Recent questions tagged true-false

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91
True/False Question :Every countable group $G$ has only countably many distinct subgroups.
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92
True/False Question :Any automorphism of the group $\mathbb{Q}$ under addition is of the form $x\mapsto qx$ for some $q\in \mathbb{Q}$.
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93
True/False Question :The equation $x^{3}+3x-4=0$ has exactly one real root.
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94
True/False Question :The equation $x^{3}+10x^{2}-100x+1729=0$ has at least one complex root $\alpha$ such that $\left | \alpha \right | 12.$
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95
True/False Question :All non-trivial proper subgroups of $\left ( \mathbb{R},+ \right )$ are cyclic.
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96
True/False Question :Every inifinte abelian group has at least one element of infinite order.
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97
True/False Question :If $A$ band $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$.
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98
True/False Question :If a real square matrix $A$ is similar to a diagonal matrix and satisfies $A^{n}=0$ for some $n$, then $A$ must be the zero matrix.
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99
True/False Question :There is an element of order $51$ in the multiplicative group $\left ( \mathbb{Z}/103\mathbb{Z} \right )^{*}$.
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100
True/False Question :Any normal subgroup of order $2$ is contained in the center of the group.
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102
True/False Question :Suppose $a, b, c$ are positive real numbers such that$$\left ( 1+a+b+c \right )\left ( 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right )=16.$$Then $a+b+...
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103
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104
True/Flase Question :Over the real line,$$\underset{x\rightarrow \infty }{lim}\: log\left ( 1+\sqrt{4+x} -\sqrt{1+x}\right )=log\left ( 2 \right ).$$
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105
True/False Question :Suppose $f$ is a continuously differentiable function on $\mathbb{R}$ such that $f\left ( x \right )\rightarrow 1$ and ${f}'\left ( x \right )\right...
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106
True/False Question :If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and bijective, then $f^{-1}$ is also differentiable.
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107
True/False Question :Let $H_{1},H_{2},H_{3},H_{4}$ be four hyperplanes in $\mathbb{R}^{3}$. The maximum possible number of connected components of $\mathbb{R}^{3}-\left (...
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108
True/False Question :Let $n\geq 2$ be a natural number. Let $S$ be the set of all $n\times n$ real matrices whose entries are only $0,1$ or $2$. Then the average determin...
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109
True/False Question :For any metric space $\left ( X,d \right )$ with $X$ finite, there exists an isometric embedding $f:X\rightarrow \mathbb{R}^{4}$.
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110
True/False Question :There exists a non-negative continuous function $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ such that $\int_{0}^{1}f^{n}dx\rightarrow 2$ as $n\rig...
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111
True/False Question :There exists a subset $A$ of $\mathbb{N}$ with exactly five elements such that the sum of any three elements of $A$ is a prime number.
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112
True/False Question :There exists a finite abelian group $G$ containing exactly $60$ elements of order $2$.
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113
True/False Question :Let $\alpha ,\beta$ be complex numbers with non-positive real parts. Then $$\left | e^{\alpha } -e^{\beta }\right |\leq \left | \alpha -\beta \right ...
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114
True/False Question :Every $2 \times 2$ matrix over $\mathbb{C}$ is a square of some matrix.
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115
True/False Question :Under the projection map $\mathbb{R}^{2}\rightarrow \mathbb{R}$ sending $\left ( x,y \right )$ to $x$, the image of any closed set is closed.
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116
True/False Question :The number of ways a $2\times 8$ rectangle can be tiled with rectangular tiles of size $2\times 1$ is $34$.
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117
True/False Question :Over the real line, $$\underset{x\rightarrow \infty }{lim}\left ( \frac{x+log\:9}{x-log\:9} \right )^{x}=81.$$
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118
True/False Question :Let $f:[0,\infty )\rightarrow \mathbb{R}$ be a continuous function with $lim_{x\rightarrow \infty }f\left ( x \right )= 0.$ Then $f$ has a maximum va...
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119
True/False Question :Given a continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$, there exists a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$ such that t...
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120
True/False Question :​​​​​​For all positive integers $m$ and $n$, if $A$ is an $m \times n$ real matrix, and $B$ is an $n \times m$ real matrix such that $AB ...