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Recent questions tagged tifrmaths2014
+3
votes
0
answers
1
TIFR2014MathsB10
How many maps $\emptyset:\mathbb{N}\cup \left\{0\right\}\rightarrow \mathbb{N} \cup \left\{0\right\}$ are there, with the property that $\emptyset(ab)=\emptyset(a)+\emptyset(b)$, for all $a, b \in \mathbb{N} \cup \left\{0\right\}$? None Finitely many Countably many Uncountably many.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

123
views
tifrmaths2014
settheory&algebra
functions
+1
vote
0
answers
2
TIFR2014MathsB9
Let $f : X \rightarrow Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if The space $X$ is compact The space $Y$ is compact The space $X$ is complete The space $Y$ is complete.
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

57
views
tifrmaths2014
vectorspace
nongate
+1
vote
0
answers
3
TIFR2014MathsB8
Let $X$ be a nonempty topological space such that every function $f : X \rightarrow \mathbb{R}$ is continuous. Then $X$ has the discrete topology $X$ has the indiscrete topology $X$ is compact $X$ is not connected.
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

83
views
tifrmaths2014
vectorspace
+1
vote
0
answers
4
TIFR2014MathsB7
$X$ is a topological space of infinite cardinality which is homomorphic to $X \times X$. Then $X$ is not connected $X$ is not compact $X$ is not homomorphic to a subset of $R$ None of the above.
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

39
views
tifrmaths2014
vectorspace
nongate
+1
vote
0
answers
5
TIFR2014MathsB6
The number of irreducible polynomials of the form $x^{2}+ax+b$, with $a, b$ in the field $\mathbb{F}_{7}$ of $7$ elements is: 7 21 35 49.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

53
views
tifrmaths2014
polynomials
nongate
+1
vote
0
answers
6
TIFR2014MathsB5
Which of the following groups are isomorphic? $\mathbb{R}$ and $C$ $\mathbb{R}^{*}$ and $C^{*}$ $S_{3}\times \mathbb{Z}/4$ and $S_{4}$ $\mathbb{Z}/2\times \mathbb{Z}/2$ and $\mathbb{Z}/4$
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

72
views
tifrmaths2014
settheory&algebra
groups
groupisomorphism
nongate
+2
votes
2
answers
7
TIFR2014MathsB4
Let $H_{1}$, $H_{2}$ be two distinct subgroups of a finite group $G$, each of order $2$. Let $H$ be the smallest subgroup containing $H_{1}$ and $H_{2}$. Then the order of $H$ is Always 2 Always 4 Always 8 None of the above.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

322
views
tifrmaths2014
settheory&algebra
groups
+1
vote
0
answers
8
TIFR2014MathsB3
Let $S_{n}$ be the symmetric group of $n$ letters. There exists an onto group homomorphism From $S_{5}$ to $S_{4}$ From $S_{4}$ to $S_{2}$ From $S_{5}$ to $\mathbb{Z}/5$ From $S_{4}$ to $\mathbb{Z}/4$
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

65
views
tifrmaths2014
settheory&algebra
groups
groupisomorphism
nongate
+1
vote
0
answers
9
TIFR2014MathsB2
Let $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous map such that $f(x) = 0$ for only finitely many values of $x$. Which of the following is true? Either $f(x)\leq 0$ for all $x$, or, $f(x) \geq 0$ for all $x$ The map $f$ is onto The map $f$ is onetoone None of the above.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

37
views
tifrmaths2014
functions
+1
vote
0
answers
10
TIFR2014MathsB1
Let $f : [0, 1] \rightarrow [0, \infty)$ be continuous. Suppose $\int_{0}^{x} f(t) \text{d}t \geq f(x)$, for all $x \in [0, 1]$. Then No such function exists There are infinitely many such functions There is only one such function There are exactly two such functions.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

62
views
tifrmaths2014
settheory&algebra
functions
+1
vote
0
answers
11
TIFR2014MathsA20
Let $C$ denote the cube $\left[1, 1\right]^{3} \subset \mathbb{R}^{3}$. How many rotations are there in $\mathbb{R}^{3}$ which take $C$ to itself? 6 12 18 24.
asked
Dec 17, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
29.5k
points)

76
views
tifrmaths2014
+1
vote
1
answer
12
TIFR2014MathsA19
For $n \in \mathbb{N}$, we define $s_{n}=1^{3}+2^{3}+3^{3}+...+n^{3}$. Which of the following holds for all $n \in \mathbb{N}$? $s_{n}$ is an odd integer $s_{n} n^{2}(n+1)^{2}/4$ $s_{n} = n(n + 1)(2n + 1)/6$ None of the above.
asked
Dec 17, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
29.5k
points)

95
views
tifrmaths2014
numericalability
+1
vote
1
answer
13
TIFR2014MathsA18
What is the last digit of $97^{2013}$? 1 3 7 9.
asked
Dec 17, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
29.5k
points)

145
views
tifrmaths2014
numericalability
unitdigit
+1
vote
1
answer
14
TIFR2014MathsA17
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $S$ be a nonempty proper subset of $R$. Which one of the following statements is always true? (Here $\bar{A}$ denotes the closure of $A$ and $A^{∘}$ denotes the interior of $A$). $f(S)^{∘} \subseteq$ f(S∘) $f(\bar{S}) \subseteq$ f(S) $f(\bar{S}) \supseteq f(S)$ $f(S)^{∘} \supseteq f(S^{∘})$.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

97
views
tifrmaths2014
functions
nongate
+1
vote
0
answers
15
TIFR2014MathsA16
$X$ is a metric space. $Y$ is a closed subset of $X$ such that the distance between any two points in $Y$ is at most $1$. Then $Y$ is compact Any continuous function from $Y \rightarrow \mathbb{R}$ is bounded $Y$ is not an open subset of $X$ none of the above.
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

126
views
tifrmaths2014
linearalgebra
vectorspace
nongate
+1
vote
0
answers
16
TIFR2014MathsA15
How many proper subgroups does the group $\mathbb{Z} ⊕ \mathbb{Z}$ have? $1$ $2$ $3$ Infinitely many
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

173
views
tifrmaths2014
groups
+2
votes
1
answer
17
TIFR2014MathsA14
Let $G$ be a group and let $H$ and $K$ be two subgroups of $G$. If both $H$ and $K$ have $12$ elements, which of the following numbers cannot be the cardinality of the set $HK = \left\{hk : h \in H, k \in K\right\}$? $72$ $60$ $48$ $36$.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

92
views
tifrmaths2014
settheory&algebra
groups
+1
vote
0
answers
18
TIFR2014MathsA13
Let $S$ be the set of all tuples $(x, y)$ with $x, y$ nonnegative real numbers satisfying $x + y = 2n$, for a fixed $n \in \mathbb{N}$. Then the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ on the set $S$ is $3n^{6}$ $2n^{6}$ $4n^{6}$ $n^{6}$
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

49
views
tifrmaths2014
settheory&algebra
+1
vote
0
answers
19
TIFR2014MathsA12
There exists a map $f : \mathbb{Z} \rightarrow \mathbb{Q}$ such that $f$ Is bijective and increasing Is onto and decreasing Is bijective and satisfies $f(n) \geq 0$ if $n \leq 0$ Has uncountable image.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

48
views
tifrmaths2014
functions
+1
vote
1
answer
20
TIFR2014MathsA11
Let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$ (0matrix), for some $k \in \mathbb{N}$. Then $A$ has to be the $0$ matrix Trace$(A)$ could be nonzero $A$ is diagonalizable $0$ is the only eigenvalue of $A$.
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

79
views
tifrmaths2014
linearalgebra
matrices
+2
votes
1
answer
21
TIFR2014MathsA10
Let $C\subset \mathbb{Z} \times\mathbb{Z}$ be the set of integer pairs $(a, b)$ for which the three complex roots $r_{1}, r_{2}$ and $r_{3}$ of the polynomial $p(x)=x^{3}2x^{2}+axb$ satisfy $r^{3}_{1}+r^{3}_{2}+r^{3}_{3}=0$. Then the cardinality of $C$ is $C = \infty$ $C = 0$ $C = 1$ $1 < C < \infty$
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

96
views
tifrmaths2014
sets
+1
vote
1
answer
22
TIFR2014MathsA9
Let $A(\theta)=\begin{pmatrix} \cos \theta& \sin \theta \\ \sin \theta& \cos \theta \end{pmatrix}$, where $\theta \in (0, 2\pi)$. Mark the correct statement below. $A(\theta)$ has eigenvectors in $\mathbb{R}2$ for all $θ \in (0, 2\pi)$ ... $A(\theta)$ has eigenvectors in $\mathbb{R} 2$ , for exactly $2$ values of $θ \in (0, 2\pi)$
asked
Dec 17, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

146
views
tifrmaths2014
linearalgebra
matrices
eigenvalue
+1
vote
0
answers
23
TIFR2014MathsA8
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x)−f(y) \geq \frac{1}{2}x−y$, for all $x, y \in \mathbb{R}$ . Then $f$ is both onetoone and onto $f$ is onetoone but may not be onto $f$ is onto but may not be onetoone $f$ is neither onetoone nor onto.
asked
Dec 17, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

115
views
tifrmaths2014
functions
+1
vote
0
answers
24
TIFR2014MathsA7
Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ such that $\lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$ Which of the following statements is always correct? $f_{n} \rightarrow 0$ ... $0$ pointwise $f_{n}$ will converge pointwise and the limit may be nonzero $f_{n}$ is not guaranteed to have a pointwise limit.
asked
Dec 14, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

55
views
tifrmaths2014
convergence
nongate
+1
vote
0
answers
25
TIFR2014MathsA6
Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$ $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1} f(x) \text{d}x)^{2}$ ... $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$
asked
Dec 14, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
29.5k
points)

111
views
tifrmaths2014
continuity
+1
vote
0
answers
26
TIFR2014MathsA5
Let $a_{n}=(n+1)^{100} e^{\sqrt{n}}$ for $n \geq 1$. Then the sequence $(a_{n})_{n}$ is Unbounded Bounded but does not converge Bounded and converges to $1$ Bounded and converges to $0$.
asked
Dec 14, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

37
views
tifrmaths2014
convergence
nongate
+1
vote
0
answers
27
TIFR2014MathsA4
Let $f$ be the real valued function on $[0, \infty)$ defined by $f(x) = \begin{cases} x^{\frac{2}{3}}\log x& \text {for x > 0} \\ 0& \text{if x=0 } \end{cases}$ Then $f$ is discontinuous at $x = 0$ $f$ ... $f$ is uniformly continuous on $[0, \infty)$ $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$.
asked
Dec 10, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
29.5k
points)

85
views
tifrmaths2014
continuity
+2
votes
1
answer
28
TIFR2014MathsA3
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $\displaystyle \lim_{x \to +\infty} f'(x)=1$, then $f$ is bounded $f$ is increasing $f$ is unbounded $f'$ is bounded.
asked
Dec 10, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
29.5k
points)

165
views
tifrmaths2014
differentiability
+1
vote
0
answers
29
TIFR2014MathsA2
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then: $f$ has to be uniformly continuous There exists an $x \in \mathbb{R}$ such that $f(x) = x$ $f$ cannot be increasing $\lim_{x \rightarrow \infty} f(x)$ exists.
asked
Dec 10, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
29.5k
points)

76
views
tifrmaths2014
continuity
+2
votes
1
answer
30
TIFR2014MathsA1
Let $A, B, C$ be three subsets of $\mathbb{R}$. The negation of the following statement For every $\epsilon > 1$, there exists $a \in A$ and $b \in B$ such that for all $c \in C, a − c < \epsilon$ and $b − c > \epsilon$ is ... and $b \in B$ there exists $c \in C$ such that $a − c \geq \epsilon$ or $b − c \leq \epsilon$ .
asked
Dec 10, 2015
in
Mathematical Logic
by
makhdoom ghaya
Boss
(
29.5k
points)

115
views
tifrmaths2014
mathematicallogic
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