right )$ . Then $\begin{pmatrix} A &A \\ 0 & A \end{pmatrix}$ is diagonalizable if and only if : $A=0$ $A=I$ $n=2$ None of the other three options

Let $A \in M_{n}\left ( \mathbb{C} \right )$ . Then $\begin{pmatrix} A &A \\ 0 & A \end{pmatrix}$ is diagonalizable if and only if :$A=0$$A=I$$n=2$None of the other three...

103 views

asked Aug 28, 2020

0 votes

0 answers

993

TIFR-2020-Maths-A: 16
Let $T:\mathbb{C}\rightarrow \mathbb{R}$ be the map defined by $T\left ( z \right )=z+\bar{z}$. For a $\mathbb{C}$-vector space $V$, consider the map $\varphi :\left \{ f:V\rightarrow \mathbb{C}|f \right.$ ... . Then this map is injective, but not surjective surjective, but not injective bijective neither injective nor surjective

Let $T:\mathbb{C}\rightarrow \mathbb{R}$ be the map defined by $T\left ( z \right )=z+\bar{z}$. For a $\mathbb{C}$-vector space $V$, consider the map $\va...

125 views

asked Aug 28, 2020

0 votes

1 answer

994

TIFR-2020-Maths-A: 17
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$? $T$ is invertible as well as diagonalizable. $T$ is invertible, but not necessearily diagonalizable. $T$ is diagonalizable, but not necessary invertible. None of the other three statements.

Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$?$T$ is invertible ...

358 views

asked Aug 28, 2020

0 votes

0 answers

995

TIFR-2020-Maths-A: 18
Let $n\geq 2$. Which of the following statements is true for every $n\times n$ real matrix $A$ of rank one? There exist matrices $P,Q \in M_{n}\left ( \mathbb{R} \right )$ such that all the entries of the matrix $PAQ$ are equal to $1$ There exists an ... matrix $A$ has a nonzero eigenvalue The vector $\left ( 1,1,\dots,1 \right )\in \mathbb{R}^{n}$ is an eigenvector for $A$

Let $n\geq 2$. Which of the following statements is true for every $n\times n$ real matrix $A$ of rank one?There exist matrices $P,Q \in M_{n}\left ( \mathbb{R} \right )...

117 views

asked Aug 28, 2020

0 votes

0 answers

996

TIFR-2020-Maths-A: 19
Let $m,n$ be positive integers. Then the greatest common divisor $(gcd)$ of the polynomials $x^{m}-1$ and $x^{n}-1$ in the ring $\mathbb{C}\left [ x \right ]$ equals $x^{min\left ( m,n \right )}-1$. $x-1$. $x^{gcd\left ( m,n \right )}-1$. None of the other three options.

Let $m,n$ be positive integers. Then the greatest common divisor $(gcd)$ of the polynomials $x^{m}-1$ and $x^{n}-1$ in the ring $\mathbb{C}\left [ x \right ]$ equals$x^{m...

126 views

asked Aug 28, 2020

0 votes

0 answers

997

TIFR-2020-Maths-A: 20
Let $A_{4}$ denote the group of even permutations of $\left \{ 1,2,3,4 \right \}$. Consider the following statements: There exists a surjective group homomorphism $A_{4}\rightarrow \mathbb{Z}/4 \mathbb{Z}$ ... II) is false (II) is true and (I) is false (I) and (II) are both true (I) and (II) are both false

Let $A_{4}$ denote the group of even permutations of $\left \{ 1,2,3,4 \right \}$. Consider the following statements:There exists a surjective group homomorphism $A_{4}\r...

263 views

asked Aug 28, 2020

tags	tag:apple
author	user:martin
title	title:apple
content	content:apple
exclude	-tag:apple
force match	+apple
views	views:100
score	score:10
answers	answers:2
is accepted	isaccepted:true
is closed	isclosed:true