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.

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 makhdoom ghaya 140 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.

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 makhdoom ghaya 86 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.

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.

asked Dec 17, 2015 in Linear Algebra makhdoom ghaya 104 views

1 vote

0 answers

4

TIFR-2014-Maths-B-7
$X$ 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.

$X$ 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 makhdoom ghaya 56 views

1 vote

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.

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 makhdoom ghaya 65 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 $\mathbb{Z}/4$

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 makhdoom ghaya 107 views

2 votes

3 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.

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 makhdoom ghaya 528 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 $\mathbb{Z}/4$

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 makhdoom ghaya 93 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.

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.

asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 48 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.

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 makhdoom ghaya 82 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.

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 makhdoom ghaya 90 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.

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 makhdoom ghaya 116 views

1 vote

1 answer

13

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

What is the last digit of $97^{2013}$? 1 3 7 9.

asked Dec 17, 2015 in Numerical Ability makhdoom ghaya 176 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 $A$). $f(S)^{∘} \subseteq$ f(S∘) $f(\bar{S}) \subseteq$ f(S) $f(\bar{S}) \supseteq f(S)$ $f(S)^{∘} \supseteq f(S^{∘})$.

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 $A$). $f(S)^{∘} \subseteq$ 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 makhdoom ghaya 124 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.

$X$ 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 makhdoom ghaya 148 views

2 votes

1 answer

16

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

How many proper subgroups does the group $\mathbb{Z} &oplus; \mathbb{Z}$ have? $1$ $2$ $3$ Infinitely many

asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 305 views

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

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 makhdoom ghaya 131 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 $3n^{6}$ $2n^{6}$ $4n^{6}$ $n^{6}$

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

asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 67 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.

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 makhdoom ghaya 70 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 $A$.

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 $A$.

asked Dec 17, 2015 in Linear Algebra makhdoom ghaya 129 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 $|C| = \infty$ $|C| = 0$ $|C| = 1$ $1 < |C| < \infty$

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 $|C| = \infty$ $|C| = 0$ $|C| = 1$ $1 < |C| < \infty$

asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 135 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 $θ \in (0, 2\pi)$ ... $A(\theta)$ has eigenvectors in $\mathbb{R} 2$ , for exactly $2$ values of $θ \in (0, 2\pi)$

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 have an ... $θ \in (0, 2\pi)$ $A(\theta)$ has eigenvectors in $\mathbb{R} 2$ , for exactly $2$ values of $θ \in (0, 2\pi)$

asked Dec 17, 2015 in Linear Algebra makhdoom ghaya 185 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.

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.

asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 135 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 $\lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$ Which of the following statements is always correct? $f_{n} \rightarrow 0$ ... $0$ 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.

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? $f_{n} \rightarrow 0$ ... to $0$ 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.

asked Dec 14, 2015 in Set Theory & Algebra makhdoom ghaya 88 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? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$ $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$ ... $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$

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 makhdoom ghaya 131 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 $0$.

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 makhdoom ghaya 59 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 $(0, \infty)$.

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 $x = 0$ $f$ is continuous on $[0, \infty)$, but not ... $f$ 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 makhdoom ghaya 105 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.

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 makhdoom ghaya 218 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 $\lim_{x \rightarrow \infty} f(x)$ exists.

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 makhdoom ghaya 94 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 $|b − c| > \epsilon$ is ... and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$ .

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 There exists $\epsilon \leq 1$, such ... $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$ .

asked Dec 10, 2015 in Mathematical Logic makhdoom ghaya 140 views

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