Recent questions tagged tifrmaths2015

1
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 $d(x, y)$. $[0, 1]\times [0, \infty)\subset \mathbb{R}^{2}$ ... $[0, 1]\times [0, 1) \subset \mathbb{R}^{2}$ with the metric $d''(x, y) = \min \left \{ 1, d(x, y) \right \}$.
2
Let $G$ be a group. Suppose $|G|= p^{2}q$, where $p$ and $q$ are distinct prime numbers satisfying $q &#8802; 1 \mod p$. Which of the following is always true? $G$ has more than one $p$-Sylow subgroup. $G$ has a normal $p$-Sylow subgroup. The number of $q$-Sylow subgroups of $G$ is divisible by $p$. $G$ has a unique $q$-Sylow subgroup.
3
Let $X=\left\{(x, y) \in \mathbb{R}^{2}: 2x^{2}+3y^{2}= 1\right\}$. Endow $\mathbb{R}^{2}$ with the discrete topology, and $X$ with the subspace topology. Then. $X$ is a compact subset of $\mathbb{R}^{2}$ in this topology. $X$ is a connected subset of $\mathbb{R}^{2}$ in this topology. $X$ is an open subset of $\mathbb{R}^{2}$ in this topology. None of the above.
4
Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$? $[0, 1]$ $(0, 1)$ $[0, 1)$ $\mathbb{R}$
1 vote
5
Let $(X, d)$ be a path connected metric space with at least two elements, and let $S=\left\{d(x, y):x, y \in X\right\}$. Which of the following statements is not necessarily true ? $S$ is infinite. $S$ contains a non-zero rational number. $S$ is connected. $S$ is a closed subset of $\mathbb{R}$
6
How many finite sequences $x_{1}, x_{2},...,x_{m}$ are there such that each $x_{i}=1$ or $2$, and $\sum_{i=1}^{m} x_{i}=10$ ? $89$ $91$ $92$ $120$
7
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
8
For a group $G$, let $F(G)$ denote the collection of all subgroups of $G$. Which one of the following situations can occur ? $G$ is finite but $F(G)$ is infinite. $G$ is infinite but $F(G)$ is finite. $G$ is countable but $F(G)$ is uncountable. $G$ is uncountable but $F(G)$ is countable.
9
A complex number $\alpha \in \mathbb{C}$ is called algebraic if there is a non-zero 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 numbers. All complex numbers are algebraic. $\sin \frac{\pi}{3}+ \cos \frac{\pi}{4}$ is algebraic. None of the above.
10
Let $f: [0, 1]\rightarrow \mathbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0, 1)$ and $f(0) = f(1) = 0$. Then the equation $f(x) = f' (x)$ admits. No solution $x \in (0, 1)$ More than one solution $x \in (0, 1)$ Exactly one solution $x \in (0, 1)$ At least one solution $x \in (0, 1)$
11
Let $n \geq 1$ and let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$, for some $k \geq 1$. Let $I$ be the identity $n \times n$ matrix. Then. $I+A$ need not be invertible. Det $(I+A)$ can be any non-zero real number. Det $(I+A) = 1$ $A^{n}$ is a non-zero matrix.
1 vote
12
Let $U_{1}\supset U_{2} \supset...$ be a decreasing sequence of open sets in Euclidean $3$-space $\mathbb{R}^{3}$. What can we say about the set $\cap U_{i}$ ? It is infinite. It is open. It is non-empty. None of the above.
1 vote
13
Let $f$ be a function from $\left \{ 1, 2,....10 \right \}$ to $\mathbb{R}$ such that $(\sum_{i=1}^{10}\frac{|f(i)|}{2^{i}})^{2} = (\sum_{i=1}^{10} |f(i)|^{2}) (\sum_{i=1}^{10}\frac{1}{4^{i}})$ Make the correct statement. There are uncountably many $f$ with this property. There are only countably infinitely many $f$ with this property. There is exactly one such $f$ There is no such $f$.
14
In how many ways can the group $\mathbb{Z}_{5}$ act on the set $\left \{ 1, 2, 3, 4, 5 \right \}$ ? $5$ $24$ $25$ $120$
15
Let $X$ be a proper closed subset of $[0, 1]$. Which of the following statements is always true? The set $X$ is countable. There exists $x \in X$ such that $X ∖ \left \{ x \right \}$ is closed The set $X$ contains an open interval. None of the above.
16
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 $x \in \mathbb{R}$
17
Let $n \in \mathbb{N}$ be a six digit number whose base $10$ expansion is of the form $abcabc$, where $a, b, c$ are digits between $0$ and $9$ and $a$ is non-zero. Then, $n$ is divisible by $5$ $n$ is divisible by $8$ $n$ is divisible by $13$ $n$ is divisible by $17$
18
For a real number $t >0$, let $\sqrt{t}$ denote the positive square root of $t$. For a real number $x > 0$, let $F(x)= \int_{x^{2}}^{4x^{2}} \sin \sqrt{t} dt$. If $F'$ is the derivative of $F$, then $F'(\frac{\pi}{2}) = 0$ $F'(\frac{\pi}{2}) = \pi$ $F'(\frac{\pi}{2}) = - \pi$ $F'(\frac{\pi}{2}) = 2\pi$
19
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function that vanishes at $10$ distinct points in $\mathbb{R}$. Suppose $f^{(n)}$ denotes the $n$-th derivative of $f$, for $n \geq 1$. Which of the following statements is always true? $f^{(n)}$ has at least $10$ ... $f^{(n)}$ has at least $10$ zeros, for $n \geq 10$ $f^{(n)}$ has at least one zero, for $n \geq 9$
20
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 $\sum a_{n}^{2}$
1 vote
21
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 $\mathbb{Z}$
22
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.
23
Let $f(x)=\frac{e^{\frac{-1}{x}}}{x}$, where $x \in (0, 1)$. Then on $(0, 1)$. $f$ is uniformly continuous. $f$ is continuous but not uniformly continuous. $f$ is unbounded. $f$ is not continuous.
24
Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct? If $g$ is continuous, then $f ∘ g$ is continuous If $f$ is continuous, then $f ∘ g$ is continuous If $f$ and $f ∘ g$ are continuous, then $g$ is continuous If $g$ and $f ∘ g$ are continuous, then $f$ is continuous
25
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix} \sin\frac{\pi}{18}&-\sin \frac{4\pi}{9} \\ \sin \frac{4\pi}{9}&\sin \frac {\pi}{18} \end{pmatrix}$. Then the smallest number $n \in \mathbb{N}$ such that $A^{n}=1$ is. $3$ $9$ $18$ $27$
26
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote the function defined by $f(x)= (1-x^{2})^{\frac{3}{2}}$ if $|x| < 1$, and $f(x)=0$ if $|x| \geq 1$. Which of the following statements is correct ? $f$ is not continuous $f$ is continuous but not differentiable $f$ is differentiable but $f'$ is not continuous. $f$ is differentiable and $f'$ is continuous.
27
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 $|S| = \infty$
1 vote
28
Let $A$ be a $10 \times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false ? There exists a matrix $B$ such that $AB-BA = B$ There exists a matrix $B$ such that $AB-BA = A$ There exists a matrix $B$ such that $AB+BA=A$ There exists a matrix $B$ such that $AB+BA=B$
29
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Which one of the following sets cannot be the image of $(0, 1]$ under $f$? {0} (0, 1) [0, 1) [0, 1]
Let $A$ be an invertible $10 \times 10$ matrix with real entries such that the sum of each row is $1$. Then The sum of the entries of each row of the inverse of $A$ is $1$ The sum of the entries of each column of the inverse of $A$ is $1$ The trace of the inverse of $A$ is non-zero None of the above