Recent questions tagged true-false

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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.
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123
True/False Question : A $p$ -Sylow subgroup of the underlying additive group of a finite commutative ring $R$ is an ideal in $R$.
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124
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...
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125
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...
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126
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 $m\leq n.$
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127
True/False Question :In the symmetric group $S_{n}$ any two elements of the same order are conjugate.
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128
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...
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129
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130
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...
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132
True/False Question :$\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.
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133
True/False Question :For any closed subset $A\subset \mathbb{R}$, there exists a continuous function $f$ on $\mathbb{R}$ which vanishes exactly on $A$.
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134
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...
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135
True/False Question :The function $f\left ( x \right )=cos\left ( e^{x} \right )$ is not uniformly continuous on $\mathbb{R}$.
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True/False Question :Let $A$ be a $3 \times 3$ real symmetric matrix such that $A^{6}=I$. Then, $A^{2}=I$.
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138
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...
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139
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 $B<H$.
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140
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141
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...
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142
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 $2$.
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146
True/False Question :Let $G$ be a finite group with a normal subgroup $H$ such that $G/H$ has order $7$. Then $G\cong H\times G/H$.
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147
True/False Question :The multiplicative group $\mathbb{F}_{7}^{\times }$ is isomorphic to a subgroup of the multiplicative group $\mathbb{F}_{31}^{\times }$.
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148
True/False Question :Any linear transformation $A:\mathbb{R}^{4}\rightarrow \mathbb{R}^{4}$ has a proper non-zero invariant subspace.
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149
True/False Question :Let $A, B \in M_{n}\left ( \mathbb{R} \right )$ be such that$A+B=AB$. Then $AB=BA$.
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150
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.