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Recent questions tagged tifrmaths2010
+3
votes
0
answers
1
TIFR2010MathsB15
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
by
Arjun
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(
348k
points)

116
views
tifrmaths2010
polynomials
+5
votes
1
answer
2
TIFR2010MathsB14
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
by
makhdoom ghaya
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(
48k
points)

229
views
tifrmaths2010
linearalgebra
systemofequations
+2
votes
1
answer
3
TIFR2010MathsB13
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
by
makhdoom ghaya
Veteran
(
48k
points)

210
views
tifrmaths2010
calculus
convergence
+3
votes
2
answers
4
TIFR2010MathsB12
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
by
makhdoom ghaya
Veteran
(
48k
points)

187
views
tifrmaths2010
numericalability
modulararithmetic
+2
votes
1
answer
5
TIFR2010MathsB11
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
by
makhdoom ghaya
Veteran
(
48k
points)

174
views
tifrmaths2010
linearalgebra
matrices
+3
votes
1
answer
6
TIFR2010MathsB10
Let $x$ and $y \in \mathbb{R}^{n}$ be nonzero 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
by
makhdoom ghaya
Veteran
(
48k
points)

151
views
tifrmaths2010
matrices
+3
votes
1
answer
7
TIFR2010MathsB9
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
by
makhdoom ghaya
Veteran
(
48k
points)

272
views
tifrmaths2010
settheory&algebra
groups
+3
votes
2
answers
8
TIFR2010MathsB8
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
by
Arjun
Veteran
(
348k
points)

141
views
tifrmaths2010
calculus
+3
votes
1
answer
9
TIFR2010MathsB7
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
by
makhdoom ghaya
Veteran
(
48k
points)

139
views
tifrmaths2010
calculus
+3
votes
3
answers
10
TIFR2010MathsB6
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
by
makhdoom ghaya
Veteran
(
48k
points)

298
views
tifrmaths2010
settheory&algebra
sets
+2
votes
0
answers
11
TIFR2010MathsB5
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
by
makhdoom ghaya
Veteran
(
48k
points)

125
views
tifrmaths2010
calculus
convergence
+3
votes
1
answer
12
TIFR2010MathsB4
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 ... 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
by
makhdoom ghaya
Veteran
(
48k
points)

104
views
tifrmaths2010
numericalability
modulararithmetic
+3
votes
1
answer
13
TIFR2010MathsB3
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
by
makhdoom ghaya
Veteran
(
48k
points)

194
views
tifrmaths2010
settheory&algebra
functions
+3
votes
1
answer
14
TIFR2010MathsB2
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
by
makhdoom ghaya
Veteran
(
48k
points)

151
views
tifrmaths2010
vectorspace
+2
votes
1
answer
15
TIFR2010MathsB1
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
by
makhdoom ghaya
Veteran
(
48k
points)

107
views
tifrmaths2010
calculus
convergence
+4
votes
2
answers
16
TIFR2010MathsA15
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
by
makhdoom ghaya
Veteran
(
48k
points)

252
views
tifrmaths2010
groups
+2
votes
1
answer
17
TIFR2010MathsA14
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
by
makhdoom ghaya
Veteran
(
48k
points)

105
views
tifrmaths2010
calculus
+3
votes
3
answers
18
TIFR2010MathsA13
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
by
makhdoom ghaya
Veteran
(
48k
points)

201
views
tifrmaths2010
limits
+2
votes
2
answers
19
TIFR2010MathsA12
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
by
makhdoom ghaya
Veteran
(
48k
points)

124
views
tifrmaths2010
+2
votes
2
answers
20
TIFR2010MathsA11
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
by
Arjun
Veteran
(
348k
points)

111
views
tifrmaths2010
numberseries
convergence
+5
votes
2
answers
21
TIFR2010MathsA10
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
by
makhdoom ghaya
Veteran
(
48k
points)

246
views
tifrmaths2010
eigenvalue
+4
votes
1
answer
22
TIFR2010MathsA9
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
by
makhdoom ghaya
Veteran
(
48k
points)

212
views
tifrmaths2010
settheory&algebra
sets
+4
votes
3
answers
23
TIFR2010MathsA8
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
by
makhdoom ghaya
Veteran
(
48k
points)

311
views
tifrmaths2010
calculus
differentiability
continuity
+2
votes
1
answer
24
TIFR2010MathsA7
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
by
makhdoom ghaya
Veteran
(
48k
points)

104
views
tifrmaths2010
numericalability
+3
votes
1
answer
25
TIFR2010MathsA6
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
by
makhdoom ghaya
Veteran
(
48k
points)

171
views
tifrmaths2010
calculus
maximaminima
+3
votes
1
answer
26
TIFR2010MathsA5
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
by
makhdoom ghaya
Veteran
(
48k
points)

505
views
tifrmaths2010
settheory&algebra
functions
+2
votes
2
answers
27
TIFR2010MathsA4
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
by
makhdoom ghaya
Veteran
(
48k
points)

203
views
tifrmaths2010
numberseries
+2
votes
2
answers
28
TIFR2010MathsA3
The last digit of $2^{80}$ is.. 2 4 6 8
asked
Oct 11, 2015
in
Numerical Ability
by
makhdoom ghaya
Veteran
(
48k
points)

152
views
tifrmaths2010
numericalability
+3
votes
1
answer
29
TIFR2010MathsA2
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
by
makhdoom ghaya
Veteran
(
48k
points)

354
views
tifrmaths2010
settheory&algebra
groups
+2
votes
2
answers
30
TIFR2010MathsA1
A cyclic group of order 60 has 12 Generators 15 Generators 16 Generators 20 Generators
asked
Oct 11, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
Veteran
(
48k
points)

510
views
tifrmaths2010
groups
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