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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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