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Recent questions tagged tifrmaths2011

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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
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 makhdoom ghaya 218 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 $\int_{-\infty}^{\infty} f(x) \text{d}x > 0$.
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 makhdoom ghaya 142 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.
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 makhdoom ghaya 261 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.
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 makhdoom ghaya 108 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 $A$.
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 makhdoom ghaya 207 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 $1/2$.
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 makhdoom ghaya 295 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 $|\alpha| > 10$.
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 makhdoom ghaya 137 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.
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 makhdoom ghaya 237 views
2 votes
0 answers
9
TIFR-2011-Maths-B-7
There is a non-trivial group homomorphism from $S_{3}$ to $\mathbb{Z}/3\mathbb{Z}$.
There is a non-trivial group homomorphism from $S_{3}$ to $\mathbb{Z}/3\mathbb{Z}$.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 288 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.
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 makhdoom ghaya 99 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 $6$.
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 makhdoom ghaya 161 views
1 vote
0 answers
12
TIFR-2011-Maths-B-4
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.
asked Dec 9, 2015 in Calculus makhdoom ghaya 114 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 $\mathbb{Q}$.
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 makhdoom ghaya 117 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.
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 makhdoom ghaya 102 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.
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 makhdoom ghaya 137 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.
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 makhdoom ghaya 173 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.
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 95 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.
The space of solutions of infinitely differentiable functions satisfying the equation $y" + y = 0$ is infinite dimensional.
asked Dec 9, 2015 in Linear Algebra makhdoom ghaya 129 views
1 vote
1 answer
19
TIFR-2011-Maths-A-22
There exists a group with a proper subgroup isomorphic to itself.
There exists a group with a proper subgroup isomorphic to itself.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 225 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.
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.
asked Dec 9, 2015 in Calculus makhdoom ghaya 158 views
1 vote
1 answer
21
TIFR-2011-Maths-20
The equation $63x + 70y + 15z = 2010$ has an integral solution.
The equation $63x + 70y + 15z = 2010$ has an integral solution.
asked Dec 9, 2015 in Numerical Ability makhdoom ghaya 179 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 $e^{-1}$ .
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 makhdoom ghaya 223 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 $3$.
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 makhdoom ghaya 102 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.
The value of the infinite product $\prod_{n=2}^{\infty} (1-\frac{1}{n^{2}})$ is 1.
asked Dec 9, 2015 in Numerical Ability makhdoom ghaya 136 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.
The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 191 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 $[0,\infty)$
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 makhdoom ghaya 118 views
1 vote
1 answer
27
TIFR-2011-Maths-A-14
$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.
$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.
asked Dec 9, 2015 in Calculus makhdoom ghaya 391 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.
$A$ 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 makhdoom ghaya 216 views
1 vote
1 answer
29
TIFR-2011-Maths-A-12
$e^{\sqrt{2}}> 3$
$e^{\sqrt{2}}> 3$
asked Dec 9, 2015 in Numerical Ability makhdoom ghaya 178 views
2 votes
1 answer
30
TIFR-2011-Maths-A-11
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 133 views
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