# Recent questions tagged tifrmaths2011

1
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
1 vote
2
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 $\int_{-\infty}^{\infty} f(x) \text{d}x > 0$.
3
Any non-singular $k \times k$-matrix with real entries can be made singular by changing exactly one entry.
1 vote
4
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.
1 vote
5
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 $A$.
1 vote
6
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 $1/2$.
1 vote
7
Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $|\alpha| > 10$.
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.
9
There is a non-trivial group homomorphism from $S_{3}$ to $\mathbb{Z}/3\mathbb{Z}$.
1 vote
10
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.
1 vote
11
The symmetric group $S_{5}$ consisting of permutations on $5$ symbols has an element of order $6$.
1 vote
12
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.
1 vote
13
There are n homomorphisms from the group $\mathbb{Z}/n\mathbb{Z}$ to the additive group of rationals $\mathbb{Q}$.
1 vote
14
In the ring $\mathbb{Z}/8\mathbb{Z}$, the equation $x^{2}=1$ has exactly $2$ solutions.
1 vote
15
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.
16
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.
1 vote
17
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.
1 vote
18
The space of solutions of infinitely differentiable functions satisfying the equation $y" + y = 0$ is infinite dimensional.
1 vote
19
There exists a group with a proper subgroup isomorphic to itself.
1 vote
20
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.
1 vote
21
The equation $63x + 70y + 15z = 2010$ has an integral solution.
1 vote
22
The derivative of the function $\int_{0}^{\sqrt{x}} e^{-t^{2}}dt$ at $x = 1$ is $e^{-1}$ .
1 vote
23
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 $3$.
1 vote
24
The value of the infinite product $\prod_{n=2}^{\infty} (1-\frac{1}{n^{2}})$ is 1.
1 vote
25
The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.
1 vote
26
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 $[0,\infty)$
1 vote
27
$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.
1 vote
$A$ is $3 \times 4$-matrix of rank $3$. Then the system of equations, $Ax = b$ has exactly one solution.
$e^{\sqrt{2}}> 3$
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.