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Recent questions and answers in Exam Queries

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1
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2
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asked Nov 16 in CBSE/UGC NET Sanjay Sharma 15 views
1 vote
1 answer
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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 $\frac{1}{2}.$
answered Nov 2 in TIFR zxy123 77 views
7 votes
6 answers
9
The following Finite Automaton recognizes which of the given languages? $\{ 1, 0 \}^* \{ 0 1 \}$ $\{ 1,0\}^*\{ 1\}$ $\{ 1 \} \{1, 0\}^*\{ 1 \}$ $1^*0^*\{0,1\}$
answered Oct 16 in GATE Musa 3.3k views
0 votes
1 answer
10
True/False Question : The equation $x^{3}+3x-4=0$ has exactly one real root.
answered Oct 12 in TIFR ayush.5 26 views
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1 answer
11
True/False Question: The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
answered Sep 15 in TIFR ObitoUchiha 57 views
0 votes
1 answer
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.
answered Sep 10 in TIFR raju6 42 views
0 votes
1 answer
13
True/False Question: The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian.
answered Sep 10 in TIFR raju6 24 views
0 votes
1 answer
14
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.
answered Sep 10 in TIFR raju6 21 views
0 votes
1 answer
15
True/False Question: The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable.
answered Sep 9 in TIFR raju6 65 views
0 votes
1 answer
16
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 $\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+\left | z_{3} \right |^{2}+\left | z_{4} \right |^{2}=1$, then the least value of $\left | z_{1} -z_{2}\right |^{2}+\left | z_{1} -z_{4}\right |^{2}+\left | z_{2}-z_{3} \right |^{2}+\left | z_{3} -z_{4}\right |^{2}$ is $2$.
answered Sep 9 in TIFR raju6 30 views
1 vote
1 answer
17
True/False Question: $f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
answered Sep 7 in TIFR ayush.5 140 views
0 votes
1 answer
18
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 $M_{2}\left ( \mathbb{R} \right )$ .
answered Sep 6 in TIFR ankitgupta.1729 27 views
0 votes
1 answer
19
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 $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
answered Sep 6 in TIFR ankitgupta.1729 21 views
0 votes
1 answer
20
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.
answered Sep 6 in TIFR ankitgupta.1729 22 views
0 votes
1 answer
21
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
answered Aug 31 in TIFR ankitgupta.1729 29 views
0 votes
1 answer
22
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
answered Aug 31 in TIFR ankitgupta.1729 38 views
0 votes
1 answer
23
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
answered Aug 31 in TIFR ankitgupta.1729 33 views
0 votes
1 answer
24
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 $2$.
answered Aug 31 in TIFR ankitgupta.1729 25 views
0 votes
0 answers
25
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.
asked Aug 30 in TIFR soujanyareddy13 61 views
0 votes
0 answers
26
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 $v_{i},w_{i}\in \mathbb{C}.$
asked Aug 30 in TIFR soujanyareddy13 38 views
0 votes
0 answers
27
True/False Question: If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
asked Aug 30 in TIFR soujanyareddy13 21 views
0 votes
0 answers
28
True/False Question: The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
asked Aug 30 in TIFR soujanyareddy13 28 views
0 votes
0 answers
29
True/False Question: Consider the differential equations (with $y$ is a function of $x$) $\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$ $\begin{matrix} \frac{dy}{dx} & = & \left | y \right |^{\frac{1}{3}}\\ y\left ( 0 \right ) & = & 0. \end{matrix}$ Then $(1)$ has infinitely many solutions but $(2)$ has finite number of solutions.
asked Aug 30 in TIFR soujanyareddy13 26 views
0 votes
0 answers
30
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.
asked Aug 30 in TIFR soujanyareddy13 19 views
0 votes
0 answers
31
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 $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
asked Aug 30 in TIFR soujanyareddy13 26 views
0 votes
0 answers
32
True/False Question: Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
asked Aug 30 in TIFR soujanyareddy13 20 views
0 votes
0 answers
33
True/False Question: The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
asked Aug 30 in TIFR soujanyareddy13 32 views
0 votes
0 answers
34
True/False Question: If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
asked Aug 30 in TIFR soujanyareddy13 21 views
0 votes
0 answers
35
True/False Question: If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
asked Aug 30 in TIFR soujanyareddy13 16 views
0 votes
0 answers
36
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 $\left [ 0,1 \right ]$.
asked Aug 30 in TIFR soujanyareddy13 22 views
0 votes
0 answers
37
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 $\mathbb{R}.$
asked Aug 30 in TIFR soujanyareddy13 27 views
0 votes
0 answers
38
True/False Question: The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
asked Aug 30 in TIFR soujanyareddy13 21 views
0 votes
0 answers
39
True/False Question: The composition of two uniformly continuous functions need not always be uniformly continuous.
asked Aug 30 in TIFR soujanyareddy13 17 views
0 votes
1 answer
40
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
answered Aug 30 in TIFR ankitgupta.1729 28 views
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