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1
TIFR2021-Maths-B: 11
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
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2
TIFR2021-Maths-B: 20
Let $c_1,c_2>0,$ and let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be functions (not assumed to be continuous) such that for all $x\in \mathbb{R}$ $f(x+c_1)=f(x) \text{ and } g(x+c_2)=g(x).$ Further, assume that $\displaystyle \lim_{x\rightarrow\infty}(f(x)-g(x))=0.$ Then $f=g.$
Let $c_1,c_2>0,$ and let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be functions (not assumed to be continuous) such that for all $x\in \mathbb{R}$$$f(x+c_1)=f(x) \text{ and }...
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3
TIFR2021-Maths-B: 19
The set of polynomials in the ring $\mathbb{Z}[x]$, the sum of whose coefficients is zero, forms an ideal of the ring $\mathbb{Z}[x]$.
The set of polynomials in the ring $\mathbb{Z}[x]$, the sum of whose coefficients is zero, forms an ideal of the ring $\mathbb{Z}[x]$.
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4
TIFR2021-Maths-B: 18
The set $\{n\in\mathbb{N} | n \text{ divides } a^3-1, \text{ for all integers } \text{$a$ such that gcd}(a,n)=1\}$ is infinite.
The set $$\{n\in\mathbb{N} | n \text{ divides } a^3-1, \text{ for all integers } \text{$a$ such that gcd}(a,n)=1\}$$is infinite.
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5
TIFR2021-Maths-B: 17
There exists an injective ring homomorphism from the ring $\mathbb{Q}[x,y]/(x^2-y^2)$ into the ring $\mathbb{Q}[x,y]/(x-y^2)$.
There exists an injective ring homomorphism from the ring $\mathbb{Q}[x,y]/(x^2-y^2)$ into the ring $\mathbb{Q}[x,y]/(x-y^2)$.
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TIFR2021-Maths-B: 16
There exists a nonzero ideal $I \subseteq \mathbb{Z}[i]$ such that the quotient ring $\mathbb{Z}[i]/I$ is infinite $($here $i$ is a square root of $-1$ in $\mathbb{C})$.
There exists a nonzero ideal $I \subseteq \mathbb{Z}[i]$ such that the quotient ring $\mathbb{Z}[i]/I$ is infinite $($here $i$ is a square root of $-1$ in $\mathbb{C})$.
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7
TIFR2021-Maths-B: 15
There exists a countable group having uncountably many subgroups.
There exists a countable group having uncountably many subgroups.
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TIFR2021-Maths-B: 14
If G is a finite group such that the group $\text{Aut}(G)$ of automorphisms of $G$ is cyclic, then $G$ is abelian.
If G is a finite group such that the group $\text{Aut}(G)$ of automorphisms of $G$ is cyclic, then $G$ is abelian.
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9
TIFR2021-Maths-B: 13
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
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10
TIFR2021-Maths-B: 12
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
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11
TIFR2021-Maths-B: 10
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 30^{\circ}$, using only a straight edge and a compass.
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 3...
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12
TIFR2021-Maths-B: 9
The set $\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$ is finite.
The set$$\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$$is finite.
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13
TIFR2021-Maths-B: 8
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such that $f(x)=x$.
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such th...
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14
TIFR2021-Maths-B: 7
There exists a continuous function $f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=\text{Id}$.
There exists a continuous function$$f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$$such that $f(0)=0$ and $f(1)=\text{Id}$.
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15
TIFR2021-Maths-B: 6
Define a metric on the set of finite subsets of $\mathbb{Z}$ as ollows: $d(A,B)=\text{the cardinality of } (A\cup B \backslash (A\cap B)).$ The resulting metric space admits an isometry into $\mathbb{R}^n,$ for some positive integer $n$.
Define a metric on the set of finite subsets of $\mathbb{Z}$ as ollows:$$d(A,B)=\text{the cardinality of } (A\cup B \backslash (A\cap B)).$$The resulting metric space adm...
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16
TIFR2021-Maths-B: 5
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
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17
TIFR2021-Maths-B: 4
If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.
If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.
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18
TIFR2021-Maths-B: 3
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
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19
TIFR2021-Maths-B: 2
Let $(X,d)$ be an infinite compact metric space. Then there exists no function $f:X\rightarrow X$, continuous or otherwise, with the property that $d(f(x),f(y))>d(x,y)$ for all $x\neq y$.
Let $(X,d)$ be an infinite compact metric space. Then there exists no function $f:X\rightarrow X$, continuous or otherwise, with the property that $d(f(x),f(y))>d(x,y)$ f...
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20
TIFR2021-Maths-B: 1
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2 dx=0,$ then $\displaystyle\lim_{n\rightarrow \infty} f_n\left(\frac{1}{2}\right)=0.$
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $$\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2...
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21
TIFR2021-Maths-A: 20
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have? $13$ $16$ $4$ $25$
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have?$13$$16$$4$$25$
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TIFR2021-Maths-A: 19
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection? $43$ $45$ $47$ none of the other three options
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection?$43$$45$$4...
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23
TIFR2021-Maths-A: 18
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition: $A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degree at most } 2\}, \text{ and} $ ... $[B:A]$ of $A$ in $B$ equals $1$ $2$ $4$ none of the other three options
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition:$$A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degre...
1 answer
25
TIFR2021-Maths-A: 16
The matrix $\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$ is diagonalizable nilpotent idempotent none of the other three options
The matrix$$\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$$isdiagonalizablenilpotentidempotentnone of the other three options
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28
TIFR2021-Maths-A: 13
$T:\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ be the $\mathbb{C}-$linear transformation defined on the complex vector space $\mathbb{C}[x]$ of one variable complex polynomials by $Tf(x)=f(x+1)$. How many eigenvalues does $T$ have? $1$ finite but more than $1$ countably infinite uncountable
$T:\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ be the $\mathbb{C}-$linear transformation defined on the complex vector space $\mathbb{C}[x]$ of one variable complex polynomial...
2 answers
30
TIFR2021-Maths-A: 11
What is the number of surjective maps from the set $\{1,\dots,10\}$ to the set $\{1,2\}$? $90$ $1022$ $98$ $1024$
What is the number of surjective maps from the set $\{1,\dots,10\}$ to the set $\{1,2\}$?$90$$1022$$98$$1024$