# Recent questions tagged tifrmaths2010

1
Which of the following statements is false? The polynomial $x^{2}+x+1$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[x]$. The polynomial $x^{2}-2$ is irreducible in $\mathbb{Q}[x]$. The polynomial $x^{2}+1$ is reducible in $\mathbb{Z}/5\mathbb{Z}[x]$. The polynomial $x^{2}+1$ is reducible in $\mathbb{Z}/7\mathbb{Z}[x]$.
2
The equations. $x_{1}+2x_{2}+3x_{3}=1$ $x_{1}+4x_{2}+9x_{3}=1$ $x_{1}+8x_{2}+27x_{3}=1$ have Only one solution. Two solutions. Infinitely many solutions. No solutions
3
Define $\left \{ x_{n} \right \}$ as $x_{1}=0.1,x_{2}=0.101,x_{3}=0.101001,\dots$ Then the sequence $\left \{ x_{n} \right \}$. Converges to a rational number. Converges to a irrational number. Does not coverage. Oscillates.
4
If $n$ and $m$ are positive integers and $n^{9}=19m+r$, then the possible values for $r$ modulo 19 are. Only 0 Only 0, $\pm$ 1. Only $\pm$ 1. None of the above.
5
Which of the following is true? The matrix $\begin{pmatrix} 1&0 \\ 1&2 \end{pmatrix}$ is not diagonalisable. The matrix $\begin{pmatrix} 1&5 \\ 0&2 \end{pmatrix}$ is diagonalisable. The matrix $\begin{pmatrix} 1&1 \\ 0&1 \end{pmatrix}$ is diagonalisable None of the above.
6
Let $x$ and $y \in \mathbb{R}^{n}$ be non-zero column vectors, from the matrix $A=xy^{T}$, where $y^{T}$ is the transpose of $y$. Then the rank of $A$ is: $2$ $0$ At least $n/2$ None of the above.
7
Let $G=\left \{ z \in \mathbb{C} \mid z^n = 1 \text{ for some positive integer } n \right \}$. Then under multiplication of complex numbers, $G$ is a group of finite order. $G$ is a group of infinite order, but every element of $G$ has finite order. $G$ is a cyclic group. None of the above.
8
The function $f(x)$ defined by $f(x)= \begin{cases} 0 & \text{if x is rational } \\ x & \text{if } x\text{ is irrational } \end{cases}$ is not continuous at any point. is continuous at every point. is continuous at every rational number. is continuous at $x=0.$
9
Number of solutions of the ordinary differential equation. $\frac{d^{2}y}{dx^{2}}-y=0, y(0)=0, y(\pi )=1$ is 0 is 1 is 2 None of the above.
10
Let $A, B$ be subsets of $\mathbb{R}$. Define $A + B$ to be the set of all sums $x +y$ with $x \in A$ and $y \in B$. Which of the following statements is false? If $A$ and $B$ are bounded, then $A + B$ is bounded. If $A$ and $B$ are open, then $A + B$ is open. If $A$ and $B$ are closed, then $A + B$ is closed. If $A$ and $B$ are connected, then $A + B$ is connected.
11
If $f_{n}(x)$ are continuous functions from [0, 1] to [0, 1], and $f_{n}(x)\rightarrow f(x)$ as $n\rightarrow \infty$, then which of the following statements is true? $f_{n}(x)$ converges to $f(x)$ uniformly on [0, 1]. $f_{n}(x)$ converges to $f(x)$ uniformly on (0, 1). $f(x)$ is continuous on [0, 1]. None of the above.
12
Which of the following statements is false? There exists a natural number which when divided by 3 leaves remainder 1 and which when divided by 4 leaves remainder 0. There exists a natural number which when divided by 6 leaves remainder 2 and when divided by 9 leaves ... remainder 3. There exists a natural number which when divided by 12 leaves remainder 7 and when divided by 8 leaves remainder 3.
13
If $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are uniformly continuous functions, then their compositions g &#8728; f is. Uniformly continuous. Continuous but not uniformly continuous. Continuous and bounded. None of the above.
14
If $V$ is a vector space over the field $\mathbb{Z}/5\mathbb{Z}$ and $\dim_{Z/5\mathbb{Z}}(V)=3$ then $V$ has. 125 elements 15 elements 243 elements None of the above.
15
Let $U_{n}=\sin(\frac{\pi }{n})$ and consider the series $\sum u_{n}$. Which of the following statements is false? $\sum u_{n}$ is convergent. $u_{n}\rightarrow 0$ as $n\rightarrow \infty$ $\sum u_{n}$ is divergent. $\sum u_{n}$ is absolutely convergent.
16
Let G be the set of all 2 x 2 symmetric, invertible matrices with real entries. Then with matrix multiplication, G is. An infinite group. A finite group. Not a group. An abelian group.
17
The solution of the ordinary differential equation. $\frac{dy}{dx}=y, y(0)=0$ Is unbounded Is positive Is negative. Is zero.
18
What is the value of $\lim_{x\to 0} \sin{\left (\frac1 x \right )}$ $1$ $0$ $\frac{1}{2}$ Does Not Exist.
19
The sum of the roots of the equation $x^{5}+3x^{2}+7=0$ is. $-3$ $\frac{3}{7}$ $\frac{-1}{7}$ $0$
20
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
21
Let $M_{n}(R)$ be the set of n x n matrices with real entries. Which of the following statements is true? Any matrix $A \in M_{4}(R)$ has a real eigenvalue. Any matrix $A \in M_{5}(R)$ has a real eigenvalue. Any matrix $A \in M_{2}(R)$ has a real eigenvalue. None of the above.
22
The total number of subsets of a set of 6 elements is. 720 $6^{6}$ 21 None of the above.
23
Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then $f$ is uniformly continuous. $f$ is continuous, but not differentiable at $x=0$. $f$ is differentiable and $f '$ is continuous. $f$ is differentiable, but $f '$ is discontinuous at $x=0$.
24
The sequence $\sqrt{7},\sqrt{7+\sqrt{7}},{\sqrt{7+\sqrt{7+\sqrt{7}}}},....$ converges to. $\frac{1+\sqrt{33}}{2}$ $\frac{1+\sqrt{32}}{2}$ $\frac{1+\sqrt{30}}{2}$ $\frac{1+\sqrt{29}}{2}$
25
The maximum value of $f(x)=x^{n}(1 - x)^{n}$ for a natural numbers $n\geq 1$ and $0\leq x\leq 1$ is $\frac{1}{2^{n}}$ $\frac{1}{3^{n}}$ $\frac{1}{5^{n}}$ $\frac{1}{4^{n}}$
26
Let $f$ be an one to one function from the closed interval [0, 1] to the set of real numbers $R$, then. $f$ must be onto. Range of $f$ must contain a rational number. Range of $f$ must contain an irrational number. Range of $f$ must contain both rational and irrational numbers.
27
The sum of the series $\frac{1}{1 \cdot 2}+ \frac{1}{2 \cdot 3}+ \frac{1}{3 \cdot 4} + \dots +\frac{1}{100 \cdot 101}$ $\frac{99}{101}$ $\frac{98}{101}$ $\frac{99}{100}$ None of the above.
The last digit of $2^{80}$ is.. 2 4 6 8