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Recent questions tagged true-false
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121
TIFR-2017-Maths-A: 21
True/False Question : There is a continuous onto function $f:S^{2}\rightarrow S^{1}$ from the unit sphere in $\mathbb{R}^{3}$ to the unit sphere in $\mathbb{R}^{2}$, where $S^{n}=\left \{ v \in \mathbb{R}^{n+1} \mid \left | \left | v \right | \right |=1\right \}$ denotes the unit sphere in $ \mathbb{R}^{n+1}$.
True/False Question :There is a continuous onto function $f:S^{2}\rightarrow S^{1}$ from the unit sphere in $\mathbb{R}^{3}$ to the unit sphere in $\mathbb{R}^{2}$, where...
soujanyareddy13
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soujanyareddy13
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Aug 30, 2020
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TIFR-2017-Maths-A: 22
True/False Question : Let $P$ be a monic, non-zero, polynomial of even degree, and $K>0$. Then the function $P\left ( x \right )-K e^{x}$ has a real zero.
True/False Question :Let $P$ be a monic, non-zero, polynomial of even degree, and $K>0$. Then the function $P\left ( x \right )-K e^{x}$ has a real zero.
soujanyareddy13
102
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soujanyareddy13
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Aug 30, 2020
TIFR
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123
TIFR-2017-Maths-A: 23
True/False Question : A $p$ -Sylow subgroup of the underlying additive group of a finite commutative ring $R$ is an ideal in $R$.
True/False Question : A $p$ -Sylow subgroup of the underlying additive group of a finite commutative ring $R$ is an ideal in $R$.
soujanyareddy13
141
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soujanyareddy13
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Aug 30, 2020
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TIFR-2017-Maths-A: 24
True/False Question : Suppose $A$ is an $n \times n$ -real matrix, all whose eigenvalues have absolute value less than $1$. Then for any $v \in \mathbb{R}^{n} ,\left \| Av \right \|\leq \left \| v \right \|.$
True/False Question :Suppose $A$ is an $n \times n$ -real matrix, all whose eigenvalues have absolute value less than $1$. Then for any $v \in \mathbb{R}^{n} ,\left \| Av...
soujanyareddy13
89
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soujanyareddy13
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Aug 30, 2020
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125
TIFR-2017-Maths-A: 25
True/False Question : For any $x \in \mathbb{R}$, the sequence $\left \{ a_{n} \right \}$, where $a_{1}=x$ and $a_{n+1}=cos\left ( a_{n} \right )$ for all $n$, is convergent.
True/False Question :For any $x \in \mathbb{R}$, the sequence $\left \{ a_{n} \right \}$, where $a_{1}=x$ and $a_{n+1}=cos\left ( a_{n} \right )$ for all $n$, is converge...
soujanyareddy13
86
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soujanyareddy13
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Aug 30, 2020
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TIFR-2017-Maths-A: 26
True/False Question : Suppose $A_{1},\cdots ,A_{m}$ are distinct $n\times n$ real matrices such that $A_{i}A_{j}=0$ for all $i\neq j$. Then $m\leq n.$
True/False Question :Suppose $A_{1},\cdots ,A_{m}$ are distinct $n\times n$ real matrices such that $A_{i}A_{j}=0$ for all $i\neq j$. Then $m\leq n.$
soujanyareddy13
70
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soujanyareddy13
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Aug 30, 2020
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TIFR-2017-Maths-A: 27
True/False Question : In the symmetric group $S_{n}$ any two elements of the same order are conjugate.
True/False Question :In the symmetric group $S_{n}$ any two elements of the same order are conjugate.
soujanyareddy13
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soujanyareddy13
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Aug 30, 2020
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TIFR-2017-Maths-A: 28
True/False Question : If a particle moving on the Euclidean line traverses distance $1$ in time $1$ starting and ending at rest, then at some time $t \in \left [ 0,1 \right ]$, at absolute value of its acceleration should be at least $4$.
True/False Question :If a particle moving on the Euclidean line traverses distance $1$ in time $1$ starting and ending at rest, then at some time $t \in \left [ 0,1 \righ...
soujanyareddy13
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soujanyareddy13
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Aug 30, 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...
soujanyareddy13
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Aug 30, 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...
soujanyareddy13
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soujanyareddy13
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Aug 30, 2020
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TIFR-2018-Maths-A: 1
True/False Question : Let $A$ be a countable subset of $\mathbb{R}$ which is well-ordered with respect to the usual ordering on $\mathbb{R}$ (where ‘well-ordered’ means that every nonempty subset has a minimum element in it). Then $A$ an order perserving bijection with a subset of $\mathbb{N}$.
True/False Question :Let $A$ be a countable subset of $\mathbb{R}$ which is well-ordered with respect to the usual ordering on $\mathbb{R}$ (where ‘well-ordered’ mean...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 2
True/False Question : $\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.
True/False Question :$\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 3
True/False Question : For any closed subset $A\subset \mathbb{R}$, there exists a continuous function $f$ on $\mathbb{R}$ which vanishes exactly on $A$.
True/False Question :For any closed subset $A\subset \mathbb{R}$, there exists a continuous function $f$ on $\mathbb{R}$ which vanishes exactly on $A$.
soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 4
True/False Question : Let $f$ be a nonnegative continuous function on $\mathbb{R}$ such that $\int_{0}^{\infty }f\left ( t \right )dt$ is finite. Then $\underset{x\rightarrow \infty }{lim}\:f\left ( x \right )=0.$
True/False Question :Let $f$ be a nonnegative continuous function on $\mathbb{R}$ such that $\int_{0}^{\infty }f\left ( t \right )dt$ is finite. Then $\underset{x\rightar...
soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 5
True/False Question : The function $f\left ( x \right )=cos\left ( e^{x} \right )$ is not uniformly continuous on $\mathbb{R}$.
True/False Question :The function $f\left ( x \right )=cos\left ( e^{x} \right )$ is not uniformly continuous on $\mathbb{R}$.
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
Calculus
tifrmaths2018
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calculus
continuity
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TIFR-2018-Maths-A: 6
True/False Question : Let $A$ be a $3 \times 3$ real symmetric matrix such that $A^{6}=I$. Then, $A^{2}=I$.
True/False Question :Let $A$ be a $3 \times 3$ real symmetric matrix such that $A^{6}=I$. Then, $A^{2}=I$.
soujanyareddy13
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soujanyareddy13
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Aug 29, 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...
soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 8
True/False Question ; Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\rightarrow \infty }{lim}\:f\left ( x \right )=\infty$.
True/False Question ;Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\r...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 9
True/False Question : Let $G,H$ be finite groups. Then any subgroup of $G \times H$ is equal to $A \times B$ for some subgroups $A<G$ and $B<H$.
True/False Question :Let $G,H$ be finite groups. Then any subgroup of $G \times H$ is equal to $A \times B$ for some subgroups $A<G$ and $B<H$.
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 10
True/False Question : Let $g$ be a continuous function on $\left [ 0,1 \right ]$ such that $g\left ( 1 \right )=0$. Then the sequence of functions $f_{n}\left ( x \right )=x^{n}g\left ( x \right )$ converges uniformly on $\left [ 0,1 \right ]$.
True/False Question :Let $g$ be a continuous function on $\left [ 0,1 \right ]$ such that $g\left ( 1 \right )=0$. Then the sequence of functions $f_{n}\left ( x \right )...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 11
True/False Question : Let $A,B,C \in M_{3} \left(\mathbb{R}\right)$ be such that $A$ commutes with $B$, $B$ commutes with $C$ and $B$ is not a scalar matrix. Then $A$ commutes with $C$.
True/False Question :Let $A,B,C \in M_{3} \left(\mathbb{R}\right)$ be such that $A$ commutes with $B$, $B$ commutes with $C$ and $B$ is not a scalar matrix. Then $A$ comm...
soujanyareddy13
148
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soujanyareddy13
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Aug 29, 2020
Linear Algebra
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matrix
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TIFR-2018-Maths-A: 12
True/False Question : If $A \in M_{n}\left ( \mathbb{R} \right )$ ( with $n\geq 2$) has rank $1$, then the minimal polynomial of $A$ has degree $2$.
True/False Question :If $A \in M_{n}\left ( \mathbb{R} \right )$ ( with $n\geq 2$) has rank $1$, then the minimal polynomial of $A$ has degree $2$.
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 13
True/False Question : Let $V$ be the vector space over $\mathbb{R}$ consisting of polynomials of degree less than or equal to $3$. Let $T:V \rightarrow V$ be the operator sending $f\left(t\right)$ to $f\left(t+1 \right)$, and $D:V \rightarrow V$ the operator sending $f\left(t \right)$ to $df\left(t \right)/dt$. Then $T$ is a polynomial in $D$.
True/False Question :Let $V$ be the vector space over $\mathbb{R}$ consisting of polynomials of degree less than or equal to $3$. Let $T:V \rightarrow V$ be the operator ...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 14
True/False Question : Let $V$ be the subspace of the real vector space of real valued functions on $\mathbb{R}$, spanned by cost and sin $t$. Let $D:V \rightarrow V$ be the linear map sending $f\left ( t \right ) \in V$ to $df\left ( t \right )/dt$. Then $D$ has a real eigenvalue.
True/False Question :Let $V$ be the subspace of the real vector space of real valued functions on $\mathbb{R}$, spanned by cost and sin $t$. Let $D:V \rightarrow V$ be th...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 15
True/False Question : The set of nilpotent matrices in $M_{3}\left ( \mathbb{R} \right )$ spans $M_{3}\left ( \mathbb{R} \right )$ considered as an $\mathbb{R}$-vector space (a matrix $A$ is said to be nilpotent if there exists $n \in \mathbb{N}$ such that $A^{n}=0$).
True/False Question :The set of nilpotent matrices in $M_{3}\left ( \mathbb{R} \right )$ spans $M_{3}\left ( \mathbb{R} \right )$ considered as an $\mathbb{R}$-vector spa...
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 16
True/False Question : Let $G$ be a finite group with a normal subgroup $H$ such that $G/H$ has order $7$. Then $G\cong H\times G/H$.
True/False Question :Let $G$ be a finite group with a normal subgroup $H$ such that $G/H$ has order $7$. Then $G\cong H\times G/H$.
soujanyareddy13
120
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 17
True/False Question : The multiplicative group $\mathbb{F}_{7}^{\times }$ is isomorphic to a subgroup of the multiplicative group $\mathbb{F}_{31}^{\times }$.
True/False Question :The multiplicative group $\mathbb{F}_{7}^{\times }$ is isomorphic to a subgroup of the multiplicative group $\mathbb{F}_{31}^{\times }$.
soujanyareddy13
71
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soujanyareddy13
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Aug 29, 2020
TIFR
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TIFR-2018-Maths-A: 18
True/False Question : Any linear transformation $A:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}$ has a proper non-zero invariant subspace.
True/False Question :Any linear transformation $A:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}$ has a proper non-zero invariant subspace.
soujanyareddy13
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 19
True/False Question : Let $A, B \in M_{n}\left ( \mathbb{R} \right )$ be such that$A+B=AB$. Then $AB=BA$.
True/False Question :Let $A, B \in M_{n}\left ( \mathbb{R} \right )$ be such that$A+B=AB$. Then $AB=BA$.
soujanyareddy13
65
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soujanyareddy13
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Aug 29, 2020
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TIFR-2018-Maths-A: 20
True/False Question : Let $A\in M_{n}\left ( \mathbb{R} \right )$ be upper triangular with all diagonal entries $1$ such that $A\neq I$. Then $A$ is not diagonalizable.
True/False Question :Let $A\in M_{n}\left ( \mathbb{R} \right )$ be upper triangular with all diagonal entries $1$ such that $A\neq I$. Then $A$ is not diagonalizable.
soujanyareddy13
86
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soujanyareddy13
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Aug 29, 2020
TIFR
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