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Recent questions and answers in TIFR
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TIFR-2012-Maths-A: 1
True/False Question: If $H_{1}$ & $H_{2}$ are subgroups of a group $G$ then $H_{1} .H_{2}=\left \{ h_{1} h_{2}\in G \mid h_{1}\in H_{1},h_{2}\in H_{2}\right \}$ is a subgroup of $G$.
True/False Question:If $H_{1}$ & $H_{2}$ are subgroups of a group $G$ then $H_{1} .H_{2}=\left \{ h_{1} h_{2}\in G \mid h_{1}\in H_{1},h_{2}\in H_{2}\right \}$ is a subg...
d0t
277
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d0t
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Aug 6, 2023
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TIFR-2012-Maths-B: 17
True/False Question: If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
True/False Question:If the equation$$xyz=1$$holds in a group $G$, does it follow that$$yxz=1$$.
Psy Duck
264
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Psy Duck
answered
Aug 5, 2023
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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 $(ABC)=1$.
True/False Question :Suppose $A,B,C$ are $3\times3$ real matrices with Rank $A =2$, Rank $B=1$, Rank $C=2$. Then Rank $(ABC)=1$.
Dimpi779
274
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Dimpi779
answered
Feb 23, 2021
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TIFR-2012-Maths-D: 34
True/False Question: If a rectangle $R:=\left \{ \left ( x,y \right ) \in \mathbb{R}^{2}\mid A\leq x\leq B,C\leq y\leq D\right \}$ can be covered (allowing overlaps ) by $25$ discs of radius $1$ then it can also be covered by $101$ dics of radius $\frac{1}{2}.$
True/False Question:If a rectangle $R:=\left \{ \left ( x,y \right ) \in \mathbb{R}^{2}\mid A\leq x\leq B,C\leq y\leq D\right \}$ can be covered (allowing overlaps ) by $...
zxy123
445
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zxy123
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Nov 2, 2020
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TIFR-2013-Maths-A: 3
True/False Question : The equation $x^{3}+3x-4=0$ has exactly one real root.
True/False Question :The equation $x^{3}+3x-4=0$ has exactly one real root.
ayush.5
311
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ayush.5
answered
Oct 12, 2020
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TIFR-2012-Maths-D: 32
True/False Question: The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
True/False Question:The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
ObitoUchiha
291
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ObitoUchiha
answered
Sep 15, 2020
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TIFR-2012-Maths-B: 12
True/False Question: Let $V$ be the vector space of consisting polynomials of $\mathbb{R}\left [ t \right ]$ of deg$\leq 2$. The map $T:V\rightarrow V$ sending $f\left ( t \right )$ to $f\left ( t \right )+{f}'\left ( t \right )$ is invertible.
True/False Question:Let $V$ be the vector space of consisting polynomials of $\mathbb{R}\left [ t \right ]$ of deg$\leq 2$. The map $T:V\rightarrow V$ sending $f\left ( t...
raju6
331
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raju6
answered
Sep 10, 2020
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TIFR-2012-Maths-B: 11
True/False Question: The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian.
True/False Question:The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian.
raju6
488
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raju6
answered
Sep 10, 2020
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TIFR-2012-Maths-B: 13
True/False Question: The polynomials $\left ( t-1 \right )\left ( t-2 \right ),\left ( t-2 \right )\left ( t-3 \right ),\left ( t-3 \right )\left ( t-4 \right ),\left ( t-4 \right )\left ( t-6 \right )\in \mathbb{R}\left [ t \right ]$ are linearly independent.
True/False Question:The polynomials $\left ( t-1 \right )\left ( t-2 \right ),\left ( t-2 \right )\left ( t-3 \right ),\left ( t-3 \right )\left ( t-4 \right ),\left ( t-...
raju6
294
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raju6
answered
Sep 10, 2020
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TIFR-2012-Maths-D: 33
True/False Question: The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable.
True/False Question:The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable.
raju6
384
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raju6
answered
Sep 9, 2020
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TIFR-2012-Maths-D: 39
True/False Question: If $z_{1},z_{2},z_{3},z_{4}\in \mathbb{C}$ satisfy $z_{1}+z_{2}+z_{3}+z_{4}=0$ and $\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+\left | z_{3} \right |^{2}+\left | z_{4} \right |^{2}=1$ ... $2$.
True/False Question:If $z_{1},z_{2},z_{3},z_{4}\in \mathbb{C}$ satisfy $z_{1}+z_{2}+z_{3}+z_{4}=0$ and $\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+\left | z...
raju6
378
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raju6
answered
Sep 9, 2020
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TIFR-2012-Maths-D: 31
True/False Question: $f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
True/False Question:$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
ayush.5
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ayush.5
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Sep 7, 2020
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TIFR-2017-Maths-A: 30
True/False Question : The matrices $\begin{pmatrix} x &0 \\ 0 & y \end{pmatrix} and \begin{pmatrix} x &1 \\ 0 & y \end{pmatrix}, x\neq y,$ for any $x,y \in \mathbb{R}$ are conjugate in $M_{2}\left ( \mathbb{R} \right )$ .
True/False Question :The matrices $$\begin{pmatrix} x &0 \\ 0 & y \end{pmatrix} and \begin{pmatrix} x &1 \\ 0 & y \end{pmatrix}, x\neq y,$$for any $x,y \in \mathbb{R}$ ar...
ankitgupta.1729
301
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ankitgupta.1729
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Sep 6, 2020
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TIFR-2017-Maths-A: 29
True/False Question : Let $y\left ( t \right )$ be a real valued function defined on the real line such that ${y}'=y \left ( 1-y \right )$, with $y\left ( 0\right ) \in \left [ 0,1 \right ]$. Then $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
True/False Question :Let $y\left ( t \right )$ be a real valued function defined on the real line such that ${y}'=y \left ( 1-y \right )$, with $y\left ( 0\right ) \in \l...
ankitgupta.1729
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ankitgupta.1729
answered
Sep 6, 2020
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TIFR-2018-Maths-A: 7
True/False Question : In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions on the closed interval $\left [ 0,1 \right ]$, the set $S=\left \{ sin\left ( x \right ) , cos\left ( x \right ),tan\left ( x \right )\right \}$ is linearly independent.
True/False Question :In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions on the closed interval $\lef...
ankitgupta.1729
331
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ankitgupta.1729
answered
Sep 6, 2020
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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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
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...
ankitgupta.1729
279
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ankitgupta.1729
answered
Aug 31, 2020
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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.
True/False Question :If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
ankitgupta.1729
328
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ankitgupta.1729
answered
Aug 31, 2020
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TIFR-2018-Maths-B: 9
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
What are the last $3$ digits of $2^{2017}$?$072$$472$$512$$912.$
ankitgupta.1729
288
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ankitgupta.1729
answered
Aug 31, 2020
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TIFR-2020-Maths-A: 2
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy: $\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right )dx=\frac{1}{4}.$ Then the cardinality of this set is: $0$. $1$. $2$. more than $2$.
Consider the set of continuous functions $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ that satisfy:$$\int_{0}^{1}f\left ( x \right )\left ( 1-f\left ( x \right ) \right...
ankitgupta.1729
490
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ankitgupta.1729
answered
Aug 31, 2020
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TIFR-2012-Maths-D: 35
True/False Question: Given any integer $n\geq 2$, we can always finds an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,\dots,m+n$ are composite.
True/False Question:Given any integer $n\geq 2$, we can always finds an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,\dots,m+n$ are composite.
soujanyareddy13
219
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-D: 36
True/False Question: The $10 \times 10 $ matrix $\begin{pmatrix} v_{1}w_{1} & \cdots&v_{1}w_{10} \\ v_{2}w_{2}& \cdots & v_{2}w_{10}\\ v_{10}w_{1}&\cdots & v_{10}w_{10} \end{pmatrix}$has rank $2$, where $v_{i},w_{i}\in \mathbb{C}.$
True/False Question:The $10 \times 10 $ matrix $\begin{pmatrix} v_{1}w_{1} & \cdots&v_{1}w_{10} \\ v_{2}w_{2}& \cdots & v_{2}w_{10}\\ v_{10}w_{1}&\cdots & v_{10}w_{10} \e...
soujanyareddy13
232
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-D: 37
True/False Question: If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
True/False Question:If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
soujanyareddy13
165
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-D: 38
True/False Question: The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
True/False Question:The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
soujanyareddy13
175
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-D: 40
True/False Question: Consider the differential equations (with $y$ is a function of $x$) $\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$ ... $(1)$ has infinitely many solutions but $(2)$ has finite number of solutions.
True/False Question:Consider the differential equations (with $y$ is a function of $x$)$\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$$\...
soujanyareddy13
195
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 21
True/False Question: Let $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous function. Then the derivative $\frac{\partial ^{2}f}{\partial x\partial y}$ can exist without $\frac{\partial f}{\partial x}$ existing.
True/False Question:Let $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous function. Then the derivative $\frac{\partial ^{2}f}{\partial x\partial y}$ can exist w...
soujanyareddy13
201
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 22
True/False Question: If $f$ is continuous on $\left [ 0,1 \right ]$ and if $\int_{0}^{1}f\left ( x \right )x^{n}dx=0$ for $n=1,2,3,\cdots .$ .Then $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
True/False Question:If $f$ is continuous on $\left [ 0,1 \right ]$ and if $\int_{0}^{1}f\left ( x \right )x^{n}dx=0$ for $n=1,2,3,\cdots .$ .Then $\int_{0}^{1}f^{2}\left...
soujanyareddy13
218
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 23
True/False Question: Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
True/False Question:Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
soujanyareddy13
211
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 24
True/False Question: The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
True/False Question:The Integral$$\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$$is convergent.
soujanyareddy13
182
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 25
True/False Question: If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
True/False Question:If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
soujanyareddy13
200
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 26
True/False Question: If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
True/False Question:If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
soujanyareddy13
208
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 27
True/False Question: Let $f:\left [ 0,1 \right ]\rightarrow \left [ 0,1 \right ]$be continuous then $f$ assumes the value $\int_{0}^{1}f^{2}\left ( t \right )dt$ somewhere in $\left [ 0,1 \right ]$.
True/False Question:Let $f:\left [ 0,1 \right ]\rightarrow \left [ 0,1 \right ]$be continuous then $f$ assumes the value $\int_{0}^{1}f^{2}\left ( t \right )dt$ somewher...
soujanyareddy13
173
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 28
True/False Question: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that $\underset{h\rightarrow 0}{lim }\:\frac{f\left ( x+h \right )-f\left ( x-h \right )}{h}$ exists for all $x \in \mathbb{R}$. Then $f$ is differentiable in $\mathbb{R}.$
True/False Question:Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that$$\underset{h\rightarrow 0}{lim }\:\frac{f\left ( x+h \right )-f\left ( x-h \right )}{...
soujanyareddy13
222
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 29
True/False Question: The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
True/False Question:The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
soujanyareddy13
169
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-C: 30
True/False Question: The composition of two uniformly continuous functions need not always be uniformly continuous.
True/False Question:The composition of two uniformly continuous functions need not always be uniformly continuous.
soujanyareddy13
180
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 14
True/False Question: $A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
True/False Question:$A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
soujanyareddy13
219
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 15
True/False Question: Let $P$ be an $n \times n $ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
True/False Question:Let $P$ be an $n \times n $ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
soujanyareddy13
181
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 16
True/False Question: There is a non trivial group homomorphism from $C$ to $R$.
True/False Question:There is a non trivial group homomorphism from $C$ to $R$.
soujanyareddy13
165
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 18
True/False Question: Any $3\times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
True/False Question:Any $3\times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
soujanyareddy13
197
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 19
True/False Question: The rank of the matrix $\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$ is $2$.
True/False Question:The rank of the matrix$$\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$$is $2$.
soujanyareddy13
469
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soujanyareddy13
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Aug 30, 2020
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TIFR-2012-Maths-B: 20
True/False Question: The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$.
True/False Question:The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$.
soujanyareddy13
172
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soujanyareddy13
asked
Aug 30, 2020
TIFR
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