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Recent questions tagged tifrmaths2012

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Recent questions tagged tifrmaths2012

1 vote
1 answer
1
TIFR-2012-Maths-D: 31
True/False Question: $f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
True/False Question: $f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.
asked Aug 30 in TIFR soujanyareddy13 140 views
0 votes
1 answer
2
TIFR-2012-Maths-D: 32
True/False Question: The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
True/False Question: The polynomial $X^{8}+1$ is irreducible in $\mathbb{R}\left [ X \right ]$.
asked Aug 30 in TIFR soujanyareddy13 57 views
0 votes
1 answer
3
TIFR-2012-Maths-D: 33
True/False Question: The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable.
True/False Question: The matrix $\begin{pmatrix} 1 & \pi &3 \\ 0& 2&4 \\ 0&0 &3 \end{pmatrix}$ is diagonalisable.
asked Aug 30 in TIFR soujanyareddy13 65 views
1 vote
1 answer
4
TIFR-2012-Maths-D: 34
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 $\frac{1}{2}.$
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 $\frac{1}{2}.$
asked Aug 30 in TIFR soujanyareddy13 78 views
0 votes
0 answers
5
TIFR-2012-Maths-D: 35
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.
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
6
TIFR-2012-Maths-D: 36
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 $v_{i},w_{i}\in \mathbb{C}.$
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
7
TIFR-2012-Maths-D: 37
True/False Question: If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
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
8
TIFR-2012-Maths-D: 38
True/False Question: The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
True/False Question: The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
asked Aug 30 in TIFR soujanyareddy13 28 views
0 votes
1 answer
9
TIFR-2012-Maths-D: 39
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 $\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+\left | z_{3} \right |^{2}+\left | z_{4} \right |^{2}=1$ ... $2$.
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 $2$.
asked Aug 30 in TIFR soujanyareddy13 30 views
0 votes
0 answers
10
TIFR-2012-Maths-D: 40
True/False Question: Consider the differential equations (with $y$ is a function of $x$) $\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$ ... $(1)$ has infinitely many solutions but $(2)$ has finite number of solutions.
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
11
TIFR-2012-Maths-C: 21
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.
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
12
TIFR-2012-Maths-C: 22
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 $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
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
13
TIFR-2012-Maths-C: 23
True/False Question: Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
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
14
TIFR-2012-Maths-C: 24
True/False Question: The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
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
15
TIFR-2012-Maths-C: 25
True/False Question: If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
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
16
TIFR-2012-Maths-C: 26
True/False Question: If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
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
17
TIFR-2012-Maths-C: 27
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 $\left [ 0,1 \right ]$.
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
18
TIFR-2012-Maths-C: 28
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 $\mathbb{R}.$
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
19
TIFR-2012-Maths-C: 29
True/False Question: The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
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 23 views
0 votes
0 answers
20
TIFR-2012-Maths-C: 30
True/False Question: The composition of two uniformly continuous functions need not always be uniformly continuous.
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
21
TIFR-2012-Maths-B: 11
True/False Question: The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian.
True/False Question: The automorphism group $Aut\left ( \mathbb{Z}/2\times \mathbb{Z}/2 \right )$ is abelian.
asked Aug 30 in TIFR soujanyareddy13 24 views
0 votes
1 answer
22
TIFR-2012-Maths-B: 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.
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.
asked Aug 30 in TIFR soujanyareddy13 43 views
0 votes
1 answer
23
TIFR-2012-Maths-B: 13
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.
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.
asked Aug 30 in TIFR soujanyareddy13 21 views
0 votes
0 answers
24
TIFR-2012-Maths-B: 14
True/False Question: $A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
True/False Question: $A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
asked Aug 30 in TIFR soujanyareddy13 15 views
0 votes
0 answers
25
TIFR-2012-Maths-B: 15
True/False Question: 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 $1$.
True/False Question: 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 17 views
0 votes
0 answers
26
TIFR-2012-Maths-B: 16
True/False Question: There is a non trivial group homomorphism from $C$ to $R$.
True/False Question: There is a non trivial group homomorphism from $C$ to $R$.
asked Aug 30 in TIFR soujanyareddy13 12 views
0 votes
0 answers
27
TIFR-2012-Maths-B: 17
True/False Question: If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
True/False Question: If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
asked Aug 30 in TIFR soujanyareddy13 15 views
0 votes
0 answers
28
TIFR-2012-Maths-B: 18
True/False Question: Any $3\times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
True/False Question: Any $3\times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
asked Aug 30 in TIFR soujanyareddy13 19 views
0 votes
0 answers
29
TIFR-2012-Maths-B: 19
True/False Question: 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 $2$.
True/False Question: 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 33 views
0 votes
0 answers
30
TIFR-2012-Maths-B: 20
True/False Question: The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$.
True/False Question: The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$.
asked Aug 30 in TIFR soujanyareddy13 17 views
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