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TIFR2015MathsA15
+3
votes
76
views
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 $x \in \mathbb{R}$
tifrmaths2015
convergence
nongate
asked
Dec 20, 2015
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Set Theory & Algebra
by
makhdoom ghaya
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+2
votes
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answers
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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 $\sum a_{n}^{2}$
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Dec 20, 2015
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tifrmaths2015
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votes
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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
(
29.5k
points)

72
views
tifrmaths2015
convergence
nongate
+1
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0
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3
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
(
29.5k
points)

92
views
tifrmaths2015
groups
nongate
+2
votes
1
answer
4
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
(
29.5k
points)

75
views
tifrmaths2015
groups
groupisomorphism
nongate
+2
votes
0
answers
5
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
(
29.5k
points)

165
views
tifrmaths2015
nongate
+2
votes
0
answers
6
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
(
29.5k
points)

108
views
tifrmaths2015
functions
nongate
+2
votes
0
answers
7
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
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29.5k
points)

75
views
tifrmaths2015
settheory&algebra
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