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Recent questions tagged tifrmaths2010

3 votes
0 answers
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 $\mathbb{Z}/7\mathbb{Z}[x]$.
asked Nov 14, 2015 in Set Theory & Algebra Arjun 191 views
6 votes
1 answer
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
asked Oct 15, 2015 in Linear Algebra makhdoom ghaya 351 views
2 votes
1 answer
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.
asked Oct 15, 2015 in Calculus makhdoom ghaya 460 views
5 votes
2 answers
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.
asked Oct 14, 2015 in Numerical Ability makhdoom ghaya 393 views
2 votes
1 answer
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.
asked Oct 14, 2015 in Linear Algebra makhdoom ghaya 288 views
3 votes
1 answer
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.
asked Oct 14, 2015 in Linear Algebra makhdoom ghaya 301 views
3 votes
1 answer
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.
asked Oct 12, 2015 in Set Theory & Algebra makhdoom ghaya 383 views
3 votes
2 answers
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 $x=0.$
asked Oct 12, 2015 in Calculus Arjun 284 views
3 votes
1 answer
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.
asked Oct 12, 2015 in Calculus makhdoom ghaya 223 views
3 votes
3 answers
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.
asked Oct 12, 2015 in Set Theory & Algebra makhdoom ghaya 447 views
2 votes
0 answers
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.
asked Oct 11, 2015 in Calculus makhdoom ghaya 194 views
4 votes
1 answer
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.
asked Oct 11, 2015 in Numerical Ability makhdoom ghaya 182 views
3 votes
1 answer
13
If $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are uniformly continuous functions, then their compositions g ∘ f is. Uniformly continuous. Continuous but not uniformly continuous. Continuous and bounded. None of the above.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 323 views
3 votes
1 answer
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.
asked Oct 11, 2015 in Linear Algebra makhdoom ghaya 454 views
2 votes
1 answer
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.
asked Oct 11, 2015 in Calculus makhdoom ghaya 191 views
4 votes
2 answers
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.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 428 views
2 votes
1 answer
17
The solution of the ordinary differential equation. $\frac{dy}{dx}=y, y(0)=0$ Is unbounded Is positive Is negative. Is zero.
asked Oct 11, 2015 in Calculus makhdoom ghaya 172 views
3 votes
3 answers
18
What is the value of $\lim_{x\to 0} \sin{\left (\frac1 x \right )}$ $1$ $0$ $\frac{1}{2}$ Does Not Exist.
asked Oct 11, 2015 in Calculus makhdoom ghaya 265 views
2 votes
2 answers
19
The sum of the roots of the equation $x^{5}+3x^{2}+7=0$ is. $-3$ $\frac{3}{7}$ $\frac{-1}{7}$ $0$
asked Oct 11, 2015 in Numerical Ability makhdoom ghaya 189 views
2 votes
2 answers
20
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
asked Oct 11, 2015 in Numerical Ability Arjun 171 views
5 votes
2 answers
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.
asked Oct 11, 2015 in Linear Algebra makhdoom ghaya 343 views
5 votes
1 answer
22
The total number of subsets of a set of 6 elements is. 720 $6^{6}$ 21 None of the above.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 355 views
4 votes
3 answers
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 $x=0$.
asked Oct 11, 2015 in Calculus makhdoom ghaya 476 views
2 votes
1 answer
24
The sequence $\sqrt{7},\sqrt{7+\sqrt{7}},{\sqrt{7+\sqrt{7+\sqrt{7}}}},....$ converges to. $\frac{1+\sqrt{33}}{2}$ $\frac{1+\sqrt{32}}{2}$ $\frac{1+\sqrt{30}}{2}$ $\frac{1+\sqrt{29}}{2}$
asked Oct 11, 2015 in Numerical Ability makhdoom ghaya 187 views
3 votes
1 answer
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 $\frac{1}{2^{n}}$ $\frac{1}{3^{n}}$ $\frac{1}{5^{n}}$ $\frac{1}{4^{n}}$
asked Oct 11, 2015 in Calculus makhdoom ghaya 252 views
4 votes
2 answers
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.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 930 views
2 votes
2 answers
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.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 308 views
3 votes
3 answers
28
4 votes
1 answer
29
Which of the following is false? Any abelian group of order 27 is cyclic. Any abelian group of order 14 is cyclic. Any abelian group of order 21 is cyclic. Any abelian group of order 30 is cyclic.
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 853 views
3 votes
2 answers
30
A cyclic group of order 60 has 12 Generators 15 Generators 16 Generators 20 Generators
asked Oct 11, 2015 in Set Theory & Algebra makhdoom ghaya 1.4k views
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