Recent questions tagged tifrmaths2011

+2 votes

1 answer

1

TIFR-2011-Maths-B-15
A gardener throws $18$ seeds onto an equilateral triangle shaped plot of land with sides of length one metre. Then at least two seeds are within a distance of $25$ centimetres. TRUE/FALSE

asked Dec 10, 2015 in Numerical Ability by makhdoom ghaya Boss (30.6k points) | 169 views

+1 vote

0 answers

2

TIFR-2011-Maths-B-14
Let $f$ be a continuous integrable function of $\mathbb{R}$ such that either $f(x) > 0$ or $f(x) + f(x + 1) > 0$ for all $x \in \mathbb{R}$. Then $\int_{-\infty}^{\infty} f(x) \text{d}x > 0$.

asked Dec 10, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 116 views

+2 votes

1 answer

3

TIFR-2011-Maths-B-13
Any non-singular $k \times k$-matrix with real entries can be made singular by changing exactly one entry.

asked Dec 10, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 193 views

+1 vote

0 answers

4

TIFR-2011-Maths-B-12
Let $S$ be a finite subset of $\mathbb{R}^{3}$ such that any three elements in $S$ span a two dimensional subspace. Then $S$ spans a two dimensional space.

asked Dec 10, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 86 views

+1 vote

1 answer

5

TIFR-2011-Maths-B-11
There exists a set $A \subset \left\{1, 2,....,100\right\}$ with $65$ elements, such that $65$ cannot be expressed as a sum of two elements in $A$.

asked Dec 10, 2015 in Numerical Ability by makhdoom ghaya Boss (30.6k points) | 180 views

+1 vote

3 answers

6

TIFR-2011-Maths-B-10
Suppose a box contains three cards, one with both sides white, one with both sides black, and one with one side white and the other side black. If you pick a card at random, and the side facing you is white, then the probability that the other side is white is $1/2$.

asked Dec 10, 2015 in Probability by makhdoom ghaya Boss (30.6k points) | 206 views

+1 vote

1 answer

7

TIFR-2011-Maths-B-9
Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $|\alpha| > 10$.

asked Dec 10, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 103 views

+2 votes

1 answer

8

TIFR-2011-Maths-B-8
If $A$ and $B$ are $3 \times 3$ matrices and $A$ is invertible, then there exists an integer $n$ such that $A + nB$ is invertible.

asked Dec 9, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 201 views

+1 vote

0 answers

9

TIFR-2011-Maths-B-7
There is a non-trivial group homomorphism from $S_{3}$ to $\mathbb{Z}/3\mathbb{Z}$.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 177 views

+1 vote

0 answers

10

TIFR-2011-Maths-B-6
Suppose $f_{n}(x)$ is a sequence of continuous functions on the closed interval $[0, 1]$ converging to $0$ point wise. Then the integral $\int_{0}^{1} f_{n}(x) \text{d}x$ converges to 0.

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 70 views

+1 vote

1 answer

11

TIFR-2011-Maths-B-5
The symmetric group $S_{5}$ consisting of permutations on $5$ symbols has an element of order $6$.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 124 views

+1 vote

0 answers

12

TIFR-2011-Maths-B-4
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 92 views

+1 vote

0 answers

13

TIFR-2011-Maths-B-3
There are n homomorphisms from the group $\mathbb{Z}/n\mathbb{Z}$ to the additive group of rationals $\mathbb{Q}$.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 97 views

+1 vote

1 answer

14

TIFR-2011-Maths-B-2
In the ring $\mathbb{Z}/8\mathbb{Z}$, the equation $x^{2}=1$ has exactly $2$ solutions.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 81 views

+1 vote

0 answers

15

TIFR-2011-Maths-B-1
State True or False Let $A$ be a $2 \times 2$-matrix with complex entries. The number of $2 \times 2$-matrices $A$ with complex entries satisfying the equation $A^{3}$ is infinite.

asked Dec 9, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 114 views

+3 votes

1 answer

16

TIFR-2011-Maths-A-25
The function $f(x) = \begin{cases} 0 & \text{if x is rational} \\ x& \text{if x is irrational} \end{cases}$ is not continuous anywhere on the real line.

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 145 views

+1 vote

0 answers

17

TIFR-2011-Maths-A-24
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 77 views

+1 vote

0 answers

18

TIFR-2011-Maths-A-23
The space of solutions of infinitely differentiable functions satisfying the equation $y" + y = 0$ is infinite dimensional.

asked Dec 9, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 96 views

+1 vote

1 answer

19

TIFR-2011-Maths-A-22
There exists a group with a proper subgroup isomorphic to itself.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 148 views

+1 vote

1 answer

20

TIFR-2011-Maths-A-21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 109 views

+1 vote

1 answer

21

TIFR-2011-Maths-20
The equation $63x + 70y + 15z = 2010$ has an integral solution.

asked Dec 9, 2015 in Numerical Ability by makhdoom ghaya Boss (30.6k points) | 154 views

+1 vote

1 answer

22

TIFR-2011-Maths-A-19
The derivative of the function $\int_{0}^{\sqrt{x}} e^{-t^{2}}dt$ at $x = 1$ is $e^{-1}$ .

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 165 views

+1 vote

0 answers

23

TIFR-2011-Maths-A-18
Consider the map $T$ from the vector space of polynomials of degree at most $5$ over the reals to $R \times R$, given by sending a polynomial $P$ to the pair $(P(3), P' (3))$ where $P'$ is the derivative of $P$. Then the dimension of the kernel is $3$.

asked Dec 9, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 74 views

+1 vote

1 answer

24

TIFR-2011-Maths-A-17
The value of the infinite product $\prod_{n=2}^{\infty} (1-\frac{1}{n^{2}})$ is 1.

asked Dec 9, 2015 in Numerical Ability by makhdoom ghaya Boss (30.6k points) | 112 views

+1 vote

2 answers

25

TIFR-2011-Maths-A-16
The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 146 views

+1 vote

1 answer

26

TIFR-2011-Maths-A-15
If $A, B$ are closed subsets of $[0,\infty)$, then $A + B = \left\{x + y | x \in A, y \in B\right\}$ is closed in $[0,\infty)$

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 91 views

+1 vote

1 answer

27

TIFR-2011-Maths-A-14
$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.

asked Dec 9, 2015 in Calculus by makhdoom ghaya Boss (30.6k points) | 229 views

+1 vote

1 answer

28

TIFR-2011-Maths-A-13
$A$ is $3 \times 4$-matrix of rank $3$. Then the system of equations, $Ax = b$ has exactly one solution.

asked Dec 9, 2015 in Linear Algebra by makhdoom ghaya Boss (30.6k points) | 159 views

+1 vote

1 answer

29

TIFR-2011-Maths-A-12
$e^{\sqrt{2}}> 3$

asked Dec 9, 2015 in Numerical Ability by makhdoom ghaya Boss (30.6k points) | 148 views

+1 vote

1 answer

30

TIFR-2011-Maths-A-11
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.

asked Dec 9, 2015 in Set Theory & Algebra by makhdoom ghaya Boss (30.6k points) | 106 views

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