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Recent questions tagged tifrmaths2014

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TIFR-2014-Maths-B-10
How many maps $\emptyset:\mathbb{N}\cup \left\{0\right\}\rightarrow \mathbb{N} \cup \left\{0\right\}$ ... ? None Finitely many Countably manyUncountably many
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335
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1 votes
TIFR-2014-Maths-B-9
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
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325
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TIFR-2014-Maths-B-8
Let $X$ be a non-empty 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
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290
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TIFR-2014-Maths-B-7
$X$ is a topological space of infinite cardinality which is homeomorphic to $X \times X$. Then $X$ is not connected $X$ is not compact $X$ is not homeomorphic to a subset of $R$ None of the above
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396
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2 votes
TIFR-2014-Maths-B-6
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
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392
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1 votes
TIFR-2014-Maths-B-5
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 $\mathbb{Z}/4$
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4 answers
2 votes
TIFR-2014-Maths-B-4
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
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596
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TIFR-2014-Maths-B-3
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 $\mathbb{Z}/4$
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245
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1 votes
TIFR-2014-Maths-B-2
Let $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous map such that $f(x) = 0$ for only finitely many values of $x$ ... $x$ The map $f$ is onto The map $f$ is one-to-one None of the above
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362
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1 votes
TIFR-2014-Maths-B-1
Let $f : [0, 1] \rightarrow [0, \infty)$ be continuous. Suppose$\int_{0}^{x} f(t) \text{d}t \geq f(x)$ ... exists There are infinitely many such functions There is only one such function There are exactly two such functions
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326
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TIFR-2014-Maths-A-20
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
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489
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1 answers
1 votes
TIFR-2014-Maths-A-19
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
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1 answers
1 votes
TIFR-2014-Maths-A-18
What is the last digit of $97^{2013}$? 1 3 7 9
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609
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1 answers
1 votes
TIFR-2014-Maths-A-17
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $S$ be a non-empty proper subset of $R$. Which one of the following statements is always true? ( ... $f(S)^{∘} \supseteq f(S^{∘})$.
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399
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1 votes
TIFR-2014-Maths-A-16
$X$ 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 ... $Y$ is not an open subset of $X$ none of the above
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TIFR-2014-Maths-A-15
How many proper subgroups does the group $\mathbb{Z} ⊕ \mathbb{Z}$ have? $1$2$ $3$ Infinitely many
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1 answers
3 votes
TIFR-2014-Maths-A-14
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$60$ $48$ $36$
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323
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0 answers
1 votes
TIFR-2014-Maths-A-13
Let $S$ be the set of all tuples $(x, y)$ with $x, y$ non-negative 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}$
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323
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0 answers
1 votes
TIFR-2014-Maths-A-12
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
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460
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1 answers
1 votes
TIFR-2014-Maths-A-11
Let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$ (0-matrix), for some $k \in \mathbb{N}$. Then$A$ has to be the $0$ matrix Trace$(A)$ could be non-zero $A$ is diagonalizable $0$ is the only eigenvalue of $A$
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481
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1 answers
2 votes
TIFR-2014-Maths-A-10
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 $r_{3}$ ... $ is $|C| = \infty$|C| = 0$ $|C| = 1$ $1 < |C| < \infty$
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591
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1 answers
1 votes
TIFR-2014-Maths-A-9
Let $A(\theta)=\begin{pmatrix}\cos \theta& \sin \theta \\-\sin \theta& \cos \theta \end{pmatrix}$, where $\theta \in (0, 2\pi)$ ... in $\mathbb{R}^2$ , for exactly $2$ values of $θ \in (0, 2\pi)$
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373
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0 answers
1 votes
TIFR-2014-Maths-A-8
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 $x, y \in \mathbb{R}$ ... be onto $f$ is onto but may not be one-to-one $f$ is neither one-to-one nor onto
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338
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0 answers
1 votes
TIFR-2014-Maths-A-7
Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ ... the limit may be non-zero $f_{n}$ is not guaranteed to have a point-wise limit
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457
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1 votes
TIFR-2014-Maths-A-6
Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ ...
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491
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1 answers
1 votes
TIFR-2014-Maths-A-5
Let $a_{n}=(n+1)^{100} e^{-\sqrt{n}}$ for $n \geq 1$. Then the sequence $(a_{n})_{n}$ isUnboundedBounded but does not converge Bounded and converges to $1$Bounded and converges to $0$
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395
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TIFR-2014-Maths-A-4
Let $f$ be the real valued function on $[0, \infty)$ ... $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$
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729
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1 answers
2 votes
TIFR-2014-Maths-A-3
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
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387
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1 votes
TIFR-2014-Maths-A-2
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then: $f$ has to be uniformly continuous There exists an $x \in \mathbb{R}$ ... $f$ cannot be increasing $\displaystyle \lim_{x \rightarrow \infty} f(x)$ exists
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558
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1 answers
2 votes
TIFR-2014-Maths-A-1
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 ... $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$
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