# Recent questions tagged tifrmaths2016

1
Let $A_{1}\supset A_{2}\supset \cdots \supset A_{n}\supset A_{n+1}\supset \cdots$ be an infinite sequence of non-empty subsets of $\mathbb{R}^{3}$. which of the following conditions ensures that their intersection is non-empty ? Each $A_{i}$ is uncountable Each $A_{i}$ is open Each $A_{i}$ is connected Each $A_{i}$ is compact.
2
Let $(X,d)$ be a metric space. Which of the following is possible ? $X$ has exactly $3$ dense subsets $X$ has exactly $4$ dense subsets $X$ has exactly $5$ dense subsets $X$ has exactly $6$ dense subsets.
3
Let $\left \{ f_{n} \right \}^{\infty }_{n=1}$ be the sequence of functions on $\mathbb{R}$ defined by $f_{n}\left ( x \right )=n^{2}x^{n}.$. Let $A$ be the set of all points $a$ in $\mathbb{R}$ ... . Then $A=\left \{0\right \}$ $A=\left [0,1\right )$ $A=\mathbb{R}\setminus \left \{-1,1\right \}$ $A=\left (-1,1\right ).$
4
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that $f\left ( i \right )=0$, for all $i \in \mathbb{Z}$. Which of the following statements is always true ? Image $(f)$ is closed in $\mathbb{R}$ Image $(f)$ is open in $\mathbb{R}$ $f$ is uniformly continuous None of the above.
5
Let $S^{1}= \left \{ z \in \mathbb{C} :\left | z \right | =1\right \}$ be the unit circle. Which of the following is false ? Any continuous function from $S^{1}$ to $\mathbb{R}$ is bounded is uniformly continuous has image containing a non-empty open subset of $\mathbb{R}$ has a point $z \in S^{1}$ such that $f\left ( z \right )=f\left ( -z \right )$.
6
Which of the following is false ? Any continuous function from $\left [ 0,1 \right ]$ to $\left [ 0,1 \right ]$ has a fixed point Any homeomorphism from $\left [ 0,1 \right )$ to $\left [ 0,1 \right )$ ... $\left ( 0,1 \right )$ to $\left ( 0,1 \right )$ has a fixed point.
7
For $n\geq 1$, let $S_{n}$ denote the group of all permutations on $n$ symbols. Which of the following statements is true? $S_{3}$ has an element of order $4$ $S_{4}$ has an element of order $6$ $S_{4}$ has an element of order $5$ $S_{5}$ has an element of order $6$.
8
Which of the following rings is an integral domain ? $\mathbb{R}\left [ x \right ]/\left ( x^{2} +x +1 \right )$ $\mathbb{R}\left [ x \right ]/\left ( x^{2} +5x +6 \right )$ $\mathbb{R}\left [ x \right ]/\left ( x^{3} -2 \right )$ $\mathbb{R}\left [ x \right ]/\left ( x^{7} +1 \right )$
9
Let $f:\mathbb{R}\rightarrow \left ( 0,\infty \right )$ be a twice differentiable function such that $f\left ( 0 \right )=1$ and $\int_{a}^{b}f\left ( x \right )dx=\int_{a}^{b}{f}'\left ( x \right )dx$, for all $a,b \in \mathbb{R}$, with $a\leq b$. Which of the following statements is false? $f$ is one to one The image of $f$ is compact $f$ is unbounded There is only one such function.
10
For $X\subset \mathbb{R}^{n}$, consider $X$ as a metric space with metric induced by the usual Euclidean metric on $\mathbb{R}^{n}$. Which of the following metric spaces $X$ is complete? $X=\mathbb{Z}\times \mathbb{Z}\subset \mathbb{R}\times \mathbb{R}$ ... $X=\left [ -\pi,\pi \right ]\cap \left ( \mathbb{R}\setminus \mathbb{Q} \right ) \subset \mathbb{R}$.
11
The value of the product $\left ( 1+\frac{1}{1!} +\frac{1}{2!}+\cdots \right )\left ( 1-\frac{1}{1!} +\frac{1}{2!}-\frac{1}{3!}+\cdots \right )$ is $1$ $e^{2}$ $0$ $log_{e} \:2$.
12
Which of the following is false ? $\sum _{n=1}^{\infty } sin \frac{1}{n}$ diverges $\sum _{n=1}^{\infty } sin \frac{1}{n^{2}}$ converges $\sum _{n=1}^{\infty } cos \frac{1}{n}$ diverges $\sum _{n=1}^{\infty } cos \frac{1}{n^{2}}$ converges.
13
The value of the series $\sum _{n=1}^{\infty }\frac{n}{2^{n}}$ is $1$ $2$ $3$ $4$.
14
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function defined by $f\left ( x \right )=\frac{sin \: x}{\left | x \right |+cos \: x}$. Then $f$ is differentiable at all $x\in\mathbb{R}$ $f$ is not differentiable at $x=0$ $f$ is differentiable at $x=0$ but ${f}'$ is not continuous at $x=0$ $f$ is not differentiable at $x=\frac{\pi }{2}.$
15
Which of the following continuous functions $f:\left ( 0,\infty \right ) \rightarrow \mathbb{R}$ can be extended to a continuous function on $\left [ 0,\infty \right )$ ? $f\left ( x \right )=sin\frac{1}{x}$ $f\left ( x \right )=\frac{1-cos\:x}{x^{2}}$ $f\left ( x \right )=cos\frac{1}{x}$ $f\left ( x \right )=\frac{1}{x}$
16
Let $V$ be the vector space over $\mathbb{R}$ consisting of polynomials $p\left ( t \right )$ over $\mathbb{R}$ of degree less than or equal to $4$. Let $D:V\rightarrow V$ be the linear operator that takes any polynomial $p\left ( t \right )$ to its derivative ${p}'\left ( t \right )$. Then the ... $D$ is $x^{4}$ $x^{5}$ $x^{3}\left ( x-1 \right )$ $x^{4}\left ( x-1 \right ).$
17
Let $A=\left \{ \sum _{i=1}^{\infty } \frac{a_{i}}{5^{i}}:a_{i}=0,1,2,3\:or \:4\right \}\subset \mathbb{R}$. Then $A$ is a finite set $A$ is countably infinite $A$ is uncountable but does not contain an open interval $A$ contains an open interval
18
The number of group homomorphisms from $\mathbb{Z}/20\mathbb{Z}$ to $\mathbb{Z}/29\mathbb{Z}$ is $1$ $20$ $29$ $580$
19
Let $p\left (x\right )$ be a polynomial of degree $3$ with real coefficients. Which of the following is possible ? $p\left ( x \right )$ has no real roots $p\left ( x \right )$ has exactly $2$ real roots $p\left ( 1\right )=-1,p\left ( 2 \right )=1, p\left ( 3 \right )=11$ and $p\left ( 4\right )=35$ $i-1$ and $i+1$ are roots of $p\left ( x\right )$, where $i$ is the square root of $-1$
20
Let $\left \{ a_{n} \right \}_{n=1}^{\infty }$ and $\left \{ b_{n} \right \}_{n=1}^{\infty }$ be two sequences of real numbers such that the series $\sum _{n=1}^{\infty }a_{n}^{2}$ and $\sum _{n=1}^{\infty }b_{n}^{2}$converge. Then the series $\sum _{n=1}^{\infty }a_{n}b_{n}$ is absolutely convergent may not converge is always convergent, but may not converge absolutely converges to $0$
21
Let $v_{i}=\left ( v_{i}^{\left ( 1 \right )},v_{i}^{\left ( 2 \right )} ,v_{i}^{\left ( 3 \right )},v_{i}^{\left ( 4 \right )}\right ),$ for $i=1,2,3,4,$ be four vectors in $\mathbb{R}^{4}$ ... over $\mathbb{R}$ is always either equal to $1$ or equal to $4$ less than or equal to $3$ greater than or equal to $2$ either equal to $0$ or equal to $4$
22
Let $A$ be a subset of $\left [ 0,1 \right ]$ with non-empty interior, and let $\mathbb{Q}+A=\left \{ q+a:q\in\mathbb{Q},a\in A \right \}$. Which of the following is true ? $\mathbb{Q}+A=\mathbb{R}$ $\mathbb{Q}+A$ can be a proper subset of $\mathbb{R}$ $\mathbb{Q}+A$ need not be closed is $\mathbb{R}$ $\mathbb{Q}+A$ need not be open in $\mathbb{R}$.
23
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function such that $\left | f\left ( x \right ) -f\left ( y \right )\right |\geq \left | x-y \right |$, for all $x,y\in\mathbb{R}$. Then the equation ${f}'\left ( x \right )=\frac{1}{2}$ has exactly one solution has no solution has a countably infinite number of solutions has uncountably many solutions
24
Let $f:\mathbb{R}\rightarrow \left [0,\infty \right )$ be a continuous function such that $g\left ( x \right )=(f\left ( x \right ))^2$ is uniformly continuous. Which of the following statements is always true ? $f$ is bounded $f$ may not be uniformly continuous $f$ is uniformly continuous $f$ is unbounded
25
Which of the following sequences of functions $\left \{ f_{n} \right \}_{n=1}^{\infty }$ converges uniformly ? $f_{n}\left ( x \right )=x^{n}\:on \: \left [ 0,1 \right ]$ $f_{n}\left ( x \right )=1-x^{n} \:on\: \left [ \frac{1}{2},1 \right ]$ ... $f_{n}\left ( x \right )=\frac{1}{1+nx^{2}}\:on\: \left [ \frac{1}{2},1 \right ]$
1 vote
26
Let $S$ be a collection of subset of $\left \{ 1,2,\dots,100 \right \}$ such that the intersection of any two sets in $S$ is non-empty. What is the maximum possible cardinality $\left | S \right |$ of $S$ ? $100$ $2^{100}$ $2^{99}$ $2^{98}$
27
Let $S$ be the set of all $3 \times3$ matrices $A$ with integer entries such that the product $AA^{t}$ is the identity matrix. Here $A^{t}$ denotes the transpose of $A$. Then $\left | S \right |$ = $12$ $24$ $48$ $60$
Let A be a $3\times3$ matrix with integer entries such that det $\left ( A \right )=1$. What is the maximum possible number of entries of $A$ that are even ? $2$ $3$ $6$ $8$
The limit $\underset{n\rightarrow \infty }{lim}\left ( \frac{1}{n} +\frac{1}{n+1}+\dots +\frac{1}{2n}\right )=$ $e$ $2$ $log_{e}2$ $e^{2}$
Let $G=\mathbb{Z}/100\mathbb{Z}$ and let $S=\left \{ h \in G : Order\left ( h \right )=50 \right \}$. Then $\left | S \right |$ equals $20$ $25$ $30$ $50$