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Recent activity by soujanyareddy13

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1
Denote by the set all $n \times n$ complex matrices $A$ ($n\geq 2$ a natural number) having the property that $4$ is the only eigenvalue of $A$. Consider the following four statements. $\left ( A-4I \right )^{n}=0,$ $A^{n}=4^{n}I,$ $\left ( A^{2}-5A+4I \right )^{n}=0,$ $A^{n}=4nI.$ How many of the above statements are true for all $A \in$ ? $0$ $1$ $2$ $3$
edited 4 days ago in TIFR 11 views
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2
Let $A$ be the set of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following two properties: $f$ has derivatives of all orders, and for all $x,y \in \mathbb{R}$, $f\left ( x+y \right )-f\left ( y-x \right )=2x{f}'\left ( y \right ).$ ... less than or equal to $2$. There exists $f \in A$ which is not a polynomial. There exists $f \in A$ which is a polynomial of degree $4$.
edited 4 days ago in TIFR 7 views
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3
Suppose $p$ is a degree $3$ polynomial such that $p\left ( 0 \right )=1,p\left ( 1 \right )=2,$ and $p\left ( 2 \right )=5$. Which of the following numbers cannot equal $p\left ( 3 \right )$ ? $0$ $2$ $6$ $10$.
edited 4 days ago in TIFR 11 views
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4
How many zeroes does the function $f\left ( x \right )=e^{x}-3x^{2}$ have in $\mathbb{R}$? $0$ $1$ $2$ $3$
edited 4 days ago in TIFR 9 views
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5
The set of real numbers in the open interval $(0,1)$ which have more than one decimal expansion is empty. non-empty but finite. countable infinite. uncountable.
edited 4 days ago in TIFR 7 views
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6
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
edited 4 days ago in Calculus 41 views
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7
A stick of length $1$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece? $1/8$ $1/4$ $1/e$ $1/\pi$
edited 4 days ago in TIFR 6 views
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8
Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ that have derivatives of all orders. Then the map $C^{\infty }\left ( 0,1 \right )\rightarrow C^{\infty }\left ( 0,1 \right )$ given by $f \mapsto f+\frac{df}{dx}$ is injective but not surjective surjective but not injective neither injective nor surjective both injective and surjective
edited 4 days ago in TIFR 14 views
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9
Consider the different ways to colour the faces of a cube with six given colours, such that each face is given exactly one colour and all the six colours are used. Define two such colouring schemes to be equivalent if the resulting configurations can be obtained from one another by a rotation of the cube. Then the number of inequivalent colouring schemes is $15$ $24$ $30$ $48$
edited 4 days ago in TIFR 6 views
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10
Let $X\subset \mathbb{R}^{2}$ be the subset $X=\left \{ \left ( x,y \right ) \left | x=0, \right |y \mid \leq 1\right \}\cup \left \{ \left ( x,y \right ) \mid 0 < x \leq 1, y=sin \frac{1}{x}\right \}.$ Consider the following statements: $X$ is compact $X$ is connected $X$ is path connected. How many of the statements (i)-(iii) is /are true? $0$ $1$ $2$ $3$
edited 4 days ago in TIFR 5 views
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11
The number of ring homomorphisms from $\mathbb{Z}\left [ x,y \right ]$ to $\mathbb{F}_{2}\left [ x \right ]/\left ( x^{3}+x^{2}+x+1 \right )$ equals $2^{6}$ $2^{18}$ $1$ $2^{9}$
edited 4 days ago in TIFR 5 views
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12
Consider functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with the property that $\left | f\left ( x \right )-f\left ( y \right ) \right |\leq 4321\left | x-y \right |$ for all real numbers $x,y$. Then which one of the following statement is true? $f$ is always ... $\underset{n\rightarrow\infty }{lim}\left | \frac{f\left ( x_{n} \right )}{x_{n}} \right |\leq 10000$
edited 4 days ago in TIFR 6 views
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13
Let the sequence $\left \{ x_{n} \right \}_{n\rightarrow 1}^{\infty }$ be defined by $x1=\sqrt{2}$ and $x_{n+1}=\left ( \sqrt{2} \right )^{x_{n}}$ for $n\geq 1$ ... increasing nor monotonically decreasing $\underset{n\rightarrow \infty }{lim}\:x_{n}$ does not exist $\underset{n\rightarrow \infty }{lim}\:x_{n}=\infty$
edited 4 days ago in TIFR 7 views
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14
Let $\left \{ f_{n} \right \}_{n=1}^{\infty}$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, defined by $f_{n}\left ( x \right )=\frac{1}{n}\:exp\left ( -n^{2} x^{2}\right ).$ Then which one of the ... converges pointwise but not uniformly on any interval containing the origin $\left \{{f}'_{n} \right \}$ converges pointwise but not uniformly on any interval containing the origin
edited 4 days ago in TIFR 7 views
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15
Let $f$ be a continuous function on $\left [ 0,1 \right ]$. Then the limit $\underset{n\rightarrow \infty }{lim}\int ^{1}_{0}nx^{n} f\left ( x \right )dx$ is equal to $f(0)$ $f(1)$ $\underset{x\in\left [ 0,1 \right ]}{sup} f\left ( x\right )$ The limit need not exist
edited 4 days ago in TIFR 7 views
1 answer
16
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
edited 4 days ago in TIFR 13 views
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17
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ is bounded and uniformly continuous bounded but not uniformly continuous not bounded but uniformly continuous not bounded and not uniformly continuous
edited 4 days ago in TIFR 6 views
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18
Consider the following two statements: $(E)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=p\left ( x \right )$). $(O)$ Continuous ... both false $(E)$ and $(O)$ are both true $(E)$ is true but $(O)$ is false $(E)$ is false but $(O)$ is true
edited 4 days ago in TIFR 4 views
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19
Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements: If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$ If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$. Then (i) and (ii) are true (i) and (ii) are false (i) is true and (ii) is false (i) is false and (ii) is true
edited 4 days ago in TIFR 12 views
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20
Let $V$ be an n-dimensional vector space and let $T:V\rightarrow V$ be a linear transformation such that $Rank\:T \leq Rank\:T^{3}$. Then which one of the following statements is necessarily true? Null space$(T)$ = Range$(T)$ Null space$(T)$ $\cap$ Range$(T)$={$0$} There exists a nonzero subspace $W$ of $V$ such that Null space$(T)$ $\cap$ range$(T)$=$W$ Null space$(T)$ $\subseteq$ Range$(T)$
edited 4 days ago in TIFR 7 views
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21
Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have: $f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$ Then which one of the following sentence is true? Every ... $a,b \in \mathbb{R}$
edited 4 days ago in TIFR 7 views
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22
Define a function: $f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$ Consider the following statements: ${f}'\left ( 0 \right )$ exists and is equal to $1$ $f$ is not increasing in any ... $f$ is increasing on $\mathbb{R}.$ How many of the above statements is/are true? $0$ $1$ $2$ $3$
edited 4 days ago in TIFR 7 views
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23
For $n\geq 1$, the sequence $\left \{ x_{n} \right \}^{\infty }_{n=1},$ where: $x_{n}=1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}-2\sqrt{n}$ is decreasing increasing constant oscillating
edited 4 days ago in TIFR 6 views
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24
The following sum of numbers (expressed in decimal notation) $1+11+111+\cdots +\underset{n}{\underbrace{11\dots1}}$ is equal to $\left ( 10^{n+1}-10-9n \right )/81$ $\left ( 10^{n+1}-10+9n \right )/81$ $\left ( 10^{n+1}-10-n \right )/81$ $\left ( 10^{n+1}-10+n \right )/81$
edited 4 days ago in TIFR 6 views
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25
True/False Question : Let $v:\mathbb{R}\rightarrow \mathbb{R}^{2}$ be $C^{\infty }$ (i.e., has derivatives of all orders). Then there exists $t_{0}\in \left ( 0,1 \right )$ such that $v\left ( 1 \right )-v\left ( 0 \right )$ is a scalar multiple of $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.
edited 4 days ago in TIFR 11 views
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26
True/False Question : There exists a uniformly continuous function $f:\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$ such that $\sum_{n=1 }^{\infty }\frac{1}{f\left ( n \right )}$ converges.
edited 4 days ago in TIFR 5 views
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27
True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.
edited 4 days ago in TIFR 5 views
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28
True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
edited 4 days ago in TIFR 7 views
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29
True/False Question : Let $H,N$ be subgroups of a finite group $G$, with $N$ a normal subgroup of $G$. If the orders of $G/N$ and $H$ are relatively prime, then $H$ is necessarily contained in $N$.
edited 4 days ago in TIFR 5 views
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30
True/False Question : Given any group $G$ of order $12$, and any $n$ that divides $12$, there exists a subgroup $H$ of $G$ of order $n$.
edited 4 days ago in TIFR 7 views
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