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Recent questions tagged tifrmaths2015
+2
votes
0
answers
1
TIFR2015MathsB15
Let $d(x, y)$ be the usual Euclidean metric on $\mathbb{R}^{2}$. Which of the following metric spaces is complete? $\mathbb{Q}^{2}\subset\mathbb{R}^{2}$ with the metric $d(x, y)$. $[0, 1]\times [0, \infty)\subset \mathbb{R}^{2}$ ... $d''(x, y) = \min \left \{ 1, d(x, y) \right \}$.
asked
Dec 21, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

163
views
tifrmaths2015
nongate
+2
votes
0
answers
2
TIFR2015MathsB14
Let $G$ be a group. Suppose $G= p^{2}q$, where $p$ and $q$ are distinct prime numbers satisfying $q ≢ 1 \mod p$. Which of the following is always true? $G$ has more than one $p$Sylow subgroup. $G$ has a normal $p$Sylow subgroup. The number of $q$Sylow subgroups of $G$ is divisible by $p$. $G$ has a unique $q$Sylow subgroup.
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

217
views
tifrmaths2015
groups
+2
votes
1
answer
3
TIFR2015MathsB13
Let $X=\left\{(x, y) \in \mathbb{R}^{2}: 2x^{2}+3y^{2}= 1\right\}$. Endow $\mathbb{R}^{2}$ with the discrete topology, and $X$ with the subspace topology. Then. $X$ is a compact subset of $\mathbb{R}^{2}$ in this topology. $X$ is a connected subset of $\mathbb{R}^{2}$ in this topology. $X$ is an open subset of $\mathbb{R}^{2}$ in this topology. None of the above.
asked
Dec 21, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

72
views
tifrmaths2015
linearalgebra
+2
votes
1
answer
4
TIFR2015MathsB12
Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$? $[0, 1]$ $(0, 1)$ $[0, 1)$ $\mathbb{R}$
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

157
views
tifrmaths2015
settheory&algebra
functions
+1
vote
0
answers
5
TIFR2015MathsB11
Let $(X, d)$ be a path connected metric space with at least two elements, and let $S=\left\{d(x, y):x, y \in X\right\}$. Which of the following statements is not necessarily true ? $S$ is infinite. $S$ contains a nonzero rational number. $S$ is connected. $S$ is a closed subset of $\mathbb{R}$
asked
Dec 21, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

88
views
tifrmaths2015
linearalgebra
nongate
+2
votes
1
answer
6
TIFR2015MathsB10
How many finite sequences $x_{1}, x_{2},...,x_{m}$ are there such that each $x_{i}=1$ or $2$, and $\sum_{i=1}^{m} x_{i}=10$ ? $89$ $91$ $92$ $120$
asked
Dec 21, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
41.1k
points)

133
views
tifrmaths2015
+2
votes
0
answers
7
TIFR2015MathsB9
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function and $A \subset \mathbb{R}$ be defined by $A=\left\{ y \in \mathbb{R}:y = \lim_{n \rightarrow \infty} f(x_{n}), \text {for some sequence} x_{n} \rightarrow +\infty\right\}$ Then the set A is necessarily. $A$ connected set $A$ compact set $A$ singleton set None of the above
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

103
views
tifrmaths2015
functions
nongate
+3
votes
2
answers
8
TIFR2015MathsB8
For a group $G$, let $F(G)$ denote the collection of all subgroups of $G$. Which one of the following situations can occur ? $G$ is finite but $F(G)$ is infinite. $G$ is infinite but $F(G)$ is finite. $G$ is countable but $F(G)$ is uncountable. $G$ is uncountable but $F(G)$ is countable.
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

235
views
tifrmaths2015
groups
+2
votes
0
answers
9
TIFR2015MathsB7
A complex number $\alpha \in \mathbb{C}$ is called algebraic if there is a nonzero polynomial $P(x) \in \mathbb{Q}\left[x\right]$ with rational coefficients such that $P(\alpha)=0$. Which of the following statements is true? There are only finitely many algebraic ... . All complex numbers are algebraic. $\sin \frac{\pi}{3}+ \cos \frac{\pi}{4}$ is algebraic. None of the above.
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

70
views
tifrmaths2015
settheory&algebra
nongate
+4
votes
1
answer
10
TIFR2015MathsB6
Let $f: [0, 1]\rightarrow \mathbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0, 1)$ and $f(0) = f(1) = 0$. Then the equation $f(x) = f' (x)$ admits. No solution $x \in (0, 1)$ More than one solution $x \in (0, 1)$ Exactly one solution $x \in (0, 1)$ At least one solution $x \in (0, 1)$
asked
Dec 21, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

136
views
tifrmaths2015
settheory&algebra
functions
+5
votes
2
answers
11
TIFR2015MathsB5
Let $n \geq 1$ and let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$, for some $k \geq 1$. Let $I$ be the identity $n \times n$ matrix. Then. $I+A$ need not be invertible. Det $(I+A)$ can be any nonzero real number. Det $(I+A) = 1$ $A^{n}$ is a nonzero matrix.
asked
Dec 20, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

264
views
tifrmaths2015
linearalgebra
matrices
+1
vote
0
answers
12
TIFR2015MathsB4
Let $U_{1}\supset U_{2} \supset...$ be a decreasing sequence of open sets in Euclidean $3$space $\mathbb{R}^{3}$. What can we say about the set $\cap U_{i}$ ? It is infinite. It is open. It is nonempty. None of the above.
asked
Dec 20, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

46
views
tifrmaths2015
linearalgebra
nongate
+1
vote
0
answers
13
TIFR2015MathsB3
Let $f$ be a function from $\left \{ 1, 2,....10 \right \}$ to $\mathbb{R}$ such that $(\sum_{i=1}^{10}\frac{f(i)}{2^{i}})^{2} = (\sum_{i=1}^{10} f(i)^{2}) (\sum_{i=1}^{10}\frac{1}{4^{i}})$ ... statement. There are uncountably many $f$ with this property. There are only countably infinitely many $f$ with this property. There is exactly one such $f$ There is no such $f$.
asked
Dec 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

69
views
tifrmaths2015
functions
+3
votes
1
answer
14
TIFR2015MathsB2
In how many ways can the group $\mathbb{Z}_{5}$ act on the set $\left \{ 1, 2, 3, 4, 5 \right \}$ ? $5$ $24$ $25$ $120$
asked
Dec 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

215
views
tifrmaths2015
permutationsandcombinations
groups
+4
votes
0
answers
15
TIFR2015B1
Let $X$ be a proper closed subset of $[0, 1]$. Which of the following statements is always true? The set $X$ is countable. There exists $x \in X$ such that $X ∖ \left \{ x \right \}$ is closed The set $X$ contains an open interval. None of the above.
asked
Dec 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

50
views
tifrmaths2015
sets
+3
votes
0
answers
16
TIFR2015MathsA15
The series $\sum_{n=1}^{\infty}\frac{\cos (3^{n}x)}{2^{n}}$ Diverges, for all rational $x \in \mathbb{R}$ Diverges, for some irrational $x \in \mathbb{R}$ Converges, for some but not all $x \in \mathbb{R}$ Converges, for all $x \in \mathbb{R}$
asked
Dec 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

72
views
tifrmaths2015
convergence
nongate
+2
votes
1
answer
17
TIFR2015MathsA14
Let $n \in \mathbb{N}$ be a six digit number whose base $10$ expansion is of the form $abcabc$, where $a, b, c$ are digits between $0$ and $9$ and $a$ is nonzero. Then, $n$ is divisible by $5$ $n$ is divisible by $8$ $n$ is divisible by $13$ $n$ is divisible by $17$
asked
Dec 20, 2015
in
Numerical Ability
by
makhdoom ghaya
Boss
(
41.1k
points)

55
views
tifrmaths2015
numericalability
+3
votes
2
answers
18
TIFR2015MathsA13
For a real number $t >0$, let $\sqrt{t}$ denote the positive square root of $t$. For a real number $x > 0$, let $F(x)= \int_{x^{2}}^{4x^{2}} \sin \sqrt{t} dt$. If $F'$ is the derivative of $F$, then $F'(\frac{\pi}{2}) = 0$ $F'(\frac{\pi}{2}) = \pi$ $F'(\frac{\pi}{2}) =  \pi$ $F'(\frac{\pi}{2}) = 2\pi$
asked
Dec 20, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
41.1k
points)

91
views
tifrmaths2015
calculus
+2
votes
0
answers
19
TIFR2015MathsA12
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function that vanishes at $10$ distinct points in $\mathbb{R}$. Suppose $f^{(n)}$ denotes the $n$th derivative of $f$, for $n \geq 1$. Which of the following statements is always true? $f^{(n)}$ has at ... $f^{(n)}$ has at least $10$ zeros, for $n \geq 10$ $f^{(n)}$ has at least one zero, for $n \geq 9$
asked
Dec 20, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
41.1k
points)

67
views
tifrmaths2015
functions
calculus
+2
votes
0
answers
20
TIFR2015MathsA11
Let $\left\{a_{n}\right\}$ be a sequence of real numbers. Which of the following is true? If $\sum a_{n}$ converges, then so does $\sum a_{n}^{4}$ If $\sum a_{n}$ converges, then so does $\sum a_{n}^{2}$ If $\sum a_{n}$ diverges, then so does $\sum a_{n}^{3}$ If $\sum a_{n}$ diverges, then so does $\sum a_{n}^{2}$
asked
Dec 20, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

62
views
tifrmaths2015
convergence
nongate
+1
vote
0
answers
21
TIFR2015MathsA10
For a group $G$, let Aut(G) denote the group of automorphisms of $G$. Which of the following statements is true? Aut$(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_{2}$ If $G$ is cyclic, then Aut $(G)$ is cyclic. If Aut (G) is trivial, then $G$ is trivial. Aut $(\mathbb{Z})$ is isomorphic to $\mathbb{Z}$
asked
Dec 19, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

88
views
tifrmaths2015
groups
nongate
+3
votes
0
answers
22
TIFR2015MathsA9
Let $\left\{a_{n}\right\}$ be a sequence of real numbers such that $a_{n+1}a_{n}\leq \frac{n^{2}}{2^{n}}$ for all $n \in \mathbb{N}$. Then The sequence $\left\{a_{n}\right\}$ may be unbounded. The sequence $\left\{a_{n}\right\}$ is bounded but may not converge. The sequence $\left\{a_{n}\right\}$ has exactly two limit points. The sequence $\left\{a_{n}\right\}$ is convergent.
asked
Dec 19, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

71
views
tifrmaths2015
convergence
nongate
+3
votes
1
answer
23
TIFR2015MathsA8
Let $f(x)=\frac{e^{\frac{1}{x}}}{x}$, where $x \in (0, 1)$. Then on $(0, 1)$. $f$ is uniformly continuous. $f$ is continuous but not uniformly continuous. $f$ is unbounded. $f$ is not continuous.
asked
Dec 19, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
41.1k
points)

188
views
tifrmaths2015
calculus
continuity
+2
votes
1
answer
24
TIFR2015MathsA7
Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct? If $g$ is continuous, then $f ∘ g$ is continuous. If $f$ is continuous, then $f ∘ g$ is ... $f ∘ g$ are continuous, then $g$ is continuous. If $g$ and $f ∘ g$ are continuous, then $f$ is continuous.
asked
Dec 19, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

118
views
tifrmaths2015
functions
continuity
+5
votes
1
answer
25
TIFR2015MathsA6
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix} \sin\frac{\pi}{18}&\sin \frac{4\pi}{9} \\ \sin \frac{4\pi}{9}&\sin \frac {\pi}{18} \end{pmatrix}$. Then the smallest number $n \in \mathbb{N}$ such that $A^{n}=1$ is. $3$ $9$ $18$ $27$
asked
Dec 19, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

143
views
tifrmaths2015
matrices
linearalgebra
+3
votes
1
answer
26
TIFR2015MathsA5
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote the function defined by $f(x)= (1x^{2})^{\frac{3}{2}}$ if $x < 1$, and $f(x)=0$ if $x \geq 1$. Which of the following statements is correct ? $f$ is not continuous $f$ is continuous but not differentiable $f$ is differentiable but $f'$ is not continuous. $f$ is differentiable and $f'$ is continuous.
asked
Dec 19, 2015
in
Calculus
by
makhdoom ghaya
Boss
(
41.1k
points)

175
views
tifrmaths2015
continuity
differentiability
+2
votes
1
answer
27
TIFR2015MathsA4
Let $S$ be the collection of (isomorphism classes of) groups $G$ which have the property that every element of $G$ commutes only with the identity element and itself. Then $S = 1$ $S = 2$ $S \geq 3$ and is finite. $S = \infty$
asked
Dec 19, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

71
views
tifrmaths2015
groups
groupisomorphism
nongate
+1
vote
0
answers
28
TIFR2015MathsA3
Let $A$ be a $10 \times 10$ matrix with complex entries such that all its eigenvalues are nonnegative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false ? There exists a matrix $B$ such that $ABBA = B$. There exists a ... $B$ such that $AB+BA=A$. There exists a matrix $B$ such that $AB+BA=B$.
asked
Dec 19, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

167
views
tifrmaths2015
linearalgebra
matrices
eigenvalue
+2
votes
0
answers
29
TIFR2015MathsA2
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Which one of the following sets cannot be the image of $(0, 1]$ under $f$? {0} (0, 1) [0, 1) [0, 1]
asked
Dec 19, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

62
views
tifrmaths2015
functions
+3
votes
1
answer
30
TIFR2015MathsA1
Let $A$ be an invertible $10 \times 10$ matrix with real entries such that the sum of each row is $1$. Then The sum of the entries of each row of the inverse of $A$ is $1$. The sum of the entries of each column of the inverse of $A$ is $1$. The trace of the inverse of $A$ is nonzero. None of the above.
asked
Dec 19, 2015
in
Linear Algebra
by
makhdoom ghaya
Boss
(
41.1k
points)

215
views
tifrmaths2015
linearalgebra
matrices
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