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Recent questions tagged tifrmaths2010
4
votes
0
answers
1
TIFR2010-Maths-B-15
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 $\mathbb{Z}/7\mathbb{Z}[x]$.
Arjun
asked
in
Set Theory & Algebra
Nov 14, 2015
by
Arjun
414
views
tifrmaths2010
polynomials
7
votes
1
answer
2
TIFR2010-Maths-B-14
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
makhdoom ghaya
asked
in
Linear Algebra
Oct 15, 2015
by
makhdoom ghaya
609
views
tifrmaths2010
linear-algebra
system-of-equations
2
votes
1
answer
3
TIFR2010-Maths-B-13
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
makhdoom ghaya
asked
in
Calculus
Oct 15, 2015
by
makhdoom ghaya
1.1k
views
tifrmaths2010
calculus
convergence
5
votes
2
answers
4
TIFR2010-Maths-B-12
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
makhdoom ghaya
asked
in
Quantitative Aptitude
Oct 14, 2015
by
makhdoom ghaya
1.0k
views
tifrmaths2010
quantitative-aptitude
modular-arithmetic
2
votes
1
answer
5
TIFR2010-Maths-B-11
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
makhdoom ghaya
asked
in
Linear Algebra
Oct 14, 2015
by
makhdoom ghaya
495
views
tifrmaths2010
linear-algebra
matrix
3
votes
1
answer
6
TIFR2010-Maths-B-10
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
makhdoom ghaya
asked
in
Linear Algebra
Oct 14, 2015
by
makhdoom ghaya
1.5k
views
tifrmaths2010
matrix
3
votes
1
answer
7
TIFR2010-Maths-B-9
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 12, 2015
by
makhdoom ghaya
1.1k
views
tifrmaths2010
set-theory&algebra
group-theory
3
votes
2
answers
8
TIFR2010-Maths-B-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 $x=0$
Arjun
asked
in
Calculus
Oct 12, 2015
by
Arjun
743
views
tifrmaths2010
calculus
3
votes
1
answer
9
TIFR2010-Maths-B-7
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
makhdoom ghaya
asked
in
Calculus
Oct 12, 2015
by
makhdoom ghaya
429
views
tifrmaths2010
calculus
3
votes
3
answers
10
TIFR2010-Maths-B-6
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 12, 2015
by
makhdoom ghaya
704
views
tifrmaths2010
set-theory&algebra
set-theory
2
votes
0
answers
11
TIFR2010-Maths-B-5
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
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
400
views
tifrmaths2010
calculus
convergence
4
votes
1
answer
12
TIFR2010-Maths-B-4
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 ... remainder 3 There exists a natural number which when divided by 12 leaves remainder 7 and when divided by 8 leaves remainder 3
makhdoom ghaya
asked
in
Quantitative Aptitude
Oct 11, 2015
by
makhdoom ghaya
508
views
tifrmaths2010
quantitative-aptitude
modular-arithmetic
3
votes
1
answer
13
TIFR2010-Maths-B-3
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
1.1k
views
tifrmaths2010
set-theory&algebra
functions
3
votes
1
answer
14
TIFR2010-Maths-B-2
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
makhdoom ghaya
asked
in
Linear Algebra
Oct 11, 2015
by
makhdoom ghaya
1.4k
views
tifrmaths2010
vector-space
2
votes
1
answer
15
TIFR2010-Maths-B-1
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
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
446
views
tifrmaths2010
calculus
convergence
4
votes
2
answers
16
TIFR2010-Maths-A-15
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
1.3k
views
tifrmaths2010
group-theory
2
votes
1
answer
17
TIFR2010-Maths-A-14
The solution of the ordinary differential equation. $\frac{dy}{dx}=y, y(0)=0$ Is unbounded Is positive Is negative Is zero
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
389
views
tifrmaths2010
calculus
3
votes
3
answers
18
TIFR2010-Maths-A-13
What is the value of $\lim_{x\to 0} \sin{\left (\frac1 x \right )}$ $1$ $0$ $\frac{1}{2}$ Does Not Exist
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
497
views
tifrmaths2010
limits
2
votes
2
answers
19
TIFR2010-Maths-A-12
The sum of the roots of the equation $x^{5}+3x^{2}+7=0$ is. $-3$ $\frac{3}{7}$ $\frac{-1}{7}$ $0$
makhdoom ghaya
asked
in
Quantitative Aptitude
Oct 11, 2015
by
makhdoom ghaya
376
views
tifrmaths2010
2
votes
2
answers
20
TIFR2010-Maths-A-11
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
Arjun
asked
in
Quantitative Aptitude
Oct 11, 2015
by
Arjun
497
views
tifrmaths2010
number-series
convergence
5
votes
2
answers
21
TIFR2010-Maths-A-10
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
makhdoom ghaya
asked
in
Linear Algebra
Oct 11, 2015
by
makhdoom ghaya
773
views
tifrmaths2010
eigen-value
5
votes
1
answer
22
TIFR2010-Maths-A-9
The total number of subsets of a set of 6 elements is. 720 $6^{6}$ 21 None of the above
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
856
views
tifrmaths2010
set-theory&algebra
set-theory
4
votes
4
answers
23
TIFR2010-Maths-A-8
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 $x=0$.
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
868
views
tifrmaths2010
calculus
differentiation
continuity
2
votes
1
answer
24
TIFR2010-Maths-A-7
The sequence $\sqrt{7},\sqrt{7+\sqrt{7}},{\sqrt{7+\sqrt{7+\sqrt{7}}}},....$ converges to. $\frac{1+\sqrt{33}}{2}$ $\frac{1+\sqrt{32}}{2}$ $\frac{1+\sqrt{30}}{2}$ $\frac{1+\sqrt{29}}{2}$
makhdoom ghaya
asked
in
Quantitative Aptitude
Oct 11, 2015
by
makhdoom ghaya
617
views
tifrmaths2010
quantitative-aptitude
3
votes
1
answer
25
TIFR2010-Maths-A-6
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 $\frac{1}{2^{n}}$ $\frac{1}{3^{n}}$ $\frac{1}{5^{n}}$ $\frac{1}{4^{n}}$
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
480
views
tifrmaths2010
calculus
maxima-minima
4
votes
2
answers
26
TIFR2010-Maths-A-5
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
2.3k
views
tifrmaths2010
set-theory&algebra
functions
2
votes
2
answers
27
TIFR2010-Maths-A-4
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
529
views
tifrmaths2010
number-series
3
votes
3
answers
28
TIFR2010-Maths-A-3
The last digit of $2^{80}$ is.. 2 4 6 8
makhdoom ghaya
asked
in
Quantitative Aptitude
Oct 11, 2015
by
makhdoom ghaya
464
views
tifrmaths2010
quantitative-aptitude
4
votes
1
answer
29
TIFR2010-Maths-A-2
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
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
1.7k
views
tifrmaths2010
set-theory&algebra
group-theory
3
votes
2
answers
30
TIFR2010-Maths-A-1
A cyclic group of order 60 has 12 Generators 15 Generators 16 Generators 20 Generators
makhdoom ghaya
asked
in
Set Theory & Algebra
Oct 11, 2015
by
makhdoom ghaya
3.5k
views
tifrmaths2010
group-theory
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