Recent questions tagged tifrmaths2010

3 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 $\mathbb{Z}/7\mathbb{Z}[x]$.

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

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

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

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.

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

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.

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

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.

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

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.

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

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.

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

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 $x=0.$

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

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.

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

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.

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

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.

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

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 divided by ... 3. There exists a natural number which when divided by 12 leaves remainder 7 and when divided by 8 leaves remainder 3.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

TIFR2010-Maths-A-12
The sum of the roots of the equation $x^{5}+3x^{2}+7=0$ is. $-3$ $\frac{3}{7}$ $\frac{-1}{7}$ $0$

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

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.

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

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.

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

TIFR2010-Maths-A-9
The total number of subsets of a set of 6 elements is. 720 $6^{6}$ 21 None of the above.

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

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 $x=0$.

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

TIFR2010-Maths-A-7
The sequence $\sqrt{7},\sqrt{7+\sqrt{7}},{\sqrt{7+\sqrt{7+\sqrt{7}}}},....$ converges to. $\frac{1+\sqrt{33}}{2}$ $\frac{1+\sqrt{32}}{2}$ $\frac{1+\sqrt{30}}{2}$ $\frac{1+\sqrt{29}}{2}$

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

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 $\frac{1}{2^{n}}$ $\frac{1}{3^{n}}$ $\frac{1}{5^{n}}$ $\frac{1}{4^{n}}$

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

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.

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

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.

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

TIFR2010-Maths-A-3
The last digit of $2^{80}$ is.. 2 4 6 8

The last digit of $2^{80}$ is.. 2 4 6 8

asked Oct 11, 2015 in Numerical Ability makhdoom ghaya 282 views

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.

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

TIFR2010-Maths-A-1
A cyclic group of order 60 has 12 Generators 15 Generators 16 Generators 20 Generators

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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