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Recent questions and answers in Exam Queries

0 votes
1 answer
1
TIFR-2012-Maths-D: 32
The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
answered Sep 15 in TIFR ObitoUchiha 42 views
0 votes
1 answer
2
TIFR-2012-Maths-B: 12
Let $V$ be the vector space of consisting polynomials of $\mathbb{R}\left [ t \right ]$ 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.
Let $V$ be the vector space of consisting polynomials of $\mathbb{R}\left [ t \right ]$ 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 19 views
0 votes
1 answer
3
TIFR-2012-Maths-B: 11
The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian
The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian
answered Sep 10 in TIFR raju6 15 views
0 votes
1 answer
4
TIFR-2012-Maths-B: 13
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.
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 12 views
0 votes
1 answer
5
TIFR-2012-Maths-D: 33
The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable
The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable
answered Sep 9 in TIFR raju6 40 views
0 votes
1 answer
6
TIFR-2012-Maths-D: 39
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 $2$.
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 22 views
1 vote
1 answer
7
TIFR-2012-Maths-D: 31
$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
answered Sep 7 in TIFR ayush.5 90 views
0 votes
1 answer
8
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 $x,y \in \mathbb{R}$ are conjugate in $M_{2}\left ( \mathbb{R} \right )$ .
answered Sep 6 in TIFR ankitgupta.1729 22 views
0 votes
1 answer
9
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 $y\left ( 0\right ) \in \left [ 0,1 \right ]$. Then $\lim_{t\rightarrow \infty }y\left ( t \right )=1$ .
answered Sep 6 in TIFR ankitgupta.1729 16 views
0 votes
1 answer
10
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 function 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 function 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 13 views
0 votes
1 answer
11
TIFR-2019-Maths-A: 10
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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
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 14 views
0 votes
1 answer
12
TIFR-2019-Maths-B: 2
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
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 26 views
0 votes
1 answer
13
TIFR-2018-Maths-B: 9
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
answered Aug 31 in TIFR ankitgupta.1729 13 views
0 votes
1 answer
14
TIFR-2020-Maths-A: 2
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 $2$.
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 15 views
0 votes
0 answers
15
TIFR-2012-Maths-D: 34
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 $\frac{1}{2}.$
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 $\frac{1}{2}.$
asked Aug 30 in TIFR soujanyareddy13 30 views
0 votes
0 answers
16
TIFR-2012-Maths-D: 35
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.
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 20 views
0 votes
0 answers
17
TIFR-2012-Maths-D: 36
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 $v_{i},w_{i}\in \mathbb{C}.$
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 18 views
0 votes
0 answers
18
TIFR-2012-Maths-D: 37
If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
asked Aug 30 in TIFR soujanyareddy13 12 views
0 votes
0 answers
19
TIFR-2012-Maths-D: 38
The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
asked Aug 30 in TIFR soujanyareddy13 18 views
0 votes
0 answers
20
TIFR-2012-Maths-D: 40
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.
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 16 views
0 votes
0 answers
21
TIFR-2012-Maths-C: 21
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.
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 15 views
0 votes
0 answers
22
TIFR-2012-Maths-C: 22
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 $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
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 17 views
0 votes
0 answers
23
TIFR-2012-Maths-C: 23
Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
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 16 views
0 votes
0 answers
24
TIFR-2012-Maths-C: 24
The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
asked Aug 30 in TIFR soujanyareddy13 22 views
0 votes
0 answers
25
TIFR-2012-Maths-C: 25
If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
asked Aug 30 in TIFR soujanyareddy13 15 views
0 votes
0 answers
26
TIFR-2012-Maths-C: 26
If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
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 13 views
0 votes
0 answers
27
TIFR-2012-Maths-C: 27
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 $\left [ 0,1 \right ]$.
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 14 views
0 votes
0 answers
28
TIFR-2012-Maths-C: 28
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 $\mathbb{R}.$
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 15 views
0 votes
0 answers
29
TIFR-2012-Maths-C: 29
The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
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 13 views
0 votes
0 answers
30
TIFR-2012-Maths-C: 30
The composition of two uniformly continuous functions need not always be uniformly continuous.
The composition of two uniformly continuous functions need not always be uniformly continuous.
asked Aug 30 in TIFR soujanyareddy13 14 views
0 votes
0 answers
31
TIFR-2012-Maths-B: 14
$A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
$A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
asked Aug 30 in TIFR soujanyareddy13 8 views
0 votes
0 answers
32
TIFR-2012-Maths-B: 15
Let $P$ be an $n \times n $ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
Let $P$ be an $n \times n $ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
asked Aug 30 in TIFR soujanyareddy13 9 views
0 votes
0 answers
33
TIFR-2012-Maths-B: 16
There is a non trivial group homomorphism from $C$ to $R$.
There is a non trivial group homomorphism from $C$ to $R$.
asked Aug 30 in TIFR soujanyareddy13 6 views
0 votes
0 answers
34
TIFR-2012-Maths-B: 17
If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
asked Aug 30 in TIFR soujanyareddy13 8 views
0 votes
0 answers
35
TIFR-2012-Maths-B: 18
Any $3 \times3 $ and $5\times5$ skew-symmetric matrices have always zero determinants.
Any $3 \times3 $ and $5\times5$ skew-symmetric matrices have always zero determinants.
asked Aug 30 in TIFR soujanyareddy13 11 views
0 votes
0 answers
36
TIFR-2012-Maths-B: 19
The rank of the matrix $\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$ is $2$.
The rank of the matrix $\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$ is $2$.
asked Aug 30 in TIFR soujanyareddy13 10 views
0 votes
0 answers
37
TIFR-2012-Maths-B: 20
The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$
The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$
asked Aug 30 in TIFR soujanyareddy13 7 views
0 votes
1 answer
38
TIFR-2020-Maths-B: 10
True/False Question : Let $A,B \in M_{3}\left ( \mathbb{R} \right ).$ Then $det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$
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 12 views
0 votes
1 answer
39
TIFR-2020-Maths-A: 17
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$? $T$ is invertible as well as diagonalizable. $T$ is invertible, but not necessearily diagonalizable. $T$ is diagonalizable, but not necessary invertible. None of the other three statements.
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$? $T$ is invertible as well as diagonalizable. $T$ is invertible, but not necessearily diagonalizable. $T$ is diagonalizable, but not necessary invertible. None of the other three statements.
answered Aug 29 in TIFR ankitgupta.1729 17 views
0 votes
1 answer
40
TIFR-2020-Maths-A: 7
Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m$. Then the sequence diverges to.$+\infty$ has more than one ... . converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$.
Let $f\left ( x \right )=\frac{log\left ( 2+x \right )}{\sqrt{1+x}}$ for $x\geq 0$, and $a_{m}=\frac{1}{m}\int_{0}^{m}f\left ( t \right )dt$ for every positive integer $m$. Then the sequence diverges to.$+\infty$ has more than one limit point. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=\frac{1}{2}$log $2$. converges and satisfies $\lim_{m\rightarrow \infty }a_{m}=0$.
answered Aug 29 in TIFR ankitgupta.1729 12 views
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