Dark Mode

Recent questions tagged tifrmaths2014

3 votes

0 answers

1

TIFR-2014-Maths-B-10
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

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

226 views

1 vote

0 answers

2

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

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

159 views

1 vote

0 answers

3

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

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

180 views

1 vote

0 answers

4

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

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

147 views

2 votes

0 answers

5

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

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

167 views

1 vote

0 answers

6

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 $\mathbb{Z}/4$

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

275 views

2 votes

4 answers

7

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

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

1.8k views

1 vote

0 answers

8

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 $\mathbb{Z}/4$

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

311 views

1 vote

0 answers

9

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 $x$. 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 one-to-one None of the above

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

144 views

1 vote

0 answers

10

TIFR-2014-Maths-B-1
Let $f : [0, 1] \rightarrow [0, \infty)$ be continuous. Suppose $\int_{0}^{x} f(t) \text{d}t \geq f(x)$, 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

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

213 views

1 vote

0 answers

11

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

makhdoom ghaya asked in Quantitative Aptitude Dec 17, 2015

by makhdoom ghaya

180 views

1 vote

1 answer

12

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

makhdoom ghaya asked in Quantitative Aptitude Dec 17, 2015

by makhdoom ghaya

199 views

1 vote

1 answer

13

TIFR-2014-Maths-A-18
What is the last digit of $97^{2013}$? 1 3 7 9

makhdoom ghaya asked in Quantitative Aptitude Dec 17, 2015

by makhdoom ghaya

387 views

1 vote

1 answer

14

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? (Here $\bar{A}$ denotes the closure of $A$ and $A^{∘}$ denotes the interior of $A$ ... $f(\bar{S}) \supseteq \overline{f(S)}$ $f(S)^{∘} \supseteq f(S^{∘})$.

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

364 views

1 vote

0 answers

15

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 function from $Y \rightarrow \mathbb{R}$ is bounded $Y$ is not an open subset of $X$ none of the above

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

237 views

2 votes

1 answer

16

TIFR-2014-Maths-A-15
How many proper subgroups does the group $\mathbb{Z} ⊕ \mathbb{Z}$ have? $1$ $2$ $3$ Infinitely many

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

787 views

3 votes

1 answer

17

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 $HK = \left\{hk : h \in H, k \in K\right\}$? $72$ $60$ $48$ $36$

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

626 views

1 vote

0 answers

18

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 $3n^{6}$ $2n^{6}$ $4n^{6}$ $n^{6}$

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

194 views

1 vote

0 answers

19

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

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

198 views

1 vote

1 answer

20

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 $A$

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

293 views

2 votes

1 answer

21

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 $r_{3}$ of the polynomial $p(x)=x^{3}-2x^{2}+ax-b$ satisfy $r^{3}_{1}+r^{3}_{2}+r^{3}_{3}=0$. Then the cardinality of $C$ is $|C| = \infty$ $|C| = 0$ $|C| = 1$ $1 < |C| < \infty$

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

287 views

1 vote

1 answer

22

TIFR-2014-Maths-A-9
Let $A(\theta)=\begin{pmatrix} \cos \theta& \sin \theta \\ -\sin \theta& \cos \theta \end{pmatrix}$, where $\theta \in (0, 2\pi)$. Mark the correct statement below. $A(\theta)$ has eigenvectors in $\mathbb{R}^2$ for all $θ \in (0, 2\pi)$ $A(\theta)$ does not ... $θ \in (0, 2\pi)$ $A(\theta)$ has eigenvectors in $\mathbb{R}^2$ , for exactly $2$ values of $θ \in (0, 2\pi)$

makhdoom ghaya asked in Linear Algebra Dec 17, 2015

by makhdoom ghaya

340 views

1 vote

0 answers

23

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 $x, y \in \mathbb{R}$ . Then $f$ is both one-to-one and onto $f$ is one-to-one but may not be onto $f$ is onto but may not be one-to-one $f$ is neither one-to-one nor onto

makhdoom ghaya asked in Set Theory & Algebra Dec 17, 2015

by makhdoom ghaya

236 views

1 vote

0 answers

24

TIFR-2014-Maths-A-7
Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ such that $\displaystyle \lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$ ... $0$ point-wise $f_{n}$ will converge point-wise and the limit may be non-zero $f_{n}$ is not guaranteed to have a point-wise limit

makhdoom ghaya asked in Set Theory & Algebra Dec 14, 2015

by makhdoom ghaya

206 views

1 vote

0 answers

25

TIFR-2014-Maths-A-6
Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$ $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$ ... $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$

makhdoom ghaya asked in Calculus Dec 14, 2015

by makhdoom ghaya

242 views

1 vote

1 answer

26

TIFR-2014-Maths-A-5
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 $0$

makhdoom ghaya asked in Set Theory & Algebra Dec 14, 2015

by makhdoom ghaya

230 views

1 vote

0 answers

27

TIFR-2014-Maths-A-4
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 $(0, \infty)$

makhdoom ghaya asked in Calculus Dec 10, 2015

by makhdoom ghaya

213 views

2 votes

1 answer

28

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

makhdoom ghaya asked in Calculus Dec 10, 2015

by makhdoom ghaya

447 views

1 vote

0 answers

29

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 $x \in \mathbb{R}$ such that $f(x) = x$ $f$ cannot be increasing $\displaystyle \lim_{x \rightarrow \infty} f(x)$ exists

makhdoom ghaya asked in Set Theory & Algebra Dec 10, 2015

by makhdoom ghaya

236 views

2 votes

1 answer

30

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 all $c \in C, |a − c| < \epsilon$ and $|b − c| > \epsilon$ ... all $a \in A$ and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$

makhdoom ghaya asked in Mathematical Logic Dec 10, 2015

by makhdoom ghaya

268 views

To see more, click for the full list of questions or popular tags.

Recent questions tagged tifrmaths2014